Chapter 2 The Fourier Transform in Clifford Analysis

CHAPTER

2 The Fourier Transform in Clifford Analysis Fred Brackx, Nele De Schepper, and Frank Sommen*

Contents

1 Introduction 56 2 The Clifford Analysis Toolkit 58 2.1 Clifford Algebra 59 2.2 Clifford Analysis 62 2.3 Some Useful Results Concerning the Fourier Transform and Spherical Harmonics 68 2.4 The Generalized Clifford–Hermite Polynomials 73 2.5 Multidimensional “Analytic Signals” 83 3 The Fractional Fourier Transform 86 3.1 Introduction 87 3.2 The Classical Fractional Fourier Transform 87 3.3 Multidimensional Fractional Fourier Transform: Definition and Operator Exponential Form 90 3.4 The Mehler Formula for the Generalized Clifford–Hermite Polynomials 93 4 The Clifford–Fourier Transform 95 4.1 Introduction 95 4.2 Clifford–Hermite Monogenic Operators 98 4.3 Alternative Representations of the Classical Fourier Transform 113 4.4 Clifford–Fourier Transform: Definition and Properties 115 4.5 The Two-Dimensional Case 122 5 Clifford Filters for Early Vision 143 5.1 Introduction 143 5.2 Generalized Clifford–Hermite Filters 145 5.3 The Two-Dimensional Clifford–Gabor Filters 159 6 The Cylindrical Fourier Transform 167 6.1 Definition 167 6.2 Properties 170 6.3 Cylindrical Fourier Spectrum of the L2 -Basis Consisting of Generalized Clifford–Hermite Functions 175 Acknowledgments 197 References 197

* Clifford Research Group, Department of Mathematical Analysis, Faculty of Engineering, Ghent University, Galglaan 2, 9000 Ghent, Belgium Advances in Imaging and Electron Physics, Volume 156, ISSN 1076-5670, DOI: 10.1016/S1076-5670(08)01402-x. Copyright © 2009 Elsevier Inc. All rights reserved.

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1. INTRODUCTION The Fourier transform is the most important integral transform. Since its introduction by Fourier in the early 1800s, it has remained an indispensable and stimulating mathematical concept at the core of the highly evolved branch of mathematics called Fourier analysis. It has found use in innumerable applications and has become a fundamental tool in engineering sciences, thanks to the generalizations extending the class of Fourier transformable functions and to the development of efficient algorithms for computing the discrete version of it. The Fourier transform provides a representation of functions defined over an infinite interval and showing no particular periodicity in terms of a superposition of periodic, say sinusoidal, functions. More precisely, if f (t) is a real- or complex-valued integrable function of the real variable t, then its Fourier transform, also called Fourier image or frequency contents or spectrum, is the result of the following integral:

1 fˆ (ω) = F [ f ](ω) = √ 2π



+∞ −∞

exp (−iωt) f (t) dt,

which is a continuous and bounded function of the frequency variable ω ∈ ] − ∞, +∞ [. Still more important is that the original function f (t) may be recovered from its spectrum on the condition that it is continuous and bounded and that its Fourier image is integrable; this is the so-called inverse Fourier transform

f (t) = F

−1

1 [ fˆ ](t) = √ 2π



+∞

−∞

exp (itω) fˆ (ω) dω.

The natural habitat of the Fourier transform, however, is the space S of rapidly decreasing functions; these are indefinite continuously differentiable functions decaying at infinity more rapidly than the inverse of any polynomial. On S the Fourier transform is a one-to-one mapping. By a density argument the Fourier transform may then be defined on the space L2 of square integrable functions—the so-called signals of finite energy—on which it becomes a homeomorphism, that is, a bicontinuous bijection, that preserves the L2 -norm. The latter is expressed by the celebrated Parseval formula: for all f , g in L2 (R) holds

 f , g = F [ f ], F [g]. For handling functions of several variables the extension of the Fourier transform to higher dimension is obtained in a straightforward tensorial

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manner, considering Fourier transforms in each of the Cartesian variables separately:

1 fˆ (ω) = F [ f ](ω) = √ ( 2π)m

 Rm

exp (−iω, x) f (x) dV(x),

where ω stands for (ω1 , . . . , ωm ), x for (x1 , . . . , xm ), and ω, x for the traditional scalar product in Euclidean space: ω, x = m j=1 ωj xj . The properties of this standard multidimensional Fourier transform are similar to those of the one-dimensional (1D) transform. The second player in this paper is Clifford analysis. This is a function theory for functions defined in Euclidean space Rm of arbitrary dimension m and taking values in the real Clifford algebra R0,m constructed over Rm . Clifford algebra, named after William Kingdon Clifford (1845– 1879) who himself talked about geometric algebra, is an associative but noncommutative algebra with zero divisors, which combines the algebraic properties of the reals, the complex numbers, and the quaternions with the geometric properties of Grassmann algebra. Clifford algebra has been rediscovered several times, among others as the algebra of Pauli matrices, which is the geometric algebra of the physical space; and the algebra of Dirac matrices, which is the geometric algebra of Minkowski space-time. Much of the recent interest in Clifford algebras can be traced to the works of David Hestenes in the 1960s, who viewed Clifford’s geometric algebra as a unifying language for mathematics and physics. During the past 50 years, Clifford analysis has gradually developed into a comprehensive theory offering a direct, elegant, and powerful generalization to higher dimension of the theory of holomorphic functions in the complex plane. In its most simple but still useful setting, flat m-dimensional Euclidean space, Clifford analysis focuses on monogenic functions, i.e., null solutions of the Clifford vector–valued Dirac operator, ∂x = m j=1 ej ∂xj , where (e1 , . . . , em ) forms an orthogonal basis for the quadratic space R0,m underlying the construction of the real Clifford algebra R0,m . Monogenic functions have a special relationship with harmonic functions of several variables in that they are refining their properties. The reason is that, as does the Cauchy–Riemann operator in the complex plane, the rotation-invariant Dirac operator factorizes the m-dimensional Laplace operator. At the same time, Clifford analysis offers the possibility of generalizing 1D mathematical analysis to higher dimension in a rather natural way by encompassing all dimensions at once, in contrast to the traditional approach, which consists of taking tensor products of 1D phenomena. This last qualification of Clifford analysis will be exploited in this paper to construct a genuine multidimensional Fourier transform within the

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context of Clifford analysis. In fact, over the past years several specific multidimensional Fourier transforms were constructed using Clifford algebra with goals focused mostly in the field of signal analysis. An overview of these constructs is given in Section 4 where our Clifford–Fourier theory is developed. We have included an introductory section on Clifford analysis, and each section starts with an introductory situation. The new Clifford–Fourier transform is given in terms of an operator exponential, or alternatively, by a series representation. Particular attention is directed to the two-dimensional (2D) case since then the Clifford–Fourier kernel can be written in a closed form. The main section is preceded by a section on the fractional Fourier transform wherein, surprisingly, it is shown that the traditional and the Clifford analysis approach coincide. Section 5 develops the theory for the Clifford–Hermite and Clifford–Gabor filters for early vision. The topic in the last section is still in an experimental stage; we try to devise a cylindrical Fourier transform. The idea is the following: For a fixed vector in the image space, the level surfaces of the traditional Fourier kernel are planes perpendicular to that fixed vector. For this Fourier kernel we now substitute a new Clifford–Fourier kernel such that, again for a fixed vector in the image space, its phase is constant on coaxial cylinders w.r.t. that fixed vector. The point is that when restricted to dimension two, this new cylindrical Fourier transform coincides with the Clifford–Fourier transform of Section 4. Thus we are faced with the following situation: In dimension greater than two, we have a first Clifford–Fourier transform with elegant properties but no kernel in closed form, and a second cylindrical one with a kernel in closed form but more complicated calculation formulae. In dimension two both transforms coincide. Our aim is to present a consistent theory on a specific multidimensional Fourier transform in the hope that it might be used in applications.

2. THE CLIFFORD ANALYSIS TOOLKIT Clifford analysis is a well-established mathematical discipline that is closely related but complementary to harmonic analysis. It has gradually developed into a comprehensive theory that offers a direct, elegant, and powerful generalization to higher dimension of the theory of holomorphic functions in the complex plane. In its most simple but still useful setting, it focuses on the null solutions of various special partial differential operators arising naturally within the Clifford algebra language, the  most important of which is the so-called Dirac operator, ∂x = m j=1 ej ∂xj . Here (e1 , . . . , em ) forms an orthonormal basis for the quadratic space R0,m underlying the construction of the real Clifford algebra R0,m . Numerous papers, conference proceedings, and books have molded this theory and

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shown its ability for applications (e.g., Brackx, Delanghe, and Sommen, 1982; Delanghe, Sommen, and Souˇcek, 1992; Gilbert and Murray, 1991; Gürlebeck and Sprößig, 1990; Gürlebeck and Sprößig, 1997; Gürlebeck, Habetha, and Sprößig, 2006; Qian et al., 2004; Ryan, 2004; Ryan, 1996; Ryan, 2000; and Ryan and Struppa, 1998). This section is included to make the chapter self-contained and readable by an audience unacquainted with Clifford analysis. It deals with the basic notions of Clifford algebra (Section 2.1) and Clifford analysis (Section 2.2) that are necessary for our purpose. Moreover, in Section 2.3 we collect some results concerning the standard tensorial Fourier transform and spherical harmonics that are used in the following sections. The Clifford–Hermite functions, forming a  generalized  basis for the space L2 Rm , dV(x) of square-integrable functions, are shown to be eigenfunctions of the standard Fourier transform (Section 2.4). The last section discusses multidimensional analytic signals.

2.1. Clifford Algebra Clifford algebra may be considered a generalization to higher dimension of the norm division algebras of the real numbers R, the complex numbers C, and the quaternions H. In these traditional algebras, R, C, and H, division by a nonzero number is always possible and in each of these cases there is a norm  .  such that λ μ = λ μ, for all λ, μ ∈ R, C or H. Such as in the skew field of quaternions H the multiplication in a Clifford algebra is noncommutative, but still associative. However, there are zero divisors making division by a nonzero Clifford number impossible in general. One of the most constructive ways to define a Clifford algebra is as follows. Let the Euclidean space Rm be endowed with a nondegenerate quadratic form of signature (p, q), p + q = m, and let (e1 , . . . , em ) be an orthonormal basis for Rp,q . The noncommutative multiplication in the universal Clifford algebra Rp,q , constructed over Rp,q , is governed by the following rules:

ej2 = 1, j = 1, . . . , p 2 ep+j = −1, j = 1, . . . , q

ej ek + ek ej = 0, j  = k, j, k = 1, . . . , m. A canonical basis for Rp,q is obtained by considering for any set A = {j1 , . . . , jh } ⊂ {1, . . . , m} = M, ordered by 1 ≤ j1 < j2 < . . . < jh ≤ m, the element eA = ej1 ej2 . . . ejh . Moreover, for the empty set ∅ one puts e∅ = 1, the latter being the identity element (i.e., the neutral element with respect to multiplication). Note that R0,1 is isomorphic with C via the correspondence

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e1 ←→ i, while R0,2 is isomorphic with H via the correspondence

e1 ←→ i e2 ←→ j e1 e2 ←→ ij = k.  Any Clifford number  λ in Rp,q may thus be written as λ = A⊂M eA λA ,  λA ∈ R, or still as λ = m |A|=k eA λA is the so-called k=0 [λ]k , where [λ]k = k-vector part of λ (k = 0, 1, . . . , m). Denoting by Rkp,q the subspace of all k-vectors in Rp,q (i.e., the image of Rp,q under the projection operator [ . ]k ), one has the multivector structure decomposition Rp,q = R0p,q ⊕ R1p,q ⊕ . . . ⊕ Rm p,q , leading to the identification of R with the subspace of real scalars R0p,q and of the Euclidean space Rm with the subspace of real Clifford vectors R1p,q . The Clifford number eM = e1 e2 . . . em is usually called the pseudoscalar; depending on the dimension m, the pseudoscalar commutes or anticommutes with the k-vectors and squares to ±1. In the following text we consider the real Clifford algebra R0,m and the complex Clifford algebra Cm , which may be seen as its complexification Cm = C ⊗ R0,m = R0,m ⊕ i R0,m (i.e., all coefficients are taken to be complex). An important anti-automorphism of Cm leaving the multivector structure invariant is the Hermitean conjugation, defined by

(λμ)† = μ† λ† † (λA eA )† = λcA eA

ej† = −ej

(A ⊂ M) ( j = 1, . . . , m).

Here λcA denotes the complex conjugate of the complex number λA . In view of the decomposition Cm = R0,m ⊕ i R0,m , any complex Clifford number λ ∈ Cm may also be written as λ = a + ib with a, b ∈ R0,m . Moreover, the restriction of the Hermitean conjugation to R0,m coincides with the usual conjugation in the Clifford algebra R0,m (i.e., the main antiinvolution for which ej = −ej , j = 1, . . . , m). Hence, one may also write λ† = (a + ib)† = a − ib. The Hermitean conjugation leads to a Hermitean inner product and its associated norm on Cm , given respectively, by

(λ, μ) = [λ† μ]0

and |λ|2 = [λ† λ]0 =

 A

|λA |2 .

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For λ, μ ∈ Cm the following properties hold:

|λμ| ≤ 2m |λ| |μ|

|λ + μ| ≤ |λ| + |μ|.

and

(1)

The Euclidean space Rm is embedded in the Clifford algebras R0,m and Cm by identifying variable x  the point (x1 , . . . , xm ) with the real vector m+1 is identified given by x = m e x , whereas the Euclidean space R j=1 j j with R00,m ⊕ R10,m by identifying (x0 , x1 , . . . , xm ) with the real paravector x0 + x . The product of two vectors divides up into a scalar part (the inner product up to a minus sign) and a 2-vector, also called bivector, part (the wedge product):

x y = x . y + x ∧ y, where

x . y = − < x, y > = −

m 

xj yj

j=1

and

x∧y =

m  m 

ei ej (xi yj − xj yi ).

i=1 j=i+1

Note that the square of a vector variable x is scalar valued and equals the norm squared up to a minus sign

x2 = − < x, x > = −|x|2 . This implies that each vector is invertible

x−1 = −

x . |x|2

In addition, each paravector x0 + x is invertible with

(x0 + x)−1 =

x0 − x . |x0 + x|2

The spin group SpinR (m) of the Clifford algebra consists of all products of an even number of unit vectors

SpinR (m) = {s = ω1 . . . ω2 ; ωj ∈ Sm−1 , j = 1, . . . , 2 ,  ∈ N},

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with Sm−1 the unit sphere in Rm . The spin group doubly covers the rotation group SOR (m): for T ∈ SOR (m) there exists s ∈ SpinR (m) such that T(x) = sxs. But then also T(x) = (−s)x(−s), explaining the double character of this covering. For a detailed study of Clifford algebra, see Porteous (1995).

2.2. Clifford Analysis Clifford analysis offers a function theory that is a higher-dimensional analog of the theory of the holomorphic functions of one complex variable. The functions considered are defined in the Euclidean space Rm or Rm+1 (m > 1) and take their values in Clifford algebra R0,m or in its complexification Cm . The central notion in Clifford analysis is the notion of monogenicity, a notion that is the multidimensional counterpart to that of holomorphy in the complex plane. A function F(x1 , . . . , xm ), respectively F(x0 , x1 , . . . , xm ), defined and continuously differentiable in an open region of Rm , respectively Rm+1 , and taking values in R0,m or Cm , is called left monogenic in that region if

∂x [F] = 0,

respectively

(∂x0 + ∂x )[F] = 0.

Here ∂x is the Dirac operator in Rm :

∂x =

m 

ej ∂xj .

j=1

This Dirac operator is an elliptic, rotation-invariant, vector differential operator of the first order, which may be viewed as the “square root” of the Laplace operator in Rm , since

m = −∂x2 .

(2)

The operator ∂x0 + ∂x is termed the Cauchy–Riemann operator in Rm+1 ; it factorizes the Laplace operator in Rm+1 :

m+1 = (∂x0 + ∂x )(∂x0 + ∂x ) = (∂x0 + ∂x )(∂x0 − ∂x ). The factorization shown in Eq. (2) of the Laplace operator establishes a special relationship between Clifford analysis and harmonic analysis in that monogenic functions refine the properties of harmonic functions. Note, for instance, that each monogenic function is harmonic and that in its turn each harmonic function h(x) can be split as h(x) = f (x) + x g(x) with f and g monogenic, and that a real harmonic function is always the real part of

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a monogenic one, which need not be the case for a harmonic function of several complex variables. The notion of right monogenicity is defined in a similar manner by letting act the Dirac operator or the Cauchy–Riemann operator from the right. It is easily seen that if a Clifford algebra–valued function F is left monogenic, its Hermitean conjugate F† is right monogenic. Introducing spherical coordinates in Rm by:

x = rω ,

r = |x| ∈ [0, +∞[,

ω ∈ Sm−1 ,

the Dirac operator takes the form

 1 ∂x = ω ∂r +  , r 

where

 = −x ∧ ∂x = −

m m  

ei ej (xi ∂xj − xj ∂xi )

i=1 j=i+1

is the angular Dirac operator acting only on the angular coordinates. In Subsection 4.5.1 we will also use the angular momentum operators

Lij = xi ∂xj − xj ∂xi ,

i, j = 1, 2, . . . , m.

Another fundamental operator is the Euler operator

E = < x , ∂x > =

m 

xi ∂xi = r∂r

i=1

which measures the degree of homogeneity of both Clifford polynomials and Clifford polynomial operators. If P s denotes the space of scalar-valued polynomials in Rm , then a Clifford algebra–valued polynomial (Clifford polynomial for short) is an element of P = P s ⊗ Cm . The inner product on the space P is defined as

  < P(x), Q(x) > = P† (∂x ) Q(x) |x=0 , with P(∂x ) the differential operator obtained by substituting for all i, ∂xi for xi in P(x). The subspaces Pk of homogeneous Clifford polynomials of degree k, k ∈ N, are the polynomial eigenspaces of the Euler operator

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(i.e., E[Rk ] = kRk , Rk ∈ Pk ). Obviously, every Clifford polynomial can be decomposed into homogeneous ones. Let End(P s ) be the algebra of endomorphisms of P s ; the elements of End(P s ) ⊗ Cm are called Clifford polynomial operators. A Clifford polynomial operator A transforms a Clifford polynomial P ∈ P into another Clifford polynomial A[P] ∈ P. Like a Clifford polynomial, every Clifford polynomial operator can be decomposed into homogeneous parts:

A=



A ,

(3)

∈Z

with A a homogeneous polynomial operator of degree  (i.e., A [Pk ] ⊂ Pk+ , k ∈ N). Moreover, the homogeneous operators A are determined by the commutation relation [E, A ] = EA − A E = A . In Section 4.2 we consider so-called Clifford differential operators with polynomial coefficients. These operators take the form

P(x, ∂x ) =



α

pα (x) ∂x ,

α α

α

with ∂x = ∂x11 . . . ∂xαmm and where pα (x) ∈ P. Let D(Cm ) denote the algebra of Clifford differential operators with polynomial coefficients. Then clearly D(Cm ) ⊂ End(P s ) ⊗ Cm . This algebra D(Cm ) is generated by the basic operators {ej , xj , ∂xj , j = 1, 2, . . . , m}. Examples of such Clifford differential operators with polynomial coefficients are the Dirac operator ∂x , the Euler operator E, the angular Dirac operator , and the left vector multiplication operator f → x f. Many classical theorems from complex analysis in the plane (e.g., Cauchy’s theorem, Cauchy’s integral theorem, Taylor series, Laurent series) have their multidimensional counterpart in Clifford analysis. However, note that due to the noncommutativity of the multiplication in the Clifford algebra, the product of two monogenic functions is, generally no longer monogenic. The natural powers of the vector variable x or the paravector variable x0 + x are not monogenic either. These drawbacks have been overcome in the following way. There is a fundamental method for constructing monogenic functions, the so-called Cauchy– Kowalewskaia (CK) extension procedure, introduced by Sommen (1981). It runs as follows. If ⊂ Rm is open, then an open neighborhood of in Rm+1 is said to be x0 -normal if for each x ∈ the line segment {x + t ; t ∈ R} ∩ is connected and contains exactly one point in (Figure 1). Considering Rm as the hyperplane x0 = 0 in Rm+1 , a real analytic function f (x) in an open connected domain in Rm can be uniquely extended to a monogenic function F(x0 , x) in an open connected and x0 -normal neighbourhood of

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65



x



m

{x t ;t  }

FIGURE 1

An x0 -normal neighborhood of in Rm+1 .

in Rm+1 . This CK extension of f (x) is given by

F(x0 , x) =

∞  x (−1) 0 ∂x [ f (x)] = exp (−x0 ∂x )[ f (x)]. !

(4)

=0

In particular, the CK extension of the real variables xj , j = 1, . . . , m, are zj = xj − x0 ej , j = 1, . . . , m, the so-called monogenic variables. The CK extension procedure leads to the CK product which, despite the noncommutativity of the Clifford algebra, preserves the monogenicity of the factors; the CK product of two monogenic functions in Rm+1 is the CK extension to Rm+1 of the product of the real analytic restrictions to Rm . For example, the CK product of the monogenic variables zj and zk (k  = j) is the CK extension of xj xk , given by

zj  zk =

1 (zj zk + zk zj ). 2

The CK products of the monogenic variables are precisely the building blocks of the Taylor series expansion of a monogenic function. In this chapter, the monogenic homogeneous Clifford polynomials and functions, or spherical monogenics, play an important role. A left, respectively right, monogenic homogeneous Clifford polynomial Pk of degree k (k ≥ 0) in Rm is called a left, respectively right, solid inner spherical monogenic of order k. A left, respectively right, monogenic homogeneous function Qk of degree −(k + m − 1) in Rm \ {0} is called a left, respectively right, solid outer spherical monogenic of order k. The set of all left, respectively right, solid inner spherical monogenics of order k will be denoted by M+ (k), respectively Mr+ (k), whereas the set of all

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left, respectively right, solid outer spherical monogenics of order k will be denoted by M− (k), respectively Mr− (k). The dimension of M+ (k) is given by

    m+k−2 (m + k − 2)! . dim M+ (k) = = (m − 2)! k! m−2 For Pk ∈ M+ (k) and s ∈ N the following fundamental formula holds:

s

∂x [x Pk ] =

−s xs−1 Pk −(s + 2k + m − 1) xs−1 Pk

for s even for s odd.

(5)

Moreover, the left solid inner spherical monogenics are polynomial eigenfunctions of the angular Dirac operator, i.e., [Pk ] = −kPk , Pk ∈ M+ (k). The set of harmonic homogeneous polynomials Sk of degree k in Rm :

m [Sk (x)] = 0

and

Sk (tx) = tk Sk (x),

usually called solid spherical harmonics, is denoted by H(k). Obviously, we have that

M+ (k) ⊂ H(k)

and Mr+ (k) ⊂ H(k).

Let H(r) be a unitary right Clifford module; that is, H(r) , + is an abelian group and a law ( f , λ) → f λ from H(r) × R0,m into H(r) is defined such that for all λ, μ ∈ R0,m and f , g ∈ H(r)

(i) f (λ + μ) = f λ + f μ (iii) ( f + g)λ = f λ + gλ

(ii) f (λμ) = ( f λ)μ (iv) fe∅ = f .

Note that H(r) becomes a real vector space if R is identified with Re∅ ⊂ R0,m . Then a function ( . , . ) : H(r) × H(r) → R0,m is said to be an inner product on H(r) if for all f , g, h ∈ H(r) and λ ∈ R0,m

(i) ( f , gλ + h) = ( f , g)λ + ( f , h); (iii) [( f , f )]0 ≥ 0;

(ii) ( f , g) = ( g, f ); (iv) [( f , f )]0 = 0 iff f = 0.

From this R0,m -valued inner product ( . , . ), one can deduce the real inner product

( f , g)R = [( f , g)]0

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67

on H(r) , the latter being considered a real vector space. Putting for each f ∈ H(r) : f 2 = [( f , f )]0 ,  .  is a norm on H(r) turning it into a normed right Clifford module. Now, let H(r) be a unitary right Clifford module provided with an inner product ( . , . ). Then it is called a right Hilbert–Clifford module if H(r) considered as a real vector space provided with the real inner product ( . , . )R is a Hilbert space (see Delanghe and Brackx, 1978). Let h be a positive function on Rm . Then we consider the Clifford algebra–valued inner product of the functions f and g defined in Rm and taking values in Clifford algebra Cm



< f,g > =

Rm

h(x) f † (x) g(x) dV(x),

where dV(x) is the Lebesgue measure on Rm , and moreover the associated norm

f 2 = [< f , f >]0 . The unitary right Clifford module of Clifford algebra–valued measurable functions on Rm for which f 2 < ∞ is a right Hilbert–Clifford module, denoted by L2 Rm , h(x) dV(x) . In particular, if we take h(x) ≡ 1, we obtain the right Hilbert–Clifford module of square integrable functions:



m



f : Lebesgue measurable in Rm for which

L2 R , dV(x) =

 f 2 =

1/2

Rm

| f (x)|2 dV(x)

 <∞ .

A representation of the spin group on this space is the H-representation given by H(s)f (x)  = sf (sxs)s,  s ∈ SpinR (m). Naturally, L1 Rm , dV(x) denotes the right Clifford module of integrable functions



m





f : Lebesgue measurable in Rm for which

L1 R , dV(x) =  f 1 =

Rm

 | f (x)| dV(x) < ∞ ,

      whereas C0 Rm , L∞ Rm and S Rm denote the right Clifford modules of Cm -valued, respectively, continuous, bounded, and rapidly decreasing functions. A basic integral formula used in Section 6.2 is the Clifford–Stokes formula.

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Let ⊂ Rm be open, let C be a compact orientable m-dimensional manifold with boundary ∂C, and define the oriented vector-valued surface element dσx on ∂C by the Clifford differential form

dσx =

m  (−1)j ej dxM\{j} , j=1

where

dxM\{j} = dx1 ∧ . . . ∧ [dxj ] ∧ . . . ∧ dxm ,

j = 1, 2, . . . , m.

We also introduce the volume element

dxM = dx1 ∧ . . . ∧ dxm . Theorem 2.1 [Clifford–Stokes theorem]. Let f , g ∈ C1 ( ). Then for each C ⊂ , one has



 ∂C

f (x) dσx g(x) =

C

  [ f ∂x g + f (∂x g)] dxM .

2.3. Some Useful Results Concerning the Fourier Transform and Spherical Harmonics In Clifford analysis extensive use is made of the standard tensorial multidimensional Fourier transform given by:

F [ f ](ξ) =

1 (2π)m/2

 Rm

exp (−i < x , ξ >) f (x) dV(x).

(6)

Note that the Fourier image inherits its Clifford algebra character only from the original function f (x) itself, because the traditional Fourier kernel is scalar valued. We now list a few fundamental results concerning the Fourier transform.

2 Proposition 2.1 The Gaussian function exp − |x|2 is an eigenfunction of the Fourier transform:

  

 2 |ξ|2 (ξ) = exp − 2 . F exp − |x|2 Theorem 2.2 The Fourier transform F is an isometry of square  on the space  integrable functions, in other words, for all f , g ∈ L2 Rm , dV(x) the Parseval

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69

formula holds

< f , g > = < F [f ], F [g] > .   In particular, for each f ∈ L2 Rm , dV(x) one has f 2 = F [f ]2 . Proposition 2.2 The Fourier transform F satisfies (i) the multiplication rule:

F [x f (x)](ξ) = i∂ξ [ F [f (x)](ξ)]; (ii) the differentiation rule:

F [∂x [ f (x)]](ξ) = iξ F [ f (x)](ξ). The properties established in Proposition 2.2 also may be expressed by stating that the vector variable x and the Dirac operator ∂x are each other’s Fourier symbol; this property is usually called Fourier duality; in Clifford analysis circles, it is often termed Fischer duality. Next, we formulate the Funk–Hecke theorem (see Hochstadt, 1971), which is the key to the calculation of certain integrals involving spherical harmonics. Theorem 2.3 [Funk–Hecke theorem on the unit sphere]. Let Sk ∈ H(k) be a spherical harmonic of degree k and η a fixed point on the unit sphere Sm−1 . Denote η) = tη for ω ∈ Sm−1 . Then < ω, η > = cos (ω,



 Sm−1

f (tη )Sk (ω)dS(ω) = Am−1

1

−1

 f (t)(1 − t2 )(m−3)/2 Pk,m (t)dt Sk (η),

where Pk,m (t) denotes the Legendre polynomial of degree k in m-dimensional Euclidean space and

Am−1 =

2 π(m−1)/2

 m−1 2

the surface area of the unit sphere Sm−2 in Rm−1 . From the above we can easily deduce a so-called Funk–Hecke theorem in space. Theorem 2.4 [Funk–Hecke theorem in space] Let Sk ∈ H(k) be a spherical harmonic of degree k and η a fixed point on the unit sphere Sm−1 . Denote

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Fred Brackx et al.

< ω, η > = tη for ω ∈ Sm−1 , then

 Rm

g(r) f (tη ) Sk (ω) dV(x) 

= Am−1



+∞

g(r) r

m−1

dr

1

−1

0

2 (m−3)/2

f (t) (1 − t )

 Pk,m (t) dt Sk (η).

As the Legendre polynomials are even or odd according to the parity of k, we can also state the following corollary, which will be frequently used in Section 6.3. Corollary 2.1 Let Sk ∈ H(k) be a spherical harmonic of degree k and η a fixed point on the unit sphere Sm−1 . Denote < ω, η > = tη for ω ∈ Sm−1 , then the 3D integral

 Rm

g(r) f (tη ) Sk (ω) dV(x)

is zero whenever • f is an odd function and k is even; • f is an even function and k is odd. Theorem 2.3 leads to the following result (see Stein and Weiss, 1971). Proposition 2.3 Let Sk ∈ H(k) be a spherical harmonic of degree k, then for r, ρ > 0 and ω, η ∈ Sm−1 , one has

 Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω)

= (−i)k (2π)m/2 Sk (η) (ρr)1−m/2 Jk+m/2−1 (ρr)

m with Jk+m/2−1 the Bessel function of the first kind of order k + − 1 . 2 Proof. Application of the Funk–Hecke theorem on the unit sphere leads to

 Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω)

= Am−1 Sk (η)



+1

−1

exp (−irρt) (1 − t2 )(m−3)/2 Pk,m (t) dt.

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71

As

Pk,m (t) =

k! (m − 3)! (m−2)/2 C (t) (k + m − 3)! k

(7)

and the Gegenbauer polynomials Ckλ satisfy

Ckλ (−x) = (−1)k Ckλ (x), we obtain

 Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω)  k

= (−1) Am−1 Sk (η)

+1 −1

exp (irρt) (1 − t2 )(m−3)/2 Pk,m (t) dt.

(8)

Taking into account (see Hochstadt, 1971)



+1

−1

exp (irρt) (1 − t2 )(m−3)/2 Pk,m (t) dt k

m/2−1

=i 2

√



m−1 π 2

 (ρr)1−m/2 Jk+m/2−1 (ρr)



m with Jk+m/2−1 the Bessel function of the first kind of order k + − 1 , 2 Eq. (8) becomes

 Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω) k

m/2−1

= (−i) 2

√



m−1 π 2

 Am−1 Sk (η) (ρr)1−m/2 Jk+m/2−1 (ρr).

Expliciting the area Am−1 of the unit sphere Sm−2 in Rm−1 in terms of the Gamma function, we finally obtain

 Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω)

= (−i)k (2π)m/2 Sk (η) (ρr)1−m/2 Jk+m/2−1 (ρr).

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By means of this result one can prove the following proposition (see Brackx, De Schepper, and Sommen, 2003). Proposition 2.4 Let Sk ∈ H(k) be a solid spherical harmonic of degree k, then



 2 2 = (−1)k Sk (x) exp − |x|2 . Sk (∂x ) exp − |x|2 Proof.

By means of Proposition 2.1 and 2.2, we have

 F Sk (x) exp



2 − |x|2





 |x|2 (ξ) = Sk (i∂ξ ) F exp − 2 (ξ) 



k

= i Sk (∂ξ ) exp

|ξ|2 − 2



or equivalently





Sk (∂ξ ) exp

|ξ|2 − 2

 |x|2 (ξ). = (−i) F Sk (x) exp − 2





k

Using spherical coordinates

x = r ω,

ξ = ρ η where

r = |x| ,

ρ = |ξ|,

ω, η ∈ Sm−1 ,

we obtain 

 2 F Sk (x) exp − |x|2 (ξ) 1 = (2π)m/2 =

1 (2π)m/2

 Rm

 0

2 exp (−i < x, ξ >) Sk (x) exp − |x|2 dV(x)

+∞

2  rk+m−1 exp − r2 dr

Sm−1

exp (−irρ < ω, η >) Sk (ω) dS(ω).

Next, Proposition 2.3 leads to

  2 (ξ) F Sk (x) exp − |x|2  = (−i)k ρ1−m/2 Sk (η)

0

+∞

2 rk+m/2 Jk+m/2−1 (ρr) exp − r2 dr.

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73

From the theory of Bessel functions (see Erdélyi et al., 1953b) we know that



+∞

0

2 2 rk+m/2 Jk+m/2−1 (ρr) exp − r2 dr = ρk+m/2−1 exp − ρ2 .

Hence, we finally obtain

  2 2 (ξ) = (−i)k ρk Sk (η) exp − ρ2 F Sk (x) exp − |x|2   |ξ|2 = (−i)k Sk (ξ) exp − 2 , which leads to the desired result.

2.4. The Generalized Clifford–Hermite Polynomials On the real line the

Hermite polynomials associated with the weight x2 function exp − 2 may be defined by the Rodrigues formula

Hen (x) = (−1)n exp

2  2 dn  x exp − x2 , 2 dxn

n = 0, 1, 2, . . .

They constitute an orthogonal

 basis for the weighted Hilbert space  x2 L2 ] − ∞, +∞[ , exp − 2 dx , and satisfy the orthogonality relation



+∞ −∞



x2 exp − 2

 Hen (x) Hen (x) dx = n!

√ 2π δn,n

and moreover the recurrence relation

Hen+1 (x) = x Hen (x) −

d [Hen (x)]. dx

Furthermore, Hen (x) is an even or an odd function according to the parity of n; that is, Hen (−x) = (−1)n Hen (x). Sommen (1988) introduced the generalized Clifford–Hermite polynomials, which are a specific generalization to Clifford analysis of the above Hermite polynomials on the real line.

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2.4.1. Definition In order to a basis for the weighted Hilbert module

obtain |x|2 m L2 R , exp − 2 dV(x) , Sommen (1988) introduced the generalized Clifford–Hermite polynomials. These generalized Clifford–Hermite ∗ polynomials

are defined by the CK-extension G of the weight function

exp − |x|2 order k :

2

Pk (x), with Pk (x) any left solid inner spherical monogenic of

∗

G (x0 , x) = exp



2 − |x|2

∞ 

 x

0

=0

!

H,k (x) Pk (x).

From the monogenicity of G∗ , we obtain

H+1,k (x) Pk (x) = (x − ∂x )[H,k (x) Pk (x)], which in its turn leads to the following recurrence relations:

H2+1,k (x) = (x − ∂x )[H2,k (x)] and

H2+2,k (x) = (x − ∂x )[H2+1,k (x)] − 2k

x H2+1,k (x). |x|2

A straightforward calculation yields

H0,k (x) = 1 H1,k (x) = x H2,k (x) = x2 + 2k + m = −|x|2 + 2k + m H3,k (x) = x3 + (2k + m + 2) x = x (−|x|2 + 2k + m + 2) H4,k (x) = x4 + 2(2k + m + 2) x2 + (2k + m)(2k + m + 2) = |x|4 − 2(2k + m + 2)|x|2 + (2k + m)(2k + m + 2) H5,k (x) = x5 + 2(2k + m + 4) x3 + (2k + m + 4)(2k + m + 2) x   = x |x|4 − 2(2k + m + 4)|x|2 + (2k + m + 4)(2k + m + 2) etc.

(9)

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The Fourier Transform in Clifford Analysis

It is important to remark that the functions H,k (x) do not depend upon the particular choice of the left solid inner spherical monogenic Pk , but only upon the order k. Note that the generalized Clifford–Hermite polynomials H,k are of degree  in the variable x with real coefficients depending on k. The polynomial H2,k (x) contains only even powers of x and so is scalar valued, while H2+1,k (x) contains only odd ones and so is vector valued. Moreover, taking into account Eq. (4), it is easily seen that the generalized Clifford–Hermite polynomials satisfy the Rodrigues formula

H,k (x) Pk (x) = exp



|x|2 2

 

2 (−∂x ) exp − |x|2 Pk (x) .

(10)

Furthermore, they can be expressed in terms of the generalized Laguerre polynomials Lα on the real line given by

Lα (u)

  = (−1)s s=0

us ( + α + 1) . ( − s + 1) (α + s + 1) s!

We indeed have:

 H2p,k (x) = 2p p!

m/2+k−1 Lp

 |x|2 , 2

 H2p+1,k (x) = 2p p!

m/2+k Lp

|x|2 2

 x, (11)

confirming that H2p,k is scalar valued, while H2p+1,k is vector valued. Moreover, it follows that

H2p+1,k (x) = x H2p,k+1 (x). In Section 6 we will need the so-called Kummer function, also termed confluent hypergeometric function ∞

1 F1 (a; c; u) =

(c)  (a + s) us , (a) (c + s) s!

c  = 0, −1, −2, . . .

s=0

The fact is that the Laguerre polynomials may be expressed in terms of Kummer’s function:

Lα (u) =

( + α + 1) 1 F1 (−; α + 1; u), ( + 1) (α + 1)

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Fred Brackx et al.

which leads to the following expression of the generalized Clifford– Hermite polynomials:

    m 2  p + + k |x| m  2  1 F1 −p; + k; H2p,k (x) = 2p 2 2  m2 + k      p + m2 + k + 1 |x|2 m   x 1 F1 −p; + k + 1; . (12) H2p+1,k (x) = 2 2 2  m2 + k + 1 p

It may also be shown that the generalized Clifford–Hermite polynomials are mutually orthogonal with respect to a Gaussian weight function, more precisely

 Rm

 † 2 H,k1 (x) Pk1 (x) Ht,k2 (x) Pk2 (x) dV(x) = γ,k1 δ,t δk1 ,k2 exp − |x|2 (13)

with

γ2p,k

22p+m/2+k p! πm/2    =  m2

m 2



+k+p

and

γ2p+1,k

22p+m/2+k+1 p! πm/2    =  m2

m 2



+k+p+1

.

Furthermore, the set



1 (j) Ht,k (x) Pk (x) ; t, k ∈ N, j ≤ dim(M+ (k)) (γt,k )1/2







2 constitutes an orthonormal basis for L2 Rm , exp − |x|2 dV(x) . Here

  (j) Pk (x) ; j = 1, 2, . . . , dim M+ (k)



denotes an orthonormal basis of the space M+ (k) of left solid inner spherical monogenics of order k.

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77

2.4.2. Differential Equations In this subsection we list a few useful differential equations satisfied by the generalized Clifford–Hermite polynomials. Proposition 2.5 The generalized Clifford–Hermite polynomials satisfy (i) the eigenvalue equation

(∂x x − x∂x )[H,k (x) Pk (x)] = a,k H,k (x) Pk (x) (ii) the annihilation equation

∂x [H,k (x) Pk (x)] = −C,k H−1,k (x) Pk (x) (iii) the second-order differential equation

∂x2 [H,k (x) Pk (x)] − x ∂x [H,k (x) Pk (x)] − C,k H,k (x) Pk (x) = 0 with



−(m + 2k) m + 2k − 2

a,k = and

C,k =

  − 1 + m + 2k

for  even for  odd

for  even for  odd.

Proof. (i) Taking into account Eq. (5), we indeed have for  even—that is,  = 2p,

∂x [x H2p,k (x) Pk (x)] = ∂x [H2p,k (x)] x Pk (x) + H2p,k (x) ∂x [x Pk (x)] = x ∂x [H2p,k (x)]Pk (x) − (m + 2k) H2p,k (x) Pk (x) = x ∂x [H2p,k (x) Pk (x)] − (m + 2k) H2p,k (x) Pk (x). Now we consider the case  odd—that is,  = 2p + 1. As H2p+1,k (x) takes the form x f (r) with f a polynomial of degree p in r2 , we obtain

∂x [x H2p+1,k (x) Pk (x)] = ∂x [ f (r) x2 Pk (x)] = ∂x [ f (r)] x2 Pk (x) + f (r) ∂x [x2 Pk (x)] = x ∂x [ f (r)] x Pk (x) − 2f (r) x Pk (x).

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Fred Brackx et al.

As

∂x [H2p+1,k (x) Pk (x)] = ∂x [f (r)] x Pk (x) − (m + 2k) f (r) Pk (x), we obtain the desired result:

∂x [x H2p+1,k (x) Pk (x)] = x ∂x [H2p+1,k (x) Pk (x)] + (m + 2k − 2) H2p+1,k (x) Pk (x). (ii) We prove the statement by induction. First, for  = 0 we have

∂x [H0,k (x) Pk (x)] = ∂x [Pk (x)] = 0, while for  = 1 ∂x [H1,k (x) Pk (x)] =∂x [x Pk (x)] = − (m + 2k) Pk (x) = − C1,k H0,k (x) Pk (x). Next, assume that the property holds for  − 1; that is,

∂x [H−1,k (x) Pk (x)] = −C−1,k H−2,k (x) Pk (x). Using Eq. (9) and the induction hypothesis, we obtain

∂x [H,k (x) Pk (x)] = ∂x [x H−1,k (x) Pk (x)] − ∂x2 [H−1,k (x) Pk (x)] = ∂x [x H−1,k (x) Pk (x)] + C−1,k ∂x [H−2,k (x) Pk (x)]. By means of (i), the induction hypothesis and Eq. (9), this becomes consecutively

∂x [H,k (x) Pk (x)] = a−1,k H−1,k (x) Pk (x) + x ∂x [H−1,k (x) Pk (x)] + C−1,k ∂x [H−2,k (x) Pk (x)] = a−1,k H−1,k (x) Pk (x) − C−1,k (x − ∂x )[H−2,k (x) Pk (x)] = a−1,k H−1,k (x) Pk (x) − C−1,k H−1,k (x) Pk (x) = − C,k H−1,k (x) Pk (x).

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(iii) By means of (ii) and Eq. (9), we have consecutively

∂x2 [H,k (x) Pk (x)] − x ∂x [H,k (x) Pk (x)] = −C,k ∂x [H−1,k (x) Pk (x)] + C,k x H−1,k (x) Pk (x) = C,k (x − ∂x )[H−1,k (x) Pk (x)] = C,k H,k (x) Pk (x). Note that formula (iii) of Proposition 2.5 generalizes the differential equation

d d2 [Hen (x)] + n Hen (x) = 0 [Hen (x)] − x 2 dx dx satisfied by the classical Hermite polynomials on the real line. Furthermore, combining formula (ii) of Proposition 2.5 and Eq. (9) yields

H+1,k (x) Pk (x) − x H,k (x) Pk (x) − C,k H−1,k (x) Pk (x) = 0,

(14)

which is a generalization of the recurrence relation

Hen+1 (x) − x Hen (x) + n Hen−1 (x) = 0 satisfied by the classical Hermite polynomials.

2.4.3. Orthonormal Basis for the Space of Square Integrable Functions By means of the generalized Clifford–Hermite polynomials we now construct an orthonormal basis for the space of square integrable functions, the finite energy signals. Proposition 2.6 The set



1 (j) |x|2 ; s, k ∈ N, j ≤ dim(M+ (k)) H (x) P (x) exp − s,k 4 k (γs,k )1/2



  constitutes an orthonormal basis for L2 Rm , dV(x) . Proof. By means of Eq. (13), the orthonormality of the set is straightforward.   Now, take f ∈ L2 Rm , dV(x) . This means that

 Rm

   f (x)2 dV(x) =

 Rm



2 2  |x|  |x|2  f (x) exp 4  exp − 2 dV(x) < ∞,

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Fred Brackx et al.



2

2 in other words, f (x) exp |x|4 ∈ L2 Rm , exp − |x|2 dV(x) . Consequently, there exists a linear combination 

N N  

 



dim M+ (k)

s=0 k=0

as,k,j (γs,k

j=1

(j)

)1/2

Hs,k Pk ,

as,k,j ∈ Cm

such that

 +   dim M (k)   N N    as,k,j ( j)   f exp |x|2 − Hs,k Pk  4    (γ )1/2 s=0 k=0



|x|2

L2 Rm ,exp − 2

s,k

j=1



dV(x)

tends to zero if N, N  → ∞. This also implies that

 +   dim M (k)  2 N N  2    as,k,j ( j) |x|  f exp − Hs,k Pk  4    1/2 (γ ) s=0 k=0

j=1



|x|2

L2 Rm ,exp − 2

s,k



dV(x)

tends to zero if N, N  → ∞ or that

 Rm

 +   dim M (k) 2  N N

    as,k,j 2 (j)  f (x) exp |x| − Hs,k (x) Pk (x)  4 1/2 (γ ) s=0 k=0

2 × exp − |x|2 dV(x)

j=1

s,k

tends to zero if N, N  → ∞. So finally we obtain that

 Rm

 +   dim M (k)   N N  

2  as,k,j (j) |x|2   f (x) − Hs,k (x) Pk (x) exp − 4  dV(x)  (γ )1/2 s=0 k=0

j=1

s,k

tends to zero if N, N  → ∞, which proves the statement.

2.4.4. Eigenfunctions of the Fourier Transform The final step is to obtain an L2 -basis consisting of eigenfunctions of the Fourier transform. In view of Proposition 2.1, we carry out the

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81

√ substitution x → 2 x, which leads to the following orthonormal basis for L2 (Rm , dV(x)):

φs,k,j (x) =



√ 2m/4 (j) √ |x|2 H ( 2x) P ( 2x) exp − s,k 2 ; k (γs,k )1/2 dim(M+ (k))

s, k ∈ N, j ≤

 .

(15)

Note that an equivalent orthogonal basis is given by





−p;

1 F1



2 m (j) + k; |x|2 exp − |x|2 Pk (x), 2 

1 F1



2 m (j) −p; + k; |x|2 exp − |x|2 x Pk−1 (x) 2

with p, k ∈ N, j ≤ dim(M+ (k)). Proposition 2.7 For each left solid inner spherical monogenic Pk of order k ∈ N and all s ∈ N one has

 F exp



2 − |x|2

 × exp

|ξ|2 − 2

 √ √   Hs,k ( 2x) Pk ( 2x) (ξ) = exp −i(s + k) π2 

√ √ Hs,k ( 2ξ) Pk ( 2ξ).

Proof. (by induction). By means of Propositions 2.1, 2.2, and 2.4, one can easily verify that the statement holds for s = 0 and s = 1. Assuming that it holds for s, we now prove it for s + 1. By means of the recurrence relation in Eq. (14) we have

√ √ 2 exp − |x|2 Hs+1,k ( 2x) Pk ( 2x) =

√ √ √ 2 2x exp − |x|2 Hs,k ( 2x) Pk ( 2x)

√ √ 2 + Cs,k exp − |x|2 Hs−1,k ( 2x) Pk ( 2x).

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Fred Brackx et al.

Taking the Fourier transform yields

 F exp



2 − |x|2

 √ √ Hs+1,k ( 2x) Pk ( 2x) (ξ)

  

√ √ √  |x|2 = 2i ∂ξ F exp − 2 Hs,k ( 2x) Pk ( 2x) (ξ)  

√ √ 2 + Cs,k F exp − |x|2 Hs−1,k ( 2x) Pk ( 2x) (ξ). In view of the induction hypothesis, this becomes

 

√ √ 2 F exp − |x|2 Hs+1,k ( 2x) Pk ( 2x) (ξ) =

    √ √ √   |ξ|2 2i exp −i(s + k) π2 ∂ξ exp − 2 Hs,k ( 2ξ) Pk ( 2ξ) + Cs,k exp



−i(s − 1 + k) π2



 exp

|ξ|2 − 2



√ √ Hs−1,k ( 2ξ) Pk ( 2ξ).

Using formula (ii) of Proposition 2.5, we have





∂ξ exp

|ξ|2 − 2



 = −ξ exp  × exp



√ √ Hs,k ( 2ξ) Pk ( 2ξ)

|ξ|2 − 2

|ξ|2 − 2





√ √ √ Hs,k ( 2ξ) Pk ( 2ξ) − 2 Cs,k

√ √ Hs−1,k ( 2ξ) Pk ( 2ξ).

Consequently, we finally obtain

 

√ √ 2 F exp − |x|2 Hs+1,k ( 2x) Pk ( 2x) (ξ) = exp ×



−i(s + 1 + k) π2

 √



 exp

|ξ|2 − 2

 

√ √ √ √ 2ξ Hs,k ( 2ξ)Pk ( 2ξ) + Cs,k Hs−1,k ( 2ξ)Pk ( 2ξ)

  √ √   |ξ|2 π = exp −i(s + 1 + k) 2 exp − 2 Hs+1,k ( 2ξ) Pk ( 2ξ).

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83

2.5. Multidimensional “Analytic Signals” Let us introduce the fundamental solution E of the Cauchy–Riemann operator in Rm+1 :

E(x0 , x) =

x0 − x . Am+1 |x0 + x|m+1 1

In the complement of the origin it is a strong null solution of the Cauchy– Riemann operator ∂x0 + ∂x , this means a C∞ -smooth function satisfying

(∂x0 + ∂x )[E(x0 , x)] = 0. Moreover, considered as a distribution it satisfies

(∂x0 + ∂x )[E(x0 , x)] = δ(x0 , x), with δ(x0 , x) the delta or Dirac distribution supported at the origin. We may now define for a square integrable function f ∈ L2 (Rm , dV(x)), its Cauchy integral (see, for example, Gilbert and Murray, 1991) in the half spaces

Rm+1 = {(x0 , x) ∈ Rm+1 : x0 > 0} ± <

by

 C[ f ](x0 , x) = E(x0 , . ) ∗ f ( . )(x) =

Rm

E(x0 , x − y) f (y) dV(y).

Clearly this Cauchy integral is left monogenic in Rm+1 w.r.t the Cauchy– ± Riemann operator ∂x0 + ∂x and has limit zero for (x0 , x) tending to infinity. Moreover, it is a linear isomorphism between L2 (Rm , dV(x)) and the so-called Hardy spaces H 2 (Rm+1 ± ), defined by

H 2 (Rm+1 ± )

= F(x0 , x) left monogenic in Rm+1 ± 

 sup x0

> <

0

2

Rm

|F(x0 , x)| dV(x) < +∞ .

such that

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Furthermore, the L2 (Rm , dV(x)) nontangential boundary values for x0 → 0 + and x0 → 0− of the Cauchy integral—these are the Cauchy transforms—take the following form:

C + [ f ](x) := lim C[ f ](x0 , x) = x0 →0 >

1 1 f (x) + H[ f ](x) 2 2

and

1 1 C − [ f ](x) := lim C[ f ](x0 , x) = − f (x) + H[ f ](x). x0 →0 2 2 < Here H[ f ] denotes the Hilbert transform of the function f given by

x 2 Pv m+1 ∗ f Am+1 r  x−y 2 = Pv f ( y) dV( y) m+1 Am+1 Rm |x − y|  x−y 2 =− lim f (y) dV(y), 0 |x−y|> |x − y|m+1 Am+1 → >

H[ f ](x) =

where Pv denotes the Cauchy principal value. The properties of this Hilbert transform are the following: (i) the Hilbert transform H is a bounded operator on L2 (Rm , dV(x)) (ii) the Hilbert transform H is a unitary operator on L2 (Rm , dV(x)), this means H ∗ H = HH ∗ = I, H ∗ being the adjoint and I the identity operator (iii) the inverse of H is H itself, or equivalently, H 2 = I (iv) the adjoint operator H ∗ is H itself, which means

H[ f ], g =  f , H[g], for all f and g in L2 (Rm , dV(x)). Next, we introduce another Hardy space, namely H 2 (Rm ), which is defined as the closure in L2 (Rm , dV(x)) of the space of the nontangential bound2 m ary values for x0 → 0+ of all functions in H 2 (Rm+1 + ). Because H (R ) is a m closed subspace of the Hilbert space L2 (R , dV(x)), we obtain the following orthogonal decomposition:

L2 (Rm , dV(x)) = H 2 (Rm ) ⊕ H 2 (Rm )⊥ .

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85

Hence, there exist two projection operators, the so-called Szegö projections, denoted by

P+ : L2 (Rm , dV(x)) → H 2 (Rm )

and

P− : L2 (Rm , dV(x)) → H 2 (Rm )⊥ .

It is clear that these Szegö projections coincide with the Hardy projections or Cauchy transforms C + and −C − mentioned above:

P+ [ f ] = C + [ f ] =

1 1 ( f + H[ f ]) and P− [ f ] = −C − [ f ] = ( f − H[ f ]). 2 2

Moreover, since the Fourier transform of the Hilbert convolution kernel, considered as a tempered distribution, is given by

 F

2 Am+1

Pv



x |x|m+1

(ξ) = i

ξ |ξ|

,

the Fourier transform of the Hilbert transform is given by

F [H[ f ]](ξ) = iη F [ f ](ξ), where we have used spherical coordinates in frequency space given by

ξ = ρ η, ρ = |ξ| ∈ [0, +∞[, η ∈ Sm−1 . The orthogonal decomposition of an L2 (Rm , dV(x))-function f :

f = P+ [ f ] + P− [ f ] thus reads in frequency space

F[ f ] =

1 1 (1 + iη) F [ f ] + (1 − iη) F [ f ]. 2 2

Here the so-called Clifford–Heaviside functions

P+ =

1 1 + iη , 2

P− =

1 1 − iη , η ∈ Sm−1 , 2

appear; they were introduced independently by Sommen in 1982 and McIntosh (Li, McIntosh, and Qian, 1994, and McIntosh, 1996). The Clifford–Heaviside functions may be considered the higher-dimensional

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analogs of the Heaviside step-function on the real axis and are a typical feature of Clifford analysis; they are self-adjoint mutually orthogonal primitive idempotents satisfying:

P+ + P− = 1; (P± )† = P± ; P+ P− = P− P+ = 0; (P± )2 = P± . It thus follows that the Hardy components of a finite energy signal f ∈ L2 (Rm , dV(x)) may be computed as follows:

    P+ [ f ] = F −1 P+ F [ f ] ; P− [ f ] = F −1 P− F [ f ] , where F −1 denotes the inverse Fourier transform. It also follows that

  P∓ F P± [f ] = 0. If we call a function g (anti-)causal when P± g = 0, then we have shown that “the Fourier spectrum of the Hardy components of a finite energy signal is (anti-)causal.” This property of the Hardy components, together with their property of being the nontangential boundary values of a monogenic function in the upper, respectively, lower half space, allows us to call

P± [ f ] =

 1 f ± H[ f ] 2

an (anti-)analytic or (anti-)monogenic signal, a mathematically sound candidate for the multidimensional counterpart to the well-known notion of (anti-)analytic signal in 1D signal analysis. For further reading on the multidimensional Hilbert transform and Hardy spaces, see Bernstein and Lanzani (2002), Brackx et al., (2006d), Brackx and Van Acker (1992), Delanghe (2002a,b, 2004), Gilbert and Murray (1991), Li, McIntosh, and Qian (1994), McIntosh (1996), Mitrea (1994) and Murray (1985).

3. THE FRACTIONAL FOURIER TRANSFORM A multidimensional fractional Fourier transform is defined in the Clifford analysis context as an operator exponential. It coincides with the tensorial fractional Fourier transform. In this way we are able to prove Mehler’s formula for the generalized Clifford–Hermite polynomials (see Brackx et al., 2007).

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3.1. Introduction The fractional Fourier transform (abbreviated FrFT) may be considered a fractional power of the classical Fourier transform. It has been intensely studied during the past decade, an attention it may have gained partially because of the vivid interest in time-frequency analysis methods of signal processing (see, e.g., Bultheel and Martinez 2002; McBride and Kerr 1987; Namias 1980). In the 1D case, an integral representation for the FrFT can be obtained by means of Mehler’s formula for the classical Hermite polynomials (see Watson 1933). Here, we proceed in the reverse manner. First, we introduce a multidimensional FrFT in the framework of Clifford analysis using the generalized Clifford–Hermite polynomials introduced in Section 2.4. Then we show that this FrFT coincides with the classical tensorial FrFT in higher dimension. Thus, we are able to prove Mehler’s formula for the generalized Clifford–Hermite polynomials. In Section 3.2 we describe the classical FrFT. Next, we introduce the FrFT in the Clifford analysis setting and show that it can be written as an operator exponential. From this operator exponential form it becomes clear that our FrFT coincides with the classical tensorial higher-dimensional FrFT (Section 3.3). This allows us to derive the so-called Mehler formula for the generalized Clifford–Hermite polynomials (Section 3.4).

3.2. The Classical Fractional Fourier Transform The concept of fractional powers of the Fourier operator appears in the mathematical literature as early as 1929 (see Condon 1937; Kober 1939; Wiener 1929). It has been rediscovered in quantum mechanics, optics, and signal processing. The boom in publications started in the early years of the 1990s and it is still ongoing. A recent state of the art can be found in Ozaktas et al. (2001). The FrFT on the real line is first defined on a basis for the space of rapidly decreasing functions S(R). For this basis, one uses a complete orthonormal set of eigenfunctions of the Fourier transform given by

1 F [f ](ξ) = √ 2π



+∞

−∞

exp (−iξx) f (x) dx,

f ∈ S(R).

A possible choice for these eigenfunctions are the normalized Hermite functions:

2 21/4 exp − x2 Hn (x), φn (x) = √ 2n n!

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where

Hn (x) = (−1)n exp (x2 )

dn [exp (−x2 )] dxn

(17)

are the Hermite polynomials associated with the weight function exp (−x2 ). These eigenfunctions satisfy the orthonormality relation

< φn , φn > = δn,n with respect to the L2 -inner product

1 < f, g > = √ 2π



+∞ −∞

f (x) g(x) dx

and the eigenvalue equation

  F [φn ] = exp −in π2 φn .

(18)

  The eigenvalue for φn is thus given by λn = λn with λ = exp −i π2 representing a rotation over an angle π2 . It is precisely that concept of rotation that is generalized by the FrFT. Just as the classical Fourier transform corresponds to a rotation in the timefrequency plane over an angle π2 , the FrFT corresponds to a rotation over an arbitrary angle α = a π2 with a ∈ R. Consequently, the FrFT is defined by

  F a [φn ] = exp −ina π2 φn = λan φn = λna φn ,

(19)

  with λa = exp −ia π2 = exp (−iα). Thus, the classical Fourier transform corresponds to F 1 . Note also that for α = 0 or a = 0, we get the identity operator F 0 = I and for α = π or a = 2 we get the parity operator F 2 [ f ](ξ) = f (−ξ) . The FrFT can be written as an operator exponential F a = exp (−iαH), so that

  2 2 exp (−iαH) exp − x2 Hn (x) = exp (−inα) exp − x2 Hn (x).

Differentiating the above relation with respect to α, setting α = 0, and then using the differential equation

d d2 [Hn (x)] + 2n Hn (x) = 0, [Hn (x)] − 2x dx dx2 one can easily verify that the operator H is given by H = − 12



d2 dx2

 − x2 +1 .

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As the set of normalized Hermite functions φn constitutes an orthonormal basis for L2 (R, dx), each function f ∈ L2 (R, dx) can be expanded in terms of these eigenfunctions φn :

f =

∞ 

an φn ,

n=0

where the coefficients an are given by

1 an = √ 2π



+∞

−∞

φn (x) f (x) dx

1

= √ 2n n! π 2



+∞

−∞

2 Hn (x)exp − x2 f (x) dx.

(20)

Applying the FrFT on this function yields a

F [f] =

∞ 

  an exp −ina π2 φn .

(21)

n=0

The calculation of FrFTs by means of Eq. (21) is usually not practical. In order to obtain the integral representation of the operator F a , a formula provided by Mehler (1866), is used: ∞  1 exp (−inα) Hn (ξ) Hn (x) n 2 n! n=0

1 2xξ exp = 1 − exp (−2iα)

exp (−iα)−exp (−2iα) (ξ 2 +x2 ) 1−exp (−2iα)

.

Inserting an from Eq. (20) into Eq. (21) and using Mehler’s formula, one obtains 1 F a [ f ](ξ) = √  π 1 − exp (−2iα) 



+∞ −∞

exp

2xξ exp (−iα)−exp (−2iα) (ξ 2 +x2 ) 1−exp (−2iα)



2 2 f (x) dx. exp − ξ +x 2

(22)

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Note that for 0 < |α| < π, this expression can also be written as



  1 exp − 2i π2 αˆ − α exp 2i ξ 2 cot (α) 2π| sin (α)|  +∞

exp −i sinxξ(α) + 2i x2 cot (α) f (x) dx,

F a [ f ](ξ) = √

−∞

where αˆ = sgn(sin (α)). It was previously mentioned that F 0 [ f ](ξ) = f (ξ) and F ±π [ f ](ξ) = f (−ξ). Furthermore, when |α| > π, the definition is taken modulo 2π and reduced to the interval [−π, π]. The FrFT can be extended to higher dimension by taking tensor products. If Ka (ξ, x) denotes the kernel of the 1D FrFT, that is,



a

F [ f ](ξ) =

+∞

−∞

Ka (ξ, x) f (x) dx,

then one defines the m-dimensional FrFT as follows:

 F a1 ,..., am [ f ](ξ1 , . . . , ξm ) =

+∞ −∞

 ...

+∞

−∞

Ka1 ,..., am (ξ1 , . . . , ξm ; x1 , . . . , xm )

f (x1 , . . . , xm ) dV(x), where

Ka1 ,..., am (ξ1 , . . . , ξm ; x1 , . . . , xm ) = Ka1 (ξ1 , x1 ) . . . Kam (ξm , xm ).

3.3. Multidimensional Fractional Fourier Transform: Definition and Operator Exponential Form In Section 2.4, we showed that

  F [φs, k, j ](ξ) = exp −i(s + k) π2 φs, k, j (ξ). Following the definition on the real line [see Eq. (19)], we define the multidimensional FrFT in Clifford analysis by

  FCa [φs, k, j ](ξ) = exp −i(s + k)a π2 φs, k, j (ξ);   = exp −i(s + k)α φs, k, j (ξ) π with α = a . 2

a∈R

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We now will show that, similar to the classical case, the FrFT FCa can be written as an operator exponential. Proposition 3.1 The FrFT FCa can be written as an operator exponential

  FCa = exp (−iαHC ) = exp −ia π2 HC , where the operator HC is given by

HC =

1 2 1 (∂x − x2 − mI) = − (m − |x|2 + mI) 2 2

with I the identity operator. Proof. First, we note that the operator exponential exp (−iαHC ) is defined as the series ∞  Hn exp (−iαHC ) = (−iα)n C . n! n=0

Differentiating the relation







√ 2 (j) √ exp (−iαHC ) Hs, k ( 2x) Pk ( 2x) exp − |x|2

√   2 (j) √ = exp −i(s + k)α Hs, k ( 2x) Pk ( 2x) exp − |x|2 with respect to α, and setting α equal to zero, yields



 √ 2 (j) √ HC Hs, k ( 2x) Pk ( 2x) exp − |x|2

√ 2 (j) √ = (s + k) Hs, k ( 2x) Pk ( 2x) exp − |x|2 . Now we will verify that the operator HC is indeed given by

HC =

1 2 (∂ − x2 − mI). 2 x

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We have

 

√ 2 ( j) √ (∂x2 − x2 − mI) exp − |x|2 Hs, k ( 2x) Pk ( 2x)

√ 2 ( j) √ = −exp − |x|2 ∂x [x Hs, k ( 2x) Pk ( 2x)]

√ 2 ( j) √ − x exp − |x|2 ∂x [Hs, k ( 2x) Pk ( 2x)]

(23)

√ 2 ( j) √ + exp − |x|2 ∂x2 [Hs, k ( 2x) Pk ( 2x)]

√ 2 ( j) √ − m exp − |x|2 Hs, k ( 2x) Pk ( 2x). From formula (i) of Proposition 2.5, we readily obtain

√ ( j) √ ∂x [x Hs, k ( 2x) Pk ( 2x)] √ √ ( j) √ ( j) √ = as, k Hs, k ( 2x) Pk ( 2x) + x ∂x [Hs,k ( 2x) Pk ( 2x)]. Consequently, Eq. (23) becomes

(∂x2

 

√ ( j) √ |x|2 − x − mI) exp − 2 Hs, k ( 2x) Pk ( 2x) 2



 √ √ 2 ( j) √ ( j) √ = exp − |x|2 ∂x2 [Hs, k ( 2x) Pk ( 2x)] − 2x ∂x [Hs, k ( 2x) Pk ( 2x)]  √ ( j) √ − (as, k + m) Hs, k ( 2x) Pk ( 2x) . Furthermore, formula (iii) of Proposition 2.5 implies

√ √ ( j) √ ( j) √ ∂x2 [Hs, k ( 2x) Pk ( 2x)] − 2x ∂x [Hs, k ( 2x) Pk ( 2x)] √ ( j) √ = 2 Cs, k Hs, k ( 2x) Pk ( 2x),

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which finally leads to

 

√ 1 2 ( j) √ |x|2 2 (∂ − x − mI) exp − 2 Hs, k ( 2x) Pk ( 2x) 2 x

√  2 1 ( j) √ = 2Cs, k − as, k − m Hs, k ( 2x) Pk ( 2x) exp − |x|2 2

√ 2 ( j) √ = (s + k) Hs, k ( 2x) Pk ( 2x) exp − |x|2 , since for all s

2Cs, k − as, k − m = 2s + 2k. From Proposition 3.1 we observe that, surprisingly, our FrFT coincides with the classical tensorial higher-dimensional FrFT F a1 ,..., am with a1 = a2 = . . . = am = a.

3.4. The Mehler Formula for the Generalized Clifford–Hermite Polynomials A Clifford algebra–valued square integrable function f can be expanded in terms of the eigenfunctions {φs, k, j }: +

f (x) =

∞  ∞ dim(M   (k)) s=0 k=0

φs, k, j (x) as, k, j ,

j=1

where the Clifford algebra–valued coefficients as, k, j are given by

 as, k, j = < φs, k, j , f > =

Rm



†

φs, k, j (x)

f (x) dV(x).

(24)

By applying the operator FCa , we get +

FCa [ f ](ξ) =

∞  ∞ dim(M   (k)) s=0 k=0

j=1 +

=

∞ dim(M ∞    (k)) s=0 k=0

FCa [φs, k, j ](ξ) as, k, j   exp −i(s + k)α φs, k, j (ξ) as, k, j .

j=1

We thus have obtained the definition of the FrFT FCa in the form of a series.

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By replacing as, k, j in the series by their integral expression in Eq. (24) it is turned into +

FCa [ f ](ξ) =

∞  ∞ dim(M   (k)) s=0 k=0

  exp −i(s + k)α

j=1

 √ |ξ| ( j) √ × Hs, k ( 2ξ) Pk ( 2ξ) exp − 2  Rm

2m/4 (γs, k )1/2  2



√ † 2m/4  ( j) √ |x|2 ( 2x) P ( 2x) exp − H f (x) dV(x) s, k 2 k (γs, k )1/2 +    ∞ dim(M ∞   (k)) exp −i(s + k)α

 m/2

=2

√ ( j) Hs, k ( 2ξ) Pk

γs, k    √  √ † |x|2 +|ξ|2 ( j) √ exp − 2 f (x) dV(x). ( 2ξ) Hs, k ( 2x) Pk ( 2x) Rm

s=0 k=0

j=1

(25) Conversely, from the previous section we know that our FrFT FCa coincides with F a,..., a . Consequently, by means of Eq. (22) we have

 FCa [ f ](ξ) =

Rm

Ka (ξ1 , x1 ) . . . Ka (ξm , xm ) f (x) dV(x)

 =

1

m

√  π 1 − exp (−2iα)      2x1 ξ1 exp (−iα)−exp (−2iα)(ξ12 +x12 ) ξ12 +x12 exp − 2 × exp 1−exp (−2iα) Rm

2 +x2 ) 2xm ξm exp (−iα)−exp (−2iα)(ξm m . . . exp 1−exp (−2iα) 2 2 ξ +x exp − m 2 m f (x) dV(x)  m  2exp (−iα) 1 = √  exp 1−exp (−2iα) π 1 − exp (−2iα) Rm     (|x|2 +|ξ|2 ) exp (−2iα) |x|2 +|ξ|2 × exp − 1−exp (−2iα) exp − 2 f (x) dV(x). (26) Comparing Eqs. (25) and (26) yields the following result.

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Theorem 3.1 The Mehler formula for the generalized Clifford–Hermite polynomials takes the following form: ∞  ∞  exp (−i(s + k)α)

γs, k

s=0 k=0 + dim(M  (k))

 =

√ Hs,k ( 2ξ)

 √ †  ( j) √ †  ( j) √ Hs, k ( 2x) Pk ( 2ξ) Pk ( 2x)

j=1

1 1 √  2π 1 − exp (−2iα)

m

 exp

2exp (−iα)−(|x|2 +|ξ|2 ) exp (−2iα) 1−exp (−2iα)

 .

4. THE CLIFFORD–FOURIER TRANSFORM Recently several generalizations to higher dimension of the Fourier transform, using Clifford algebra, have been introduced, including our Clifford– Fourier transform, which we defined in Brackx, De Schepper, and Sommen (2005) as an operator exponential with a Clifford algebra–valued kernel. This section provides an overview of all these generalizations. Moreover, an in-depth study of our Clifford–Fourier transform is presented. Particular attention is paid to the 2D situation, since in this case we succeed in finding a closed form for the integral kernel of the Clifford–Fourier transform leading to further properties, in both the L1 and the L2 context (see Brackx, De Schepper, and Sommen, 2006b).

4.1. Introduction The Fourier transformation is, next to convolution, one of the two robust and frequently used tools for the analysis of scalar fields and image processing as computer vision. Quite naturally, attempts have been made to extend these methods to analyze 2D and 3D vector fields and even multivector fields. This introduction sketches an overview of these generalizations of the Fourier transform. Bülow and Sommer (2001) define a quaternionic Fourier transform of 2D signals f (x1 , x2 ) taking their values in the algebra H of real quaternions. Recall that the quaternion algebra H is nothing but the Clifford algebra R0,2 where, traditionally, the basis vectors are denoted by i and j, with i2 = j2 = −1, and the bivector by k = ij. In terms of these basis vectors, this quaternionic Fourier transform takes the form



q

F [ f ](u1 , u2 ) =

R2

exp (−2πiu1 x1 ) f (x1 , x2 ) exp (−2πju2 x2 ) dV(x).

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Due to the noncommutativity of the multiplication in H, the convolution theorem for this quaternionic Fourier transform takes a rather complicated form, which is moreover the case for its higher-dimensional analogue, the Clifford–Fourier transform given by

 cl

F [ f ](u) =

Rm

f (x) exp (−2πe1 u1 x1 ) . . . exp (−2πem um xm ) dV(x).

Note that for m = 1, this Clifford–Fourier transform reduces to the standard Fourier transform on the real line, whereas for m = 2, the quaternionic Fourier transform is reobtained when restricting to real signals. In addition Bülow and Sommer introduce a commutative hypercomplex Fourier transform given by

 F h [ f ](u) =

Rm

 ˜ f (x) exp −2π m u x e j=1 j j j dV(x),

where the basis vectors (˜e1 , . . . , e˜m ) obey the commutative multiplication rules e˜j e˜k = e˜k e˜j , j, k = 1, . . . , m, while still e˜j2 = −1, j = 1, . . . , m. This commutative hypercomplex Fourier transform offers the advantage that the corresponding convolution theorem has a rather simple outlook. The hypercomplex Fourier transforms F q , F cl , and F h enable Bülow and Sommer to establish a theory of multidimensional signal analysis, and in particular, to introduce the notions of multidimensional analytic signal, Gabor filter (see the next section), instantaneous and local amplitude and phase, and so on. Using the low-dimensional Clifford algebras R2,0 and R3,0 , Felsberg (2002) defines his Clifford–Fourier transform as

 F fe [ f ](u) =

exp (−2πI < u, x >) f (x) dV(x),

where I denotes the pseudoscalar e1 e2 in the case of 1D signals, or e1 e2 e3 in the case of 2D signals. It is used among others to introduce a concept of 2D analytic signal. Ebling and Scheuermann (2003, 2005) studied convolution and Clifford– Fourier transformation of 2D and 3D signals, using the respective Fourier kernels exp (−e1 e2 (ξ1 x1 + ξ2 x2 )) and exp (−e1 e2 e3 (ξ1 x1 + ξ2 x2 + ξ3 x3 )), where again e1 e2 and e1 e2 e3 are the pseudoscalars in the Clifford algebras R2,0 and R3,0 , respectively. Note that the latter Fourier kernel is also used in Mawardi and Hitzer (2006) to define a Clifford–Fourier transform of 3D signals. These Clifford–Fourier transforms and the corresponding convolution theorems allow Ebling and Scheuermann for, among others, the analysis of vector-valued patterns in the frequency domain.

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Note that the above-mentioned Clifford–Fourier kernel of Bülow and Sommer exp (−2πe1 u1 x1 ) . . . exp (−2πem um xm ) was already introduced by Sommen (1981) and Brackx, Delanghe, and Sommen (1982) as a theoretical concept in the framework of Clifford analysis. This generalized Fourier transform was further elaborated by Sommen (1982b, 1983) in connection with similar generalizations of the Cauchy, Hilbert, and Laplace transforms. In this context, the work of Li, McIntosh, and Qian (1994) should also be mentioned; they generalize the standard multidimensional Fourier transform of a function in Rm by extending the Fourier kernel exp (i < ξ, x >) to a function that is holomorphic in Cm and monogenic in Rm+1 . Recall that one of the most fundamental features of Clifford analysis is the factorization of the Laplace operator: m = −∂x2 . Whereas in general the square root of the Laplace operator is only a pseudodifferential operator, by √ embedding Euclidean space into a Clifford algebra, −m can be realized as the Dirac operator ∂x . In the same order of ideas, our first purpose is neither to replace nor to improve the classical multidimensional Fourier transform by a Clifford analysis alternative, but rather to refine it within the language of Clifford analysis, in much the same way as the notion of electron spin appears in the Pauli matrix formalism—it is what we call the Clifford–Fourier transform. The key step in its construction is to interpret the standard Fourier transform as an operator exponential (see Section 4.3): ∞    1 π k k −i F = exp −i π2 H = H , k! 2 k=0

where H is the scalar operator

H=

1 (−m + r2 − m). 2

By means of two commuting operators, O1 and O2 , which are introduced while studying (anti-) monogenic operators in the generalized Clifford– Hermite polynomial setting (see Section 4.2), H can be split into a sum of Clifford algebra–valued second-order operators containing the angular Dirac operator . This leads in a natural way to a pair of transforms FH± , the harmonic average of which is precisely the standard Fourier transform. Moreover, the 2D case of this Clifford–Fourier transform is special in that we succeed in finding a closed form for the kernel of the integral representation. This closed form enables us to generalize the well-known results for the standard Fourier transform both in the L1 and in the L2 context. Let us end this introductory section with an overview of the content of this chapter. We  start with  considering operators acting on a subspace M of the space L2 Rm , dV(x) of square integrable functions and, in particular,

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Clifford differential operators with polynomial coefficients. This subspace M is defined as the orthogonal  sum of spaces Ms, k of specific Clifford basis functions of L2 Rm , dV(x) (see subsection 4.2.1). In subsection 4.2.2, we show that these spaces Ms, k are simultaneous eigenspaces of two commuting operators, O1 and O2 . Every Clifford endomorphism of M can be decomposed into Clifford–Hermite monogenic operators. These Clifford– Hermite monogenic operators are characterized in terms of commutation relations involving O1 and O2 and they transform a space Ms, k into a similar space Ms , k (subsection 4.2.3). Hence, once the Clifford–Hermite monogenic decomposition of an operator is obtained, its action on the space M is known. Furthermore, in subsection 4.2.4, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is studied in detail. In Section 4.3 we then recall two alternative approaches to the classical Fourier transform. Next, we split the scalar-valued kernel operator H by means of the commuting operators, O1 and O2 , which leads to a new Fourier transform in the Clifford analysis setting (subsection 4.4.1). The eigenfunctions of this Clifford–Fourier transform are computed and its relation with the standard Fourier transform is established. Furthermore, we develop an adequate operational calculus in subsection 4.4.2. Next, we thoroughly study the Clifford–Fourier transform in the specific case of two dimensions. We start with the computation of its integral kernel (subsection 4.5.1). In subsection 4.5.2, we examine the 2D Clifford–Fourier transform as a linear operator in, respectively, the space of integrable functions, the space of rapidly decreasing functions and the space of square integrable functions. Furthermore, in subsection 4.5.3, we give an explicit connection between the 2D Clifford–Fourier transform and the standard tensorial Fourier transform, and a surprising connection with the Clifford–Fourier transform of Ebling and Scheuermann (2005). We end this section by calculating, as an example, the 2D Clifford–Fourier transform of the box function.

4.2. Clifford–Hermite Monogenic Operators In this section we study Clifford–Hermite monogenic operators. To introduce this notion, we first explain the already existing notion of monogenic operator in the polynomial framework.

4.2.1. Monogenic Operators Every homogeneous Clifford polynomial Rk of degree k admits a canonical decomposition of the form

Rk (x) =

k  s=0

xs Pk−s (x),

Pk−s ∈ M+ (k − s).

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This monogenic decomposition also yields a monogenic decomposition of the space P of Clifford polynomials

P=

∞  ∞ 

⊕⊥ Ms, k ,

s=0 k=0

where

Ms, k = {xs Pk (x); Pk ∈ M+ (k)}. Similar to this polynomial setting, the decomposition of Eq. (3) of a Clifford polynomial operator into homogeneous ones can be further refined to a decomposition into monogenic operators A± λ, κ :

A=

 − (A+ λ, κ + Aλ, κ ).

(27)

λ, κ

These monogenic operators A± λ, κ transform each space Ms, k into a similar space Ms , k . Hence, once the monogenic decomposition in Eq. (27) of a Clifford polynomial operator is obtained, its action on the space P of Clifford polynomials is known. As the spaces Ms, k are the simultaneous eigenspaces of the operators E and , the monogenic operators are characterized in terms of commutation relations involving E and . Sommen and Van Acker (1992) studied the monogenic decomposition of differential operators acting on Clifford polynomials in detail. In what follows we study the action of operators on the space M given by the algebraic orthogonal sum

M=

∞  ∞ 

⊕⊥ Ms, k .

s=0 k=0

Each f ∈ M can be decomposed into a finite sum: f =   function +  f , where f s, k s, k ∈ Ms, k = span{ψs, k, j (x); j = 1, 2, . . . , dim (M (k))} s k with

√ 2 ( j) √ ψs, k, j (x) = exp − |x|2 Hs, k ( 2x) Pk ( 2x),

s, k ∈ N ∪ {0}, j = 1, 2, . . . , dim(M+ (k)). From Section 2.4, we know that the set



  ψs, k, j (x); s, k ∈ N ∪ {0}, j = 1, 2, . . . , dim M+ (k)

(28)

  constitutes an orthogonal basis for the space L2 Rm , dV(x) of square integrable functions. Hence, this space is precisely the closure of M: M =   L2 Rm , dV(x) .

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As in the polynomial framework, every Clifford endomorphism of M can be decomposed into Clifford–Hermite monogenic (CH-monogenic) operators. These CH-monogenic operators transform a space Ms, k into another such space Ms , k . As the spaces Ms, k are simultaneous eigenspaces of the operators

O1 =

1 (∂x − x)(∂x + x) 2

and

O2 =

1 (∂x + x)(∂x − x), 2

our CH-monogenic operators are characterized in terms of commutation relations involving O1 and O2 .

4.2.2. The Operators O1 and O2 In this subsection we consider the operators O1 and O2 introduced above. They will play a crucial role not only in our study of CH-(anti-)monogenic operators, but also in defining a new Clifford–Fourier transform (see Section 4.4). They satisfy the following properties. Proposition 4.1 One has (i)

O1 =

1 2 m (∂x − x2 ) +  − 2 2

O2 =

1 2 m (∂x − x2 ) −  − 2 2

(ii)

(iii)

O1 + O2 = ∂x2 − x2 (iv)

m O1 − O 2 = 2  − 2

(v) O1 and O2 are commuting operators (vi)

O1 [ψs, k, j (x)] = Cs, k ψs, k, j (x) (vii)

O2 [ψs, k, j (x)] = Cs+1, k ψs, k, j (x).

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Proof. (i)(ii) Taking into account that the angular Dirac operator  may be written as

1  = − (x ∂x − ∂x x − m), 2 the results follow from a straightforward computation. (iii)(iv) Trivial. (v) As  commutes with the Laplace operator m and with the multiplication operator r, we have that



   1 1 2 2 2 (∂ − x ),  = (−m + r ),  = 0 2 x 2

which, in view of (i) and (ii) yields [O1 , O2 ] = 0. (vi)(vii) First, we have that

√ 2 ( j) √ (∂x − x)[ψs, k, j (x)] = exp − |x|2 (∂x − 2x)[Hs, k ( 2x) Pk ( 2x)]. (29) Formula (9) implies that

√ √ √ ( j) √ ( j) √ (2x − ∂x )[Hs,k ( 2x) Pk ( 2x)] = 2 Hs+1,k ( 2x) Pk ( 2x). Consequently, Eq. (29) becomes

√ (∂x − x)[ψs,k,j (x)] = − 2 ψs+1,k,j (x).

(30)

Next, it is immediately verified that



√ 2 ( j) √ (∂x + x)[ψs,k,j (x)] = exp − |x|2 ∂x [Hs,k ( 2x) Pk ( 2x)]. Moreover, from formula (ii) of Proposition 2.5, we readily obtain that

√ √ √ ( j) √ ( j) √ ∂x [Hs,k ( 2x) Pk ( 2x)] = − 2 Cs,k Hs−1,k ( 2x) Pk ( 2x). Hence, we find that

√ (∂x + x)[ψs,k,j (x)] = − 2 Cs,k ψs−1,k,j (x).

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By combining the results of Eqs. (30) and (31), the basis functions ψs,k,j are found to be eigenfunctions of O1 and O2 . Remark 4.1 Note that (∂x − x) increases the degree of the generalized Clifford– Hermite polynomial, so that it may be qualified as a creation operator. In the same order of ideas, (∂x + x) is an annihilation operator.

4.2.3. CH-Monogenic Operators: Definition and Properties As mentioned in subsection 4.2.1, the CH-monogenic operators we now introduce will transform elements of a space Ms,k into elements of a space Ms ,k for some s and k  . As the spaces Ms,k are simultaneous eigenspaces of the operators O1 and O2 , the CH-monogenicity property is expressed in terms of commutation relations involving these operators. Definition 4.1 (i) A Clifford endomorphism A of M is called CH-monogenic of degree (λ, κ), + notation: A ∈ χλ,κ , if

[O1 , A] = λA

and

[O2 , A] = κA.

(ii) A Clifford endomorphism B of M is called CH-anti-monogenic of degree − (λ, κ), notation: B ∈ χλ,κ , if

O1 B = BO2 + λB

and

O2 B = BO1 + κB.

Remark 4.2 (i) As the set

√   ( j) √ Hs, k ( 2x) Pk ( 2x); s, k ∈ N, j = 1, 2, . . . , dim M+ (k)



constitutes a basis for the space of Clifford polynomials and as D(Cm ) ⊂ End(P s ) ⊗ Cm , Clifford differential operators with polynomial coefficients belong to the set of endomorphisms of M. (ii) The operators O1 and O2 themselves are CH-monogenic of degree (0, 0). (iii) A Clifford endomorphism A of M is CH-monogenic of degree (λ, κ) if and only if



 1 λ+κ 2 (−m + r ), A = A 2 2

 and

 m λ−κ A.  − ,A = 2 2

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(iv) A Clifford endomorphism B of M is CH-anti-monogenic of degree (λ, κ) if and only if



 1 λ+κ 2 (−m + r ), B = B 2 2

and

 m λ−κ  − ,B = B 2 2

with {A, B} = AB + BA the anti-commutator of two operators. CH-monogenic and CH-anti-monogenic operators are closed under composition as shown in the next proposition. Proposition 4.2 + + + (i) If A ∈ χλ,κ and B ∈ χλ+ ,κ then AB ∈ χλ+λ  ,κ+κ and BA ∈ χλ+λ ,κ+κ . − + + (ii) If A ∈ χλ,κ and B ∈ χλ− ,κ then AB ∈ χλ+κ  ,λ +κ and BA ∈ χλ +κ,λ+κ . + − − (iii) If A ∈ χλ,κ and B ∈ χλ ,κ then AB ∈ χλ+λ ,κ+κ and BA ∈ χλ− +κ,λ+κ .

Proof. The proof of (ii) is as follows:

O1 AB = (AO2 + λA)B = A(BO1 + κ B) + λAB = ABO1 + (λ + κ )AB and

O2 AB = (AO1 + κA)B = A(BO2 + λ B) + κAB = ABO2 + (λ + κ)AB. Similarly, we find for the operator BA:

O1 BA = BAO1 + (λ + κ)BA

and

O2 BA = BAO2 + (λ + κ )BA.

The proofs of (i) and (iii) are similar. We now prove that the CH-monogenic operators indeed transform Ms,k into an Ms ,k . + Proposition 4.3 Let A ∈ χλ,κ and f ∈ Ms, k . Then A[ f ] ∈ Ms ,k for some   s and k depending on s, k, λ, κ, and m.

Proof. We start by observing that A[ f ] is a simultaneous eigenfunction of O1 and O2

O1 [A[ f ]] = (AO1 + λA)[ f ] = (Cs, k + λ) A[ f ] O2 [A[ f ]] = (AO2 + κA)[ f ] = (Cs+1,k + κ) A[ f ].

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As A[ f ] belongs to the space M, it can be written as

A[ f ] =

∞ ∞   



dim(M+ (t))



j=0 t=0

ψj,t,i (x) aj,t,i ;

aj,t,i ∈ Cm .

i=1

Hence, we also have ∞  ∞  

O1 [A[ f ]] =



dim(M+ (t))



j=0 t=0

Cj,t ψj,t,i (x) aj,t,i .

i=1

Comparing the above expression with

O1 [A[ f ]] =

∞ ∞   



dim(M+ (t))



j=0 t=0

(Cs,k + λ) ψj,t,i (x) aj,t,i ,

i=1

we obtain that either aj,t,i = 0 or Cj,t = Cs,k + λ. Similarly, comparing ∞  ∞  

O2 [A[ f ]] =



dim(M+ (t))



j=0 t=0

Cj+1,t ψj,t,i (x) aj,t,i

i=1

with

O2 [A[ f ]] =

∞  ∞  

j=0 t=0



dim(M+ (t))



(Cs+1,k + κ) ψj,t,i (x) aj,t,i ,

i=1

yields that either aj,t,i = 0 or Cj+1,t = Cs+1,k + κ. Consequently, we must prove that at most one pair of indices ( j, t) satisfies the set of equations



Cj,t = Cs,k + λ Cj+1,t = Cs+1,k + κ.

(32)

To that end, we must distinguish several cases.

CASE A If s is even, the set of equations (32) becomes

Cj,t = s + λ Cj+1,t = s + m + 2k + κ.

(33)

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CASE A.1: m odd, λ odd. From the first equation, we obtain that j must be odd. For j odd, the set of equations in (33) becomes

j − 1 + m + 2t = s + λ j + 1 = s + m + 2k + κ,

leading to

j = s + m + 2k + κ − 1

and

2t = λ + 2 − 2m − 2k − κ.

In this case, the second equation implies that κ must be odd. As t must be positive, we thus have

A : Ms,k →

0 Ms+m+2k+κ−1,(λ+2−2m−2k−κ)/2

for 2k > λ + 2 − 2m − κ for 2k ≤ λ + 2 − 2m − κ.

CASE A.2: m odd, λ even. Now the first equation of (33) implies that j must be even. For j even, the set of equations in (33) becomes

j =s+λ j + m + 2t = s + m + 2k + κ,

which implies

j =s+λ

and

2t = 2k + κ − λ.

Now κ must be even and we have

A : Ms,k →

0 Ms+λ,(2k+κ−λ)/2

for 2k < λ − κ for 2k ≥ λ − κ.

CASE A.3: m even, λ odd. In this case, s + λ is odd. As Cj,t is always even, we have A : Ms,k → 0. CASE A.4: m even, λ even. As to the first equation of (33), both j even and j odd are possible. Hence, we need to make a distinction between κ even and κ odd. In the case where κ is even, both j even and j odd are possible for the second equation.

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For j even, the set of equations in (33) becomes



j =s+λ j + m + 2t = s + m + 2k + κ,

and hence,

j =s+λ

and

2t = 2k + κ − λ.

For j odd, we have



j − 1 + m + 2t = s + λ

j + 1 = s + m + 2k + κ, thus,

j = s + m + 2k + κ − 1

and

2t = λ − 2m + 2 − 2k − κ.

As t must be positive, we have that for j even: 2k ≥ λ − κ; while for j odd: 2k ≤ λ − κ − (2m − 2) < λ − κ. This implies

⎧ ⎪ Ms+λ,(2k+κ−λ)/2 for 2k ≥ λ − κ ⎪ ⎪ ⎨0 for λ − κ − (2m − 2) < A : Ms,k → ⎪ 2k < λ − κ ⎪ ⎪ ⎩ Ms+m+2k+κ−1,(λ−2m+2−2k−κ)/2 for 2k ≤ λ − κ − (2m − 2). In the case where κ is odd, the second equation of (33) implies that neither j even, nor j odd is possible. Hence, we have A : Ms,k → 0.

CASE B The case where s is odd is treated in a similar way. In a completely analogous manner, we obtain the following result for the CH-anti-monogenic operators. − Proposition 4.4 Let B ∈ χλ,κ and f ∈ Ms,k . Then B[ f ] ∈ Ms ,k for some s and k  depending on s, k, λ, κ, and m.

Now we are able to prove that every Clifford endomorphism of M admits a decomposition into CH-(anti-)monogenic operators, which we call the Clifford–Hermite monogenic decomposition (CHM decomposition).

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Theorem 4.1 Every Clifford endomorphism of M admits a CHM decomposition. Proof. It is clear that anarbitrary ∞ Clifford endomorphism A of M can be written as follows: A = ∞ s=0 k=0 As,k , where, by definition,

As,k |Ms,k = A|Ms,k

whenever (s, k)  = (s , k  ).

and As,k |Ms ,k = 0

As A is an endomorphism of M, we also have As,k [Ms,k ] = A[Ms,k ] ∈ M and hence,

As,k [Ms,k ] =

∞ ∞  

Ms ,k, .

s =0 k  =0

This implies that every As,k , in its turn, can be decomposed as

As,k =

∞ ∞   s =0 k  =0





Ass,k,k ,

with 







Ass,k,k : Ms,k → Ms ,k , and Ass,k,k |Ms ,k = 0 whenever (s, k)  = (s , k  ). Consequently, for the endomorphism A, we obtain A = 





s,k



s ,k 





Ass,k,k .

It is easily seen that every operator Ass,k,k is both CH-monogenic and CHanti-monogenic. For example, if s and s are even, we have 



Ass,k,k ∈ χs+ −s,s −s+2(k −k)





and Ass,k,k ∈ χs− −s−m−2k,s −s+m+2k .

Collecting the CH-monogenic and CH-anti-monogenic operators of the same degree yields a decomposition of A of the form

A=

 − (A+ λ,κ + Aλ,κ ) λ,κ

with

± A± λ,κ ∈ χλ,κ .

4.2.4. CHM Decomposition of Clifford Differential Operators With Polynomial Coefficients In this subsection, we study the CHM decomposition of Clifford differential operators with polynomial coefficients. It is sufficient to search for the CHM decomposition of the basic operators {ej , xj , ∂xj ; j = 1, 2, . . . , m}, since they are generating the algebra D(Cm ).

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The operators f → ej f , j = 1, 2, . . . , m For the special case where m = 2, we have that



 1 2 2 (∂ − x ), ej = 0 2 x

and

ej = −ej .

Consequently, we find



O1 ej =

   1 2 1 2 m m 2 2 (∂ − x ) +  − ej = ej (∂x − x ) −  + − mej 2 x 2 2 2 = ej O2 − mej

and similarly

O2 ej = ej O1 + mej . − Hence, for m = 2, ej ∈ χ−2,2 . For the general case where m > 2, we first introduce the operators

  τj = − ej + ej ( − m + 2) and δj = [, ej ] = ej − ej ,

for which we prove the following lemma. + − and δj ∈ χ−2,2 . Lemma 4.1 One has τj ∈ χ0,0

Proof. Naturally we have that



 1 2 2 (∂ − x ), τj = 0. 2 x

Furthermore, as the Laplace–Beltrami operator ∗ω = (m − 2 − ) is a scalar operator, we find

[, τj ] = −2 ej − ej ( − m + 2) + ej  + ej ( − m + 2) = −( − m + 2)ej + ej ( − m + 2) = [∗ω , ej ] = 0. This implies

O1 τj = τj O1

and O2 τj = τj O2 .

The operator δj satisfies



 1 2 2 (∂ − x ), δj = 0 2 x

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and

δj = 2 ej − ej  = (m − 2)ej − ∗ω ej − ej  = (m − 2)ej − ej (m − 2 − ) − ej  = (m − 2)(ej − ej ) − (ej − ej ) = (m − 2)δj − δj . Hence, we obtain

 O1 δj =

   1 1 2 m m (∂x − x2 ) +  − δj = δj (∂x2 − x2 ) −  + − 2δj 2 2 2 2 = δj O2 − 2δj ,

and similarly

 O2 δj =

   1 2 1 2 m m 2 2 (∂ − x ) −  + δj = δj (∂x − x ) +  − + 2δj 2 x 2 2 2 = δj O1 + 2δj .

Next we have that τj + δj = ej (m − 2 − 2). The operator (m − 2 − 2) is invertible in the set of endomorphisms of M, since it has eigenvalues 2k + m − 2 (for s even) and −(m + 2k) (for s odd) which, for m > 2, are never zero. Consequently, we can write

ej = (τj + δj )(m − 2 − 2)−1 := E0j + E1j , where

  E0j = τj (m − 2 − 2)−1 = − ej + ej ( − m + 2) (m − 2 − 2)−1 and

E1j = δj (m − 2 − 2)−1 = (ej − ej )(m − 2 − 2)−1 . Naturally, the operator (m − 2 − 2) is CH-monogenic of degree (0, 0) and + . hence (m − 2 − 2)−1 ∈ χ0,0

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+ − In view of Proposition 4.2, we obtain that E0j ∈ χ0,0 and E1j ∈ χ−2,2 . Summarizing, the CHM decomposition of ej takes the form

+ − ej = E0j + E1j with E0j ∈ χ0,0 and E1j ∈ χ−2,2 .

Finally, with regard to the action on the spaces Ms,k , we have by means of Propositions 4.3 and 4.4 that

E0j : Ms,k → Ms,k and

E1j : Ms,k

⎧ ⎪ ⎨Ms+1,k−1 → 0 ⎪ ⎩M s−1,k+1

for s even and k ≥ 1 for s even and k = 0 for s odd.

The operators f → x f and f → ∂x f The operators x and ∂x satisfy (see, e.g., Van Acker, 1991)

x = x(m − 1 − ) and ∂x = ∂x (m − 1 − ).

(34)

This result will be combined with the following lemma. Lemma 4.2 One has



Proof.



 1 2 (∂x − x2 ), x = −∂x 2

 and

 1 2 (∂x − x2 ), ∂x = −x. 2

We obtain consecutively

 m 1 



1 2 1 (∂x − x2 ), x = (∂x2 x − x ∂x2 ) = − 2 2 2 =− =−

1 2

k,j

k=1

⎞ ⎛ m  1 ∂x2k ⎝ xj ej ⎠ − x ∂x2 2 j=1

1 ∂xk (δk,j ej + xj ej ∂xk ) − x ∂x2 2

1 1 (2δk,j ej ∂xk + xj ej ∂x2k ) − x ∂x2 2 2 k,j

1 1 = −∂x + x ∂x2 − x ∂x2 = −∂x 2 2

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and 

⎞ ⎛   m m  1 2 1 ⎝ 1 2 1 2 2 2 2 ej ∂xj ⎠ xk (∂ − x ), ∂x = − (x ∂x − ∂x x ) = − x ∂x − 2 x 2 2 2 j=1

1

1 = − x 2 ∂x − 2 2

k=1

ej (2xk δk,j + xk2 ∂xj )

j,k

1 1 = − x2 ∂x − x + x2 ∂x = −x. 2 2 In view of the above, we now have

 O1 x =

   1 2 1 2 m m 2 2 (∂ − x ) +  − x = x (∂x − x ) −  + − x − ∂x 2 x 2 2 2 = xO2 − x − ∂x

and

 O2 x =

   1 1 2 m m (∂x − x2 ) −  + x = x (∂x2 − x2 ) +  − + x − ∂x 2 2 2 2 = xO1 + x − ∂x ,

whereas for the operator ∂x , we obtain

O1 ∂x = ∂x O2 − ∂x − x

and

O2 ∂x = ∂x O1 + ∂x − x.

Hence, x and ∂x are neither CH-monogenic nor CH-anti-monogenic. However, we do have the following result: − − and x − ∂x ∈ χ0,2 . Lemma 4.3 One has x + ∂x ∈ χ−2,0

Proof.

Straightforward.

We now readily obtain the CHM decomposition of x and ∂x : x = X 0 + X 1 with

X0 =

1 − (x − ∂x ) ∈ χ0,2 , 2

X1 =

1 − (x + ∂x ) ∈ χ−2,0 2

D1 =

1 − (x + ∂x ) ∈ χ−2,0 . 2

and ∂x = D0 + D1 with

1 − D0 = − (x − ∂x ) ∈ χ0,2 , 2

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The operators f → xj f , j = 1, 2, . . . , m. Once the CHM decomposition of ej , j = 1, 2, . . . , m and x is obtained, the CHM decomposition of xj easily follows from

  1 1 0 1 0 1 0 1 0 1 xj = − (xej + ej x) = − (X + X )(Ej + Ej ) + (Ej + Ej )(X + X ) 2 2 = xj0 + xj1 + xj2 + xj3 + xj4 + xj5 ,

with

1 + xj0 = − X 0 E1j ∈ χ2,0 2 1 + xj1 = − E1j X 0 ∈ χ0,2 2 1 + xj2 = − E1j X 1 ∈ χ−2,0 2 1 + xj3 = − X 1 E1j ∈ χ0,−2 2 1 − xj4 = − (X 0 E0j + E0j X 0 ) ∈ χ0,2 2 1 − xj5 = − (X 1 E0j + E0j X 1 ) ∈ χ−2,0 . 2 Here we have used the composition rules for CH-monogenic and CH-antimonogenic operators derived in Proposition 4.2. Finally, for the action on the spaces Ms,k we have

xj0 : Ms,k

xj1 : Ms,k

xj2

: Ms,k

⎧ ⎪ ⎨Ms+2,k−1 → 0 ⎪ ⎩M s,k+1

for s even and k ≥ 1 for s even and k = 0 for s odd.

⎧ ⎪ ⎨Ms,k+1 → Ms+2,k−1 ⎪ ⎩0

for s even for s odd and k ≥ 1 for s odd and k = 0

⎧ ⎪ ⎨Ms−2,k+1 → Ms,k−1 ⎪ ⎩0

for s even for s odd and k ≥ 1 for s odd and k = 0

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xj3 : Ms,k

⎧ ⎪ ⎨Ms,k−1 → 0 ⎪ ⎩M s−2,k+1

113

for s even and k ≥ 1 for s even and k = 0 for s odd.

xj4 : Ms,k → Ms+1,k xj5 : Ms,k → Ms−1,k . The operators f → ∂xj f , j = 1, 2, . . . , m. By means of

1 ∂xj = − (∂x ej + ej ∂x ), 2 the CHM decomposition of ∂xj follows at once from the CHM decomposition of ej and ∂x . The results are completely similar to these for the operators considered in the previous subsection.

4.3. Alternative Representations of the Classical Fourier Transform The idea behind the definition of our Clifford–Fourier transform originates from the operator exponential representation of the classical Fourier transform

F [ f ] = exp



∞  1 π n n −i [f] = H [ f ], n! 2



−i π2 H

(35)

n=0

with H the scalar-valued differential operator given by

1 2 1 (∂x − x2 − m) = (−m + r2 − m). 2 2   Note that the operators H and exp −i π2 H are Fourier invariant; that is, H=

F [H[ f ]] = H[F [ f ]]

      and F exp −i π2 H [ f ] = exp −i π2 H [F [ f ]].

The equivalence of this operator exponential form in Eq. (35) with the traditional integral form in Eq. (6) may be proved rather simply in the framework of Clifford analysis. To that  end, we again use the orthogonal basis of Eq. (28) of the space L2 Rm , dV(x) . The basis functions ψs,k,j satisfy the orthogonality

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relation

< ψs, k1 , j1, ψt, k2 , j2 > =

γs, k1 δs, t δk1 , k2 δj1 , j2 . 2m/2

(36)

Hence, a square integrable function f can be expanded as follows 

f (x) =

+



M (k) ∞  ∞ dim  s=0 k=0

ψs,k,j (x) bs,k,j .

(37)

j=1

The orthogonality relation in Eq. (36) implies that the Clifford algebra– valued coefficients bs,k,j are given by the integrals

bs,k,j =

2m/2 γs,k



 Rm

†

ψs,k,j (x)

f (x) dV(x).

(38)

  In the foregoing, we have shown that these L2 Rm , dV(x) -basis functions ψs,k,j are simultaneous eigenfunctions of the Fourier transform operator F in integral form and of the kernel operator H. We thus have at the same time (see Proposition 2.7)

 1 exp (−i < x , ξ >) ψs,k,j (x) dV(x) (2π)m/2 Rm   = exp −i(s + k) π2 ψs,k,j (ξ)

F [ψs,k,j ](ξ) =

and (see the proof of Proposition 3.1)

H[ψs,k,j (x)] = (s + k) ψs,k,j (x). It then follows that ∞    1 π n n −i exp −i π2 H [ψs,k,j ] = H [ψs,k,j ] n! 2 n=0

∞  1 π n = −i (s + k)n ψs,k,j n! 2 n=0   = exp −i π2 (s + k) ψs,k,j = F [ψs,k,j ],

  which immediately gives rise to the desired equivalence in L2 Rm , dV(x) .

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  Moreover, if the function f ∈ L2 Rm , dV(x) is developed in terms of the basis functions ψs,k,j according to Eq. (37), then its Fourier transform takes the series expansion form +

F [ f ](ξ) =

∞ dim(M ∞    (k)) s=0 k=0

  exp −i(s + k) π2 ψs,k,j (ξ) bs,k,j .

j=1

4.4. Clifford–Fourier Transform: Definition and Properties Note that due to the scalar character of the standard Fourier kernel, the Fourier spectrum inherits its Clifford algebra character from the original signal with no interaction with the Fourier kernel. So in order to genuinely introduce the Clifford analysis character in the Fourier transform, it occurred to us to replace in the operator exponential in Eq. (35) the scalarvalued operator H with a Clifford algebra–valued one. To that end, we aim at factorizing the operator H, making use of the factorization of the Laplace operator by the Dirac operator. This leads us to again consider the commuting operators (see Subsection 4.2.2)

O1 =

1 (∂x − x)(∂x + x) 2

and

O2 =

1 (∂x + x)(∂x − x), 2

which turn out to be crucial in our approach.

4.4.1. Definition In view of Proposition 4.1 (vi) and (vii), we define the Clifford–Fourier transform as the pair of transformations

  FH+ = exp −i π2 H+

and

  FH− = exp −i π2 H−

with the operators H+ and H− closely linked to the operators O1 and O2 . As we want the classical Fourier transform to be the harmonic average of the Clifford–Fourier transform pair {FH+ , FH− }; that is, F 2 = FH+ FH− with F 2 the parity operator: F 2 [ f ](x) = f (−x), the operators H+ and H− must satisfy

H+ + H− = 2 H = ∂x2 − x2 − m = O1 + O2 − m. This inspires the following definition of H+ and H− .

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Definition 4.2 One puts

H+ = O1

H− = O2 − m.

and

Note that the operators H+ and H− contain a scalar part and a bivector part. The following properties are easily proved. Proposition 4.5 One has (i)

H± = H ± . (ii) H+ and H− are Fourier invariant operators. (iii) ± H± [ψs,k,j (x)] = Cs,k ψs,k,j (x) + with Cs,k := Cs,k and

− Cs,k



:= Cs+1,k − m =

s + 2k s+1−m

for s even for s odd.

Proof. (i)(iii) Trivial. (ii) This property follows directly from the Fourier invariance of the operators H and . Corollary 4.1 The basis functions ψs,k,j are eigenfunctions of the Clifford– Fourier transform:



± FH± [ψs,k,j ](ξ) = exp −i π2 Cs,k ψs,k,j (ξ).    Now if f ∈ L2 Rm , dV(x) is expanded w.r.t. the basis ψs,k,j (x); s, k ∈   N ∪ {0}, j = 1, . . . , dim M+ (k) , the eigenvalue equation of Corollary 4.1 immediately yields the series representation of the Clifford–Fourier transform as follows: +

FH± [ f ](ξ) =

∞  ∞ dim(M   (k)) s=0 k=0

± ψs,k,j (ξ) bs,k,j , exp −i π2 Cs,k

j=1

the coefficients bs,k,j being given by Eq. (38).

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 Moreover, as the orthogonal L2 -basis ψs,k,j (x); s, k ∈ N ∪ {0}, j =  +  1, . . . , dim M (k) consists of eigenfunctions of both the operators H and , one can easily verify the following properties. Proposition 4.6 + − (i) The operators  m H, , O1 , O2 , H and H are self-adjoint; that is, for all f , g ∈ L2 R , dV(x) and T any of the mentioned operators, one has

< T[ f ], g >=< f , T[ g] > .   (ii) The operators H, O1 , and O2 are nonnegative—for each f ∈ L2 Rm , dV(x) and T any of the mentioned operators, one has

[< T[ f ], f >]0 ≥ 0. Next, by means of Proposition 4.5 (i), we obtain in terms of operator exponentials

        FH± = exp −i π2 (H ± ) = exp ∓i π2  exp −i π2 H = exp ∓i π2  F .

(39) This establishes the relationship between the classical Fourier transform and the newly introduced Clifford–Fourier transform. Note that the commuting property of the operators H and  has been used, so that indeed           exp −i π2 (H ± ) = exp −i π2 H exp ∓i π2  = exp ∓i π2  exp −i π2 H . Thus, the Clifford–Fourier transform is obtained as the composition of the classical Fourier transform with the operator exponential ∞    1 π k k ∓i exp ∓i π2  =  . k! 2 k=0

As an immediate consequence, we obtain an integral representation for the Clifford–Fourier transform as follows:

1 FH± [ f ](ξ) = (2π)m/2



Rm

exp ∓i π2 ξ [ exp (−i < x , ξ >)] f (x) dV(x).

Introducing the square root of the Clifford–Fourier transforms, in the sense of the FrFT (see Namias, 1980, and Ozaktas, Zalevsky, and Kutay, 2001), by

   FH± = exp −i π4 H± ,

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we also obtain that

       FH+ FH− = FH− FH+ = exp −i π4 H+ + H−   = exp −i π2 H , leading to the factorization of the standard Fourier transform:

F=

    FH+ FH− = FH− FH+ .

Note that each operator that is anti-invariant under the classical Fourier transform and commutes with the angular Dirac operator , is also antiinvariant under the Clifford–Fourier transform. For example, the operators m m  − and E + are, respectively, invariant and anti-invariant under the 2 2 classical Fourier transform. Because they both commute with the angular Dirac operator , they show the anti-invariance property w.r.t the Clifford– Fourier transform. For the inversion of the Clifford–Fourier transform, it suffices to observe that

    (FH± )−1 = exp i π2 H± = exp ±i π2  F −1 .   Finally, using the notation TT = exp −i π2 T , we can draw the following diagram: (see next page)

4.4.2. Operational Calculus As is the case for the classical Fourier transform, an operational calculus may be based on the Clifford–Fourier transform. The operational formulas are derived from Eq. (39) expressing the Clifford–Fourier transform in terms of the classical Fourier transform F . Proposition 4.7 The Clifford–Fourier transform satisfies the following: (i) the linearity property

FH± [ f λ + gμ] = FH± [ f ] λ + FH± [ g]μ

for λ, μ ∈ Cm ,

(ii) the change of scale property

  ξ 1 FH± [ f (ax)](ξ) = m FH± [ f (x)] a a

for a ∈ R+ ,

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FH [f ] 1/2

F H

F F

FH

1/2 H [f ]

T

T 1/2

1/2 H

F H

F [f ]

f

F2 [f ]

F

F 1/2

1/2

F H

F H

F

1/2 H [f ]

T

F

T

FH

1/2

H

FH [f ]

Decomposition of the Fourier transform in term of the Clifford–Fourier transforms.

(iii) the multiplication rule

FH± [x f (x)](ξ) = ∓ (∓i)m ∂ξ FH∓ [ f (x)](ξ) and more generally

FH± [x2n f (x)](ξ) = (−1)n ∂ξ2n FH± [ f (x)](ξ) FH± [x2n+1 f (x)](ξ) = ∓ (−1)n (∓i)m ∂ξ2n+1 FH∓ [ f (x)](ξ), (iv) the differentiation rule

FH± [∂x f (x)](ξ) = ∓ (∓i)m ξ FH∓ [ f (x)](ξ) and more generally

FH± [∂x2n f (x)](ξ) = (−1)n ξ 2n FH± [ f (x)](ξ) FH± [∂x2n+1 f (x)](ξ) = ∓ (−1)n (∓i)m ξ 2n+1 FH∓ [ f (x)](ξ),

and

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(v) the mixed product rule

FH± [(x∂x )n f (x)](ξ) = (−1)n (∂ξ ξ)n FH± [ f (x)](ξ) FH± [(∂x x)n f (x)](ξ) = (−1)n (ξ∂ξ )n FH± [ f (x)](ξ).

Proof. (i) Immediate. Note, however, that the Clifford algebra–valued coefficients λ and μ must be at the right of the functions f and g. (ii) As the classical Fourier transform F satisfies the change of scale property

      ξ 1 , a ∈ R+ , F f (ax) (ξ) = m F f (x) a a we obtain

FH± [f (ax)](ξ) =

 

ξ 1 π F [f (x)] . exp ∓i  ξ 2 m a a

The angular Dirac operator ξ is homogeneous of degree zero, since it commutes with the Euler operator E. Consequently, we have ξ = ξ/a . Hence, the change of scale property also holds for the Clifford-Fourier transform

 

ξ 1 π FH± [f (ax)](ξ) = m exp ∓i 2 ξ/a F [f (x)] a a   ξ 1 = m FH± [f (x)] . a a (iii) From the multiplication rule for the classical Fourier transform (see Proposition 2.2), we obtain at once that



FH+ [xf (x)](ξ) = i exp −i π2 ξ ∂ξ F [f (x)](ξ). Repeated application of the commutation relation [see Eq. (34)]

ξ ∂ξ = ∂ξ (m − 1 − ξ ),

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yields

kξ ∂ξ = ∂ξ (m − 1 − ξ )k , which enables us to prove that

exp

−i π2 ξ

∞ ∞   1 π k k 1 π k ∂ξ = −i −i ξ ∂ξ = ∂ξ (m−1−ξ )k k! 2 k! 2 k=0 k=0

= ∂ξ exp −i π2 (m − 1 − ξ ) .

Consequently, we obtain



FH+ [x f (x)](ξ) = i ∂ξ exp −i π2 (m − 1 − ξ ) F [ f (x)](ξ)

  = i exp −i π2 m i ∂ξ exp i π2 ξ F [f (x)](ξ) = −(−i)m ∂ξ FH− [f (x)](ξ).

The analogous result for the Clifford–Fourier transform involving the operator H− is derived in a similar way. The more general formulas are now proved by induction. For example, the formula

FH+ [x2n f (x)](ξ) = (−1)n ∂ξ2n FH+ [ f (x)](ξ) holds for n = 1, since FH+ [x2 f (x)](ξ) = −(−i)m ∂ξ FH− [xf (x)](ξ) = −(−i)m im ∂ξ2 FH+ [f (x)](ξ) = −∂ξ2 FH+ [f (x)](ξ). Assuming that it holds for n, we now prove it for n + 1: FH+ [x2(n+1) f (x)](ξ) = FH+ [x2n x2 f (x)](ξ) = (−1)n ∂ξ2n FH+ [x2 f(x)](ξ) 2(n+1)

= (−1)n+1 ∂ξ

FH+ [ f (x)](ξ).

(iv) By means of the differentiation rule for the classical Fourier transform and the commutation relation

ξ ξ = ξ (m − 1 − ξ ),

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a similar argument may be developed as in the proof of the multiplication rule. (v) This result follows by combining the multiplication and differentiation rules. For example, we have FH+ [x∂x f (x)](ξ) = −(−i)m ∂ξ FH− [∂x f (x)](ξ) = −(−i)m im ∂ξ ξFH+ [ f (x)](ξ) = − ∂ξ ξFH+ [ f (x)](ξ) and hence, by repeated use,

FH+ [(x∂x )n f (x)](ξ) = (−1)n (∂ξ ξ)n FH+ [ f (x)](ξ). Because the Fourier transform of a radial function remains radial, and the angular Dirac operator  does not affect radial functions, the next result follows readily. Proposition 4.8 For a radial function f ; that is, f only depends on |x| = r, one has FH± [ f ] = F [ f ] and in particular, FH± [δ] = √ 1 m and FH± [1] = 2π √ m 2π δ.

4.5. The Two-Dimensional Case The purpose of this section is twofold: (1) to present an in-depth study of our Clifford–Fourier transform in the specific case of two dimensions, thus providing a theoretical background for the use of this integral transformation, and (2) to show how our 2D Clifford–Fourier transform fits in the picture of all already existing Clifford–Fourier transforms described in Section 4.1 and in this way to promote our higher-dimensional Clifford– Fourier transform as a possible tool for multidimensional signal analysis. The 2D case of the Clifford–Fourier transform is special in that we are able to obtain a closed form for the kernel of the integral representation.

4.5.1. The Integral Kernel In the sequel, the following Clifford numbers play a crucial role:

P± =

1 (1 ± ie12 ). 2

They are self-adjoint mutually orthogonal idempotents which, by multiplication, transform e12 into the imaginary unit i.

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Lemma 4.4 The Clifford numbers P+ and P− satisfy the following properties: (i)

P+ + P− = 1 ;

P+ P− = P− P+ = 0 ;

(P± )2 = P±

(ii)

P+ (ie12 ) = P+

or

P− (ie12 ) = −P−

P+ i = P+ (−e12 ) = (−e12 )P+ ; P− i = P− e12 = e12 P−

or

(iii) for k ∈ N:

P+ (ie12 )k = P+

or

P− (ie12 )k = (−1)k P−

P+ (e12 )k = P+ (−i)k ; or

P− (e12 )k = P− ik .

Proof. (i) By a straightforward computation, we find

P+ P− =

1 1 (1 + ie12 )(1 − ie12 ) = (1 − ie12 + ie12 − 1) = 0 4 4

(P± )2 =

1 1 (1 ± ie12 )(1 ± ie12 ) = (1 ± 2 ie12 + 1) = P± . 4 4

and

(ii) We easily obtain

P+ (ie12 ) =

1 1 (1 + ie12 )(ie12 ) = (ie12 + 1) = P+ , 2 2

which by multiplication with (−e12 ) yields

P+ i = P+ (−e12 ). The result for P− is derived similarly. (iii) In view of the above, we have

P+ (ie12 )k = P+ (ie12 )(ie12 ) . . . (ie12 ) = P+ (ie12 ) . . . (ie12 ) = . . . = P+

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or, by multiplication with (−i)k ,

P+ (e12 )k = P+ (−i)k . The result for P− is proved in an analogous manner.

A. Calculation of the kernel for FH+ We calculate the kernel of the Clifford–Fourier transform involving the operator H+ . This kernel is given by

exp −i π2 ξ [exp (−i < x , ξ >)]. By means of Lemma 4.4 (i) and  = −e12 L12 , we now have

exp −i π2 ξ [exp (−i < x , ξ >)]     = P+ exp i π2 e12 L12 [exp (−i < x, ξ >)] + P−exp i π2 e12 L12 ×[exp (−i < x , ξ >)].

(40)

Moreover, again using the properties of the Clifford numbers P+ and P− , we obtain ∞    (ie12 )k π k P+ exp i π2 e12 L12 = P+ (L12 )k k! 2 k=0

=P

+

∞    1 π k (L12 )k = P+ exp π2 L12 k! 2 k=0

and similarly

    P− exp i π2 e12 L12 = P− exp − π2 L12 . Hence, Eq. (40) becomes π exp −i ξ [exp (−i < x , ξ >)] 2

π π + L12 [exp (−i < x , ξ >)] + P− exp − L12 [exp (−i < x, ξ >)]. = P exp 2 2 (41)

We now prove the following intermediate result.

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Lemma 4.5 Let f be a real-analytic function in R2 , let L12 = ξ1 ∂ξ2 − ξ2 ∂ξ1 be the angular momentum operator, and let R± be the operator exponential given by

  R± := exp ± π2 L12 . Then one has

R+ [ f (ξ1 , ξ2 )] = f (−ξ2 , ξ1 )

and

R− [f (ξ1 , ξ2 )] = f (ξ2 , −ξ1 ).

Proof. In terms of polar coordinates



ξ1 = r cos (θ) ξ2 = r sin (θ)

with r = |ξ| ∈ [0, +∞[ and θ ∈ [0, 2π[, the operator exponential R+ takes the form

Rθ+ := exp

π 2

 ∂θ .

We have + Rθ+ [ f (r, θ)] = Rψ [ f (r, θ + ψ)]ψ=0 =

∞  1 π k k ∂ψ [ f (r, θ + ψ)]ψ=0 . k! 2 k=0

As we assume f to be real-analytic in R2 , this becomes

π , Rθ+ [ f (r, θ)] = f r, θ + 2 which leads to the desired result. The result for the operator exponential R− is proved in a similar way. Remark 4.3 The operator exponentials R+ and R− may be qualified, respectively, as an anti-clockwise, and a clockwise rotation by a right angle. Returning to the calculation of the 2D Clifford–Fourier kernel, we obtain the following by applying Lemma 4.5 to Eq. (41):



   exp −i π2 ξ exp (−i < x, ξ >) = P+ exp −i(x2 ξ1 − x1 ξ2 )   + P− exp −i(x1 ξ2 − x2 ξ1 ) . This expression for the kernel of FH+ can be further simplified using the following result.

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Lemma 4.6 One has

    P+ exp −i(x2 ξ1 − x1 ξ2 ) = P+ exp (ξ ∧ x) and P− exp −i(x1 ξ2 −x2 ξ1 ) = P− exp (ξ ∧ x). Proof.

By means of Lemma 4.4 (iii), we have consecutively

∞    (−i)k (x2 ξ1 − x1 ξ2 )k P+ exp −i(x2 ξ1 − x1 ξ2 ) = P+ k! k=0

=P

+

∞  (e12 )k k=0

k!

(x2 ξ1 −x1 ξ2 )k

  = P+ exp e12 (ξ1 x2 −ξ2 x1 ) = P+ exp (ξ∧x). The second statement is proved similarly. In view of the foregoing lemma, we finally obtain a closed form for the kernel of FH+ :

exp −i π2 ξ [exp (−i < x , ξ >)] = P+ exp (ξ ∧ x) + P− exp (ξ ∧ x) = exp (ξ ∧ x).

Hence, the Clifford–Fourier transform involving the operator H+ has the following integral representation:

FH+ [f ](ξ) =

1 2π

 R2

exp (ξ ∧ x) f (x) dV(x).

B. Calculation of the kernel for FH− The computation of the kernel of the 2D Clifford–Fourier transform involving the operator H− follows the same lines. It is given by

  exp i π2 ξ [exp (−i < x , ξ >)] = exp −(ξ ∧ x) = exp (x ∧ ξ). The integral representation for FH− then takes the form

FH− [ f ](ξ) =

1 2π

 R2

exp (x ∧ ξ) f (x) dV(x).

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Remark 4.4 1. Note that the 2D Clifford–Fourier kernels consist of a scalar and a bivector part—they are so-called parabivectors. 2. The Clifford–Fourier transform FH± may be qualified as a coaxial Fourier transform, since its integral kernel may also be rewritten as

  exp ±(ξ ∧ x) = cos (ξ1 x2 − ξ2 x1 ) ± e12 sin (ξ1 x2 − ξ2 x1 ),

(42)

where ξ1 x2 − ξ2 x1 takes constant values on coaxial cylinders, which in two dimensions take the form of two lines parallel and symmetric w.r.t. the fixed vector ξ (see also Section 6). Hence, in terms of the Fourier cosine and the Fourier sine transform,

1 Fcos [ f ](ξ) = 2π Fsin [ f ](ξ) =

1 2π

 R2

cos (ξ1 x2 − ξ2 x1 ) f (x) dV(x),



R2

sin (ξ1 x2 − ξ2 x1 ) f (x) dV(x),

the Clifford–Fourier transform FH± takes the form

FH± [ f ](ξ) = Fcos [ f ](ξ) ± e12 Fsin [ f ](ξ).

4.5.2. The Two-Dimensional Clifford–Fourier Transform as a Linear Operator   1. The Clifford–Fourier Transform in L1 R2 , dV(x)

The 2D Clifford–Fourier  transform  FH+ [ f ] is well defined for each integrable function f ∈ L1 R2 , dV(x) . Indeed, by means of the properties in Eq. (1) of the Clifford norm, we have

   

R

   exp (ξ ∧ x) f (x) dV(x) ≤ 2

R2

≤4

| exp (ξ ∧ x) f (x)| dV(x)



R2

| exp (ξ ∧ x)| | f (x)| dV(x).

Furthermore, using Eq. (42), we obtain:

| exp (ξ ∧ x)| = | cos (ξ1 x2 − ξ2 x1 ) + e12 sin (ξ1 x2 − ξ2 x1 )| ≤ | cos (ξ1 x2 − ξ2 x1 )| + | sin (ξ1 x2 − ξ2 x1 )| ≤ 2.

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Hence, we have

   

R

    exp (ξ ∧ x) f (x) dV(x) ≤ 8 2

R2

   f (x) dV(x) < ∞

and thus also

  exp (ξ ∧ x) f (x) ∈ L1 R2 , dV(x) . A similar argument holds for the Clifford–Fourier transform involving the operator H− . The above reasoning leads to the following theorem.       Theorem 4.2 Let f ∈ L1 R2 , dV(x) . Then FH± [f ] ∈ L∞ R2 ∩ C0 R2 , and moreover,

  FH± [ f ]

∞

Proof.

≤

 4 f . 1 π

In view of the above, we have

    4     1  ≤  f (x) dV(x) exp ±(ξ ∧ x) f (x) dV(x)   2π R2 π R2 4  =  f 1 . π

|FH± [ f ](ξ)| =

Moreover, one can easily verify that

|FH± [ f ](ξ) − FH± [ f ](ξ  )| ≤

 8 f . 1 π

Hence, |ξ − ξ  | → 0 implies

|FH± [ f ](ξ) − FH± [ f ](ξ  )| → 0; in other words, FH± [ f ] is continuous. In Subsection 4.4.2 some operational formulas (viz., the right Cm linearity, change of scale, multiplication, differentiation, and mixed product rule) were derived for the Clifford–Fourier transform in arbitrary dimension. In the special 2D case, the obtained closed forms for the integral kernels enable some further results for the Clifford–Fourier  us to prove  transform in L1 R2 , dV(x) .

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  Proposition 4.9 Let f , g ∈ L1 R2 , dV(x) . The 2D Clifford–Fourier transform satisfies the following (i) the shift theorem

  FH± [τh f (x)](ξ) = exp ±(ξ ∧ h) FH± [ f (x)](ξ), where τh denotes the translation by h , i.e. τh f (x) = f (x − h) (ii) frequency reversion

FH± [ f ](−ξ) = FH∓ [ f ](ξ) (iii) Hermitean conjugation

†



FH+ [ f ](ξ)

†



FH− [ f ](ξ)

=

1 2π

1 = 2π

 R2

f † (x) exp (x ∧ ξ) dV(x),

 R2

f † (x) exp (ξ ∧ x) dV(x)

(iv) the modulation theorem

  FH± exp (x ∧ h) f (x) (ξ) = τ±h FH± [ f (x)](ξ) (v) the transfer formula

 R2



†

FH± [ f ](ξ)

 g(ξ) dV(ξ) =

R2

f † (ξ) FH± [ g](ξ) dV(ξ)

(vi) the convolution theorem

FH± [ f p ∗ g](ξ) = 2π FH± [ f p ](ξ) FH± [ g](ξ), FH± [ f ∗ g](ξ) = 2π FH± [ f ](ξ) FH∓ [ g](ξ), where ∗ denotes the Clifford convolution given by

 ( f ∗ g)(x) =

R2

f (x − x ) g(x ) dV(x ),

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and f p , respectively f , denotes, respectively, a parabivector-valued and vector-valued function; that is,

f p (x) = f0 (x) + f12 (x) e12 ;

f = f1 (x) e1 + f2 (x) e2

( f0 , f1 , f2 , f12 : R2 → C). (vii) the multiplication theorem

 1  FH± [ f p ] ∗ FH± [ g] (ξ) 2π

FH± [ f p g](ξ) =

 1  FH± [ f g](ξ) = FH± [ f ] ∗ FH∓ [ g] (ξ), 2π where again f p , respectively f , denotes, respectively, a parabivector-valued and vector-valued function, which is Fourier invertible. (viii) the rotation rule

FH± [ f (sxs)](ξ) = FH± [ f (x)](sξs), where s ∈ SpinR (m). Proof. We restrict ourselves to the proofs for the Clifford–Fourier transform involving the operator H+ , the proofs for FH− being similar. (i) By means of the substitution u = x − h, we obtain

FH+ [τh f (x)](ξ) =

1 2π

1 = 2π

 R2

exp (ξ ∧ x) f (x − h) dV(x)



R2

exp (ξ ∧ u + ξ ∧ h) f (u) dV(u)

1 = exp (ξ ∧ h) 2π

 R2

exp (ξ ∧ u) f (u) dV(u)

= exp (ξ ∧ h) FH+ [ f ](ξ). Here we have used the fact that

exp (ξ∧u + ξ∧h) = exp (ξ∧u) exp (ξ∧h) = exp (ξ∧h) exp (ξ∧u), since the exponentials exp (ξ ∧ u) and exp (ξ ∧ h) commute.

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(ii) This property follows at once from

  exp (−ξ) ∧ x = exp (x ∧ ξ). (iii) Taking into account that ξ ∧ x = x ∧ ξ and hence also that

exp (ξ ∧ x) = exp (x ∧ ξ), the result follows immediately. (iv) We have consecutively

   1 FH+ exp (x∧h) f (x) (ξ) = exp (ξ∧x) exp (x∧h) f (x) dV(x) 2π R2    1 = exp (ξ − h) ∧ x f (x) dV(x) 2 2π R = FH+ [ f (x)](ξ − h). (v) First note that both integrals in formula (v) are well defined, †  since FH± [ f ](ξ) g(ξ) and f † (ξ) FH± [ g](ξ) belong to the space   L1 R2 , dV(x) of integrable functions. Moreover, using property (iii) and changing the order of integration yields



†

 R2

= =

FH+ [ f ](ξ) 1 2π  R2

 =

R2

  R2

f † (x)

R2



g(ξ) dV(ξ)

 f † (x) exp (x ∧ ξ) dV(x) g(ξ) dV(ξ)

1 2π

 R2

 exp (x ∧ ξ) g(ξ) dV(ξ) dV(x)

f † (x) FH+ [ g](x) dV(x).

(vi) Let us first consider the case of a parabivector-valued function f p . Changing the order of integration and applying the substitution u = x − x , yields, consecutively,

FH+ [ f p ∗ g](ξ)    1 p    = exp (ξ ∧ x) f (x − x ) g(x ) dV(x ) dV(x) 2π R2 R2

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=

1 2π

1 = 2π



 R2

  R2

R2

 exp (ξ ∧ x) f p (x − x ) dV(x) g(x ) dV(x ) 

R2



exp (ξ∧u) exp (ξ∧x ) f (u) dV(u) g(x ) dV(x ). p

As a parabivector-valued function f p commutes with the Clifford– Fourier kernels; that is,

    exp ±(ξ ∧ x) f p (u) = f p (u) exp ±(ξ ∧ x) , we indeed obtain

FH+ [ f p ∗ g](ξ)     1 = exp (ξ ∧ u)f p (u) dV(u) exp (ξ ∧ x )g(x )dV(x ) R2 2π R2   p    = FH+ [ f ](ξ) exp (ξ ∧ x ) g(x ) dV(x ) R2

p

= 2π FH+ [ f ](ξ) FH+ [ g](ξ). On the other hand a vector-valued function f , by commutation, transforms the kernel of FH+ into the kernel of FH− and vice versa; that is,

    exp ±(ξ ∧ x) f (u) = f (u) exp ∓(ξ ∧ x) . Hence, for a vector-valued function f we find

FH+ [ f ∗ g](ξ)    1 = exp (ξ ∧ u) exp (ξ ∧ x ) f (u) dV(u) g(x ) dV(x ) 2π R2 R2     1 = exp (ξ ∧ u) f (u) dV(u) exp (x ∧ ξ) g(x ) dV(x ) R2 2π R2       = FH+ [ f ](ξ) exp (x ∧ ξ) g(x ) dV(x ) R2

= 2π FH+ [ f ](ξ) FH− [ g](ξ).

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Note that in case of a general C2 -valued function f = f p + f , the convolution theorem inevitably leads to two terms:

FH± [ f ∗ g](ξ) = 2π FH± [ f p ](ξ) FH± [ g](ξ) + 2π FH± [ f ](ξ) FH∓ [ g](ξ). (vii) First, recall from Subsection 4.4.1 that

  (FH± )−1 = exp ±i π2  F −1 or in integral form (FH± )−1 [ f ](ξ) =

1 2π

 R2

  exp ±i π2 ξ exp (i < x, ξ >) f (x) dV(x).

Similarly, as in Subsection 4.5.1, one can prove that

π     exp ±i ξ exp (i < x, ξ >) = exp ±(ξ ∧ x) , 2 from which we obtain that (FH± )−1 = FH± . Hence, we also have that

1 FH+ [ f g](ξ) = 2π



p

R2

−1 p exp (ξ ∧ x) FH + [FH+ [ f ]](x) g(x) dV(x)

 1 exp (ξ ∧ x) (2π)2 R2   p × exp (x ∧ u) FH+ [ f ](u) dV(u) g(x) dV(x). =

R2

The Clifford–Fourier spectrum of a parabivector-valued function is again parabivector valued as follows:

FH+ [ f p ](u) = Fcos [ f p ](u) + e12 Fsin [ f p ](u) = Fcos [ f0 ](u) + Fcos [ f12 ](u) e12 + e12 Fsin [ f0 ](u) −Fsin [ f12 ](u),

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and thus commutes with the Clifford–Fourier kernels, which yields

FH+ [ f p g](ξ)      1 p FH+ [ f ](u) exp (ξ − u) ∧ x g(x)dV(x) dV(u) = (2π)2 R2 R2  1 FH+ [ f p ](u) FH+ [ g](ξ − u) dV(u) = 2π R2 =

 1  FH+ [ f p ] ∗ FH+ [ g] (ξ). 2π

As the Clifford–Fourier transform of a vector-valued function f is again vector valued,

FH+ [ f ](u) = Fcos [ f1 ](u) e1 + Fcos [ f2 ](u) e2 + e2 Fsin [ f1 ](u) − e1 Fsin [ f2 ](u), the multiplication theorem in case of a vector-valued function f reads as follows:

FH+ [ f g](ξ)    1  exp (ξ∧x) exp (x∧u)FH+ [ f ](u)dV(u) g(x)dV(x) = (2π)2 R2 R2      1  + F [ f ](u) exp x ∧ (ξ − u) g(x)dV(x) dV(u) = H (2π)2 R2 R2  1 FH+ [ f ](u) FH− [ g](ξ − u) dV(u) = 2π R2 =

 1  FH+ [ f ] ∗ FH− [ g] (ξ). 2π

Again, in case of a general C2 -valued function f = f p + f , the multiplication theorem divides into two pieces

FH± [ f g](ξ) =

 1  FH± [ f p ] ∗ FH± [ g] (ξ) 2π  1  + FH± [ f ] ∗ FH∓ [ g] (ξ). 2π

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(viii) By means of the substitution u = sxs, we have



1 FH+ [ f (sxs)](ξ) = 2π =

R2

exp (ξ ∧ x) f (sxs) dV(x)



1 2π

R2

exp (ξ ∧ sus) f (u) dV(u).

Taking into account that s(u ∧ ξ)s = (sus) ∧ (sξs) yields

FH+ [ f (sxs)](ξ) =

1 2π

 R2

1 = s 2π

  exp s(sξs ∧ u)s f (u) dV(u)

 R2

exp (sξs ∧ u) s f (u) dV(u).

As s ∈ SpinR (m) takes the form s = ω1 ω2 . . . ω2 with ωj ∈ Sm−1 , j = 1, . . . , 2, and moreover, e12 ωj = −ωj e12 , we find that

e12 s = e12 ω1 . . . ω2 = (−1)2 ω1 . . . ω2 e12 = s e12 . Hence, s commutes with the parabivector-valued exponential kernel, which finally leads to 1 FH+ [ f (sxs)](ξ) = ss 2π

 R2

exp (sξs ∧ u) f (u) dV(u) = FH+ [ f ](sξs),

since ss = 1. Remark 4.5 1. Property (ii) implies that it is sufficient to compute one of the Clifford–Fourier transforms. 2. As the Clifford algebra is not commutative, property (iii) cannot be rewritten in the more elegant form



†



†

FH± [ f ](ξ)

= FH∓ [ f † ](ξ).

However, we have

FH± [ f ](ξ)

= [ f † ]FH∓ (ξ),

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where the notation [ f † ]FH∓ means that the Clifford–Fourier transform is now acting from the right on the function f † . 3. If f is rotation invariant (i.e., f (sxs) = f (x) for each s ∈ SpinR (m)), then the rotation rule implies that FH± [ f (x)](sξs) = FH± [ f (x)](ξ) for each s ∈ SpinR (m). Hence, FH± [ f ] is rotation invariant as well.

  2. The Clifford–Fourier Transform in S R2 We now investigate the Clifford–Fourier transform in a dense subspace of  L1 R2 , dV(x) , viz. the right C2 -module of rapidly decreasing C2 -valued functions.     Theorem 4.3 Let ϕ ∈ S R2 . Then FH± [ϕ] ∈ S R2 .   Proof. We will show that for every ϕ ∈ S R2 , the following inequality holds:      β  p∗,k FH± [ϕ] = sup sup sup ξ α ∂ξ [FH± [ϕ](ξ)] ξ∈R2 |α|≤k |β|≤



2 +2

≤ C sup (1 + |x| ) x∈R2

  γ  ∂x ϕ(x) = C p+2,k (ϕ).

(43)

|γ|≤k

Here {p∗,k ; , k ∈ N} and {p,k ; , k ∈ N} are two equivalent systems   α α of seminorms on S R2 . Moreover, we use the notation ξ α = ξ1 1 ξ2 2 and β β β ∂ξ = ∂ξ11 ∂ξ22 . One can easily verify that for every multi-index α ∈ N2 : α

α

∂ξα [FH± [ϕ](ξ)] = (±e12 )α1 (∓e12 )α2 FH± [x1 2 x2 1 ϕ(x)](ξ) and α

α

FH± [∂xα ϕ(x)](ξ) = (±e12 )α1 (∓e12 )α2 ξ1 2 ξ2 1 FH± [ϕ(x)](ξ). For arbitrary ξ ∈ R2 and multi-indices α, β ∈ N2 with |α| ≤ k, and |β| ≤ , we thus have  β β   β α α ξ α ∂ξ FH± [ϕ](ξ) = (±e12 )β1 (∓e12 )β2 ξ1 1 ξ2 2 FH± x1 2 x2 1 ϕ(x) (ξ)    α α  β β = (±e12 )β1 +α1 (∓e12 )β2 +α2 FH± ∂x12 ∂x21 x1 2 x2 1 ϕ(x) (ξ).

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By means of  the properties of the Clifford norm and the estimate  | exp ±(ξ ∧ x) | ≤ 2 , we obtain      1     α2 α1  β2 β1   α β  exp ±(ξ ∧ x) ∂x1 ∂x2 x1 x2 ϕ(x) dV(x) ξ ∂ξ FH± [ϕ](ξ)  ≤ 4  2π R2    16   α2 α1 β2 β1 (44) ≤ ∂x1 ∂x2 [x1 x2 ϕ(x)] dV(x). π R2 Furthermore, using the estimate

        α2 α1 β2 β1  α −γ α −γ β β γ γ Cγα ∂x12 1 ∂x21 2 [x1 2 x2 1 ] ∂x11 ∂x22 [ϕ(x)] ∂x1 ∂x2 [x1 x2 ϕ(x)] ≤ |γ|≤|α|

≤ C (1 + |x|2 )|β|

  γ  ∂x ϕ(x), |γ|≤|α|

the inequality in Eq. (44) still for |α| ≤ k and |β| ≤  becomes

    γ  16  α β  ∂x ϕ(x) dV(x) (1 + |x|2 ) ξ ∂ξ [FH± [ϕ](ξ)] ≤ C π R2 |γ|≤k    γ   16 2 +2   ≤C sup (1 + |x| ) ∂x ϕ(x) π x∈R2 |γ|≤k  1 × dV(x) 2 2 R2 (1 + |x| )    γ  2 +2   = C sup (1 + |x| ) ∂x ϕ(x) . x∈R2

|γ|≤k

Hence, we have proved the desired inequality in Eq. (43). Inequality (43) immediately yields the following result, where  use  we the notation ϕj → ϕ for the convergence of the sequence ϕj in S R2 . S

Corollary 4.2 If ϕj → ϕ, then FH± [ϕj ] → FH± [ϕ] . S

S

  Remark 4.6 The 2D Clifford–Fourier transform in S R2 is thus a continuous operator. For the inversion of the Clifford–Fourier transform, one now has the following proposition.

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  Proposition 4.10 For each ϕ ∈ S R2 , one has

FH± [FH± [ϕ]] = ϕ, or, in other words, (FH± )2 = IS R2  . By Corollary 4.2 and Proposition 4.10 we finally obtain the following fundamental result. Theorem 4.4 The 2DClifford–Fourier transform FH± is a homeomorphism of  2 the right C2 -module S R .

  3. The Clifford–Fourier Transform in L2 R2 , dV(x)   The definition of the Clifford–Fourier transform in L2 R2 , dV(x) follows classical lines and makes use of the results obtained in the foregoing  subsection on the Clifford–Fourier transform in the dense subspace S R2 . We start with the following lemma.   Lemma 4.7 For all ϕ, ψ ∈ S R2 , one has < ϕ, ψ > = FH± [ϕ], FH± [ψ] . Proof. Taking into account Proposition 4.10 and the transfer formula (see Proposition 4.9 (v)), we have consecutively

 < ϕ, ψ > =

ϕ(x)

R2

 =

†



R2



 †  ψ(x) dV(x) = FH± [FH± [ϕ]](x) ψ(x) dV(x) R2

†

FH± [ϕ](x)

& ' FH± [ψ](x) dV(x) = FH± [ϕ], FH± [ψ] .

In particular we have the following result.   Lemma 4.8 For each ϕ ∈ S R2 , one has ϕ2 = FH± [ϕ]2 . This  implies that the operator FH± can  be uniquely extended to L2 R2 , dV(x) . Indeed, take f ∈ L2 R2 , dV(x) . By means of the density       of S R2 in L2 R2 , dV(x) , there exists a sequence (ϕj )j∈N ∈ S R2 that  2  converges in L2 R , dV(x) to f . Hence, (ϕj )j∈N is a Cauchy sequence     in L2 R2 , dV(x) . In view of Lemma 4.8, FH± [ϕj ] j∈N is also a Cauchy     sequence in L2 R2 , dV(x) and thus convergent in L2 R2 , dV(x) . The limit  of FH± [ϕj ] j∈N , which is independent of the chosen sequence (ϕj )j∈N , is called the Clifford–Fourier transform of f . We denote it by FH± [ f ].

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  The above-introduced Clifford–Fourier transform in L2 R2 , dV(x) has the following properties.   Proposition 4.11 The 2D Clifford–Fourier transform FH± in L2 R2 , dV(x) satisfies the following properties:   (i) FH± is right C2 -linear; that is, for all f , g ∈ L2 R2 , dV(x) and for all λ, μ ∈ C2 , one has

FH± [ f λ + gμ] = FH± [ f ] λ + FH± [ g] μ.   (ii) The restriction of FH± to S R2 is FH± .  2  (iii) FH± is bounded on L2 R , dV(x) . (iv) The inverse of FH± is precisely FH± , or (FH± )2 = IL R2 ,dV(x) . 2

(v) The adjoint (FH± )∗ is given by (FH± )∗ = FH± = (FH± )−1 .

Proof. (i) Trivial. (ii) Immediate.    (iii) Take f ∈ L2 R2 , dV(x) and a sequence ϕj ∈ S R2 that converges in      2  L2 R , dV(x) to f ; that is, ϕj 2 →  f 2 . Furthermore, by definition  R    FH± [ϕj ] → FH± [ f ]; that is, FH± [ϕj ]2 → FH± [ f ]2 . Taking into L2   R  account Lemma 4.8, we thus obtain  f 2 = FH± [ f ]2 , which proves the statement.     (iv) Take again f ∈ L2 R2 , dV(x) and a sequence ϕj ∈ S R2 that converges   in L2 R2 , dV(x) to f . By definition, we have consecutively

FH± [ϕj ] → FH± [ f ] L2

and

FH± [FH± [ϕj ]] → FH± [FH± [ f ]]. L2

Hence, by means of Proposition 4.10 we find

ϕj → FH± [FH± [ f ]], L2

which leads to

f = FH± [FH± [ f ]].   (v) To compute the adjoint (FH± )∗ , we start with f , g ∈ L2 R2 , dV(x) and  2  2  sequences ϕj and ψj in S R that converge in L2 R , dV(x) to f and

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g, respectively. By definition, we have

FH± [ϕj ] → FH± [ f ]

and

L2

FH± [ψj ] → FH± [ g]. L2

  In terms of the inner product on L2 R2 , dV(x) , it follows that < ϕj , ψj >→< f , g > and C2

& ' ' & FH± [ϕj ], FH± [ψj ] → FH± [ f ], FH± [ g] . C2

Moreover, applying Lemma 4.7 gives

& ' < f , g >= FH± [ f ], FH± [ g] and hence also

(

'

)

& ' FH± [ f ], g = FH± [FH± [ f ]], FH± [ g] = f , FH± [ g] .

&

Summarizing we obtain the following fundamental result. Theorem 4.5 The 2D Clifford–Fourier transform FH± is a unitary operator on  the right C2 -module L2 R2 , dV(x) .

4.5.3. Connection With the Classical Fourier Transform and the Clifford–Fourier Transform of Ebling and Scheuermann In this subsection we first derive an explicit connection between the 2D Clifford–Fourier transform pair {FH+ , FH− } and the classical tensorial Fourier transform F in the plane. By means of Lemma 4.6 we can rewrite the Clifford–Fourier transform FH+ as follows:

FH+ [ f ](ξ) = P+



1 2π

+ P−



 R2

1 2π

  exp −i(ξ1 x2 − ξ2 x1 ) f (x) dV(x)

 R2

  exp −i(ξ2 x1 − ξ1 x2 ) f (x) dV(x).

Furthermore, one easily verifies that

1 2π

 R2

  exp −i(ξ1 x2 − ξ2 x1 ) f (x) dV(x) = F [ f ](e12 ξ)

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and similarly



1 2π

R2

  exp −i(ξ2 x1 − ξ1 x2 ) f (x) dV(x) = F [ f ](−e12 ξ).

This yields the following relation between the Clifford–Fourier transform involving the operator H+ and the standard Fourier transform:

FH+ [ f ](ξ) = P+ F [ f ](e12 ξ) + P− F [ f ](−e12 ξ).

(45)

Similarly, one obtains

FH− [ f ](ξ) = P+ F [ f ](−e12 ξ) + P− F [ f ](e12 ξ).

(46)

Remark 4.7 1. The transformations ξ → e12 ξ and ξ → −e12 ξ represent, respectively, an anti-clockwise and a clockwise, rotation by a right angle. 2. For a radial function f , expressions (45) and (46) reduce to FH± [ f ] = F [ f ], since the Fourier transform of a radial function remains radial. Note that we already obtained this result for the Clifford–Fourier transform in arbitrary dimension (see Proposition 4.8). Moreover, the Clifford–Fourier transform of Ebling and Scheuermann (see Section 4.1)

 e

F [ f ](ξ) =

exp (−e12 (x1 ξ1 + x2 ξ2 )) f (x) dV(x)

R2

can be expressed in terms of the Clifford–Fourier transform:

FH± [ f ](ξ) = FH± [ f ](ξ1 , ξ2 ) 1 = 2π

 R2

  exp ±e12 (ξ1 x2 − ξ2 x1 ) f (x) dV(x).

Indeed, we have

F e [ f ](ξ) = 2π FH± [ f ](∓ξ2 , ±ξ1 ) = 2π FH± [ f ](±e12 ξ),

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taking into account that under the isomorphism between the Clifford algebras R2,0 and R0,2 , both pseudoscalars e12 are isomorphic images of each other. Note that in Ebling and Scheuermann (2005) the Fourier kernel is at the right-hand side of the function f instead of at the left.

4.5.4. Example: The Box Function As an illustration, in this subsection we calculate the Clifford–Fourier transform pair of the box function:



A if a ≤ x1 ≤ b and c ≤ x2 ≤ d 0 if otherwise

f (x1 , x2 ) =

with A a constant. Its classical Fourier transform reads



  1 exp (−ibξ1 ) − exp (−iaξ1 ) ξ1 ξ2   × exp (−idξ2 ) − exp (−icξ2 ) . 

F [ f ](ξ) = −

A 2π

In view of relation (45), this yields the following expression for the Clifford– Fourier transform involving the operator H+ :





  1 exp (ibξ2 ) − exp (iaξ2 ) FH+ [ f ](ξ) = P ξ1 ξ2     1 A × exp (−idξ1 ) − exp (−icξ1 ) + P− 2π ξ1 ξ2    × exp (−ibξ2 ) − exp (−iaξ2 ) exp (idξ1 ) − exp (icξ1 ) . +

A 2π

(47) By means of, inter alia,

P

+

exp (ibξ2 ) = P

+

∞ k ∞   i (−e12 )k k + (bξ2 ) = P (bξ2 )k k! k! k=0

= P+ exp (−e12 bξ2 ),

k=0

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expression (47) can be simplified to FH+ [ f ](ξ)       1 A P+ exp (−e12 bξ2 )−exp (−e12 aξ2 ) exp (e12 dξ1 )−exp (e12 cξ1 ) = 2π ξ1 ξ2    + P− exp (−e12 bξ2 )−exp (−e12 aξ2 ) exp (e12 dξ1 ) − exp (e12 cξ1 )     1 A exp (−e12 bξ2 ) − exp (−e12 aξ2 ) exp (e12 dξ1 )−exp (e12 cξ1 ) . 2π ξ1 ξ2

 =

By a similar compution, one obtains

   1 A exp (e12 bξ2 ) − exp (e12 aξ2 ) 2π ξ1 ξ2   × exp (−e12 dξ1 ) − exp (−e12 cξ1 ) . 

FH− [ f ](ξ) =

5. CLIFFORD FILTERS FOR EARLY VISION Among the mathematical models for the receptive field profiles of the human visual system, the Gabor model is well known and widely used. Another lesser used model that agrees with the Gaussian derivative model for human vision is the Hermite model. It is based on analysis filters of the Hermite transform, which was introduced by Martens (1990b), and offers some advantages such as being an orthogonal basis and better matching with experimental physiological data. In this section we expand the filter functions of the classical Hermite transform into the generalized Clifford–Hermite polynomials. Moreover, we construct a new multidimensional Hermite transform within Clifford analysis using the generalized Clifford–Hermite polynomials and, we compare this newly introduced Clifford–Hermite transform with the Clifford– Hermite continuous wavelet transform. Next, we introduce Gabor filters in the Clifford analysis setting. These Clifford–Gabor filters are based on the Clifford–Fourier transform discussed in the previous section. Finally, we present additional properties of the Clifford–Gabor filters, such as their relationship with other types of Gabor filters and their localization in the spatial and frequency domain formalized by the uncertainty principle.

5.1. Introduction Image processing has been much inspired by human vision, particularly with regard to early vision. The latter refers to the earliest stage of visual

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processing responsible for the measurement of local structures, such as points, lines, edges, and textures, in order to facilitate subsequent interpretation of these structures in higher stages (known as high-level vision) of the human visual system. According to the Gaussian derivative theory (see Young, 1991), the receptive field profiles of the human visual system can be approximated quite well by derivatives of Gaussians. Two mathematical models suggested for these receptive field profiles are the Gabor model and the Hermite model, which is based on analysis filters of the Hermite transform. The Hermite filters are derivatives of Gaussians, whereas Gabor filters, which are defined as harmonic modulations of Gaussians, provide a good approximation to these derivatives. It is important to note that, even if the Gabor model is more widely used than the Hermite model, the latter offers some advantages, such as being an orthogonal basis and better matching with experimental physiological data. In this section we establish the construction of the Hermite and Gabor filters both in the classical and in the Clifford analysis setting. We begin by describing the classical Hermite transform, in both the 1D and multidimensional case (see Subsection 5.2.1). Next, in Subsection 5.2.2, we expand the filter functions of the classical multidimensional Hermite transform into the generalized Clifford–Hermite polynomials introduced in Section 2.4. Furthermore, we construct a new higher-dimensional Hermite transform within the framework of Clifford analysis using the generalized Clifford–Hermite polynomials (see Subsection 5.2.3). In Subsection 5.2.4 we compare this Clifford–Hermite transform with the Clifford–Hermite continuous wavelet transform. The topic of Section 5.3 is Gabor filters, a prominent tool for local spectral image processing and analysis. First, in Subsection 5.3.1, we discuss classical 1D complex and real Gabor filters. They are closely related to Fourier analysis, since the impulse response of a complex Gabor filter is the conjugated integral kernel of the complex Fourier transform at a certain frequency, multiplied by a Gaussian. In Subsection 5.3.2 we examine two nonclassical 2D Gabor filters. First, we briefly describe the so-called quaternionic Gabor filters of Bülow and Sommer, followed by the Clifford–Gabor filters of Ebling and Scheuermann. Next, we proceed with developing our 2D Clifford–Gabor filters (see Subsection 5.3.3). This new types of Gabor filters arises quite naturally from our study of the 2D Clifford–Fourier transform. Indeed, using the conjugated kernels of the 2D Clifford–Fourier transform modulated by a Gaussian gives rise to the 2D Clifford–Gabor filters. We also give an explicit connection between these new filters and the standard complex Gabor filters and the Clifford–Gabor filters of Ebling and Scheuermann. An often-cited property of Gabor filters is their optimal simultaneous localization in the spatial and the frequency domain, which is formalized by the uncertainty principle. This property makes Gabor filters suitable for local frequency analysis. We end this section by showing that

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our 2D Clifford–Gabor filters also exhibit the best possible joint localization in position and frequency space.

5.2. Generalized Clifford–Hermite Filters 5.2.1. The Classical Hermite Transform The Hermite transform was introduced by Martens (1990b) as a signal expansion technique in which a signal is windowed by a Gaussian at equidistant positions and is locally described by a weighted sum of polynomials. In van Dijk and Martens (1997) an image compression scheme based on an orientation-adaptive steered Hermite transform is presented. Comparison with other compression techniques show that the proposed scheme performs very well at high compression ratios, not only in terms of peak signal to noise ratio (the commonly used objective measure for the quality of coded images) but also in terms of perceptual image quality. Moreover, in Martens (1990c), it is demonstrated how the Hermite transform can be used for image coding and analysis. In the image coding application, the relation with existing pyramid coders is described. A new coding scheme, based on local 1D image approximations, is introduced. In the image analysis application, the relation between the Hermite transform and existing line/edge detection schemes is described. An algorithm based on the Hermite transform was applied to astronomical images in Venegas-Martinez, Escalente-Ramirez, and Garcia-Barreto (1997). Hermite transforms have been used in applications such as image deblurring (see Martens, 1990a), noise reduction (see Escalante-Ramirez and Martens, 1992) and also estimation of perceived noise and blur (see Kayargadde and Martens, 1994).

A. The Classical One-Dimensional Hermite Transform The 1D Hermite transform first localizes the original signal f (x) by multiplying it by a Gaussian window function

x2 *σ (x) = √1 V exp − 2σ 2 . πσ In order to obtain a complete description of the signal f (x), the localization process should be repeated at a sufficient number of window positions with the spacing between the windows chosen equidistant. In this way, the following expansion of the original signal f (x) is obtained:

f (x) =

1

+∞ 

* σ (x) W k=−∞

*σ (x − kT), f (x) V

(48)

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where

* σ (x) = W

+∞ 

*σ (x − kT), V

k=−∞

is the so-called weight function, which is positive for all x. The next step in the Hermite transform is the decomposition of the *σ (x − kT) into a series of orthonormal functions. localized signal f (x)V Fundamental for this expansion are the polynomials Gnσ (x), which are *σ (x))2 ; that is, orthonormal with respect to (V



+∞

−∞

*σ (x))2 Gσ (x) Gσ (x) dx = δn ,n . (V n n

These uniquely determined polynomials have the following form:

Gnσ (x) = √

1 2n n!

Hn

x σ

,

where Hn is the standard Hermite polynomial of order n (n = 0, 1, 2, . . .) associated with the weight function exp (−x2 ) [see Eq. (17)]. Under very general conditions (see Boas and Buck, 1985) for the original signal f (x), we get the following decomposition of the localized signal into *σ (x) Gσ (x): the orthonormal functions Knσ (x) = V n

*σ (x − kT) f (x) = V

∞ 

cnσ (kT) Knσ (x − kT),

(49)

n=0

where

cnσ (kT)

 =

∞

−∞

*σ (x − kT))2 dx f (x) Gnσ (x − kT) (V

(50)

are the Hermite coefficients. As the Hermite transform is locally (i.e., within each window function), a unitary transformation, we have the following generalization of Parseval’s theorem:



+∞ 

−∞

f (x)

2

*σ (x − kT))2 dx = (V

∞  (cnσ (kT))2 . n=0

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In other words, the energy of each local signal can be expressed in terms of the Hermite coefficients cnσ (kT) of the expansion. The defining relation in Eq. (50) of the Hermite coefficients cnσ (kT) can be rewritten as the convolution of the original signal f (x) with the functions

x 2 (−1)n 1 *σ (−x))2 = √ exp − σx 2 , Dσn (x) = Gnσ (−x) (V √ Hn σ 2n n! σ π followed by a downsampling by a factor T. The functions Dσn (x) are called the Hermite filters. By means of the Rodrigues formula Eq. (17) for the classical Hermite polynomials, they can be expressed as Gaussian derivatives

Dσn (x)

 2  σn dn 1 =√ √ exp − σx 2 . 2n n! dxn σ π

The Fourier transform of the Hermite filter Dσn takes the form of a Gaussian modulated by a monomial of degree n

2 2 1 σn F [Dσn ](ξ) = √ √ (iξ)n exp − σ 4ξ . 2n n! 2π The mapping from the original signal f (x) to the Hermite coefficients cnσ (kT) is called the forward Hermite transform. The signal reconstruction from the Hermite coefficients is called the inverse Hermite transform. Combining Eqs. (48) and (49), we get the expansion of the complete signal into the so-called pattern functions Qσn :

f (x) =

∞  +∞ 

cnσ (kT) Qσn (x − kT),

n=0 k=−∞

where

Qσn (x) =

Knσ (x) . * σ (x) W

This formula implies that the inverse Hermite transform consists of interpolating the Hermite coefficients {cnσ (kT); k integer} with the pattern function Qσn (x) and summing up over all orders n. As mentioned in Section 5.1, the Hermite transform models the information analysis carried out by the visual receptive fields. Because these receptive fields vary in size, each field is suited for detecting the presence of a specific spatial frequency. With the Hermite transform, field sizes can be modeled by varying the standard deviation σ of the Gaussian envelope,

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while orientation selectivity can be obtained by rotation of the Hermite filters. In practice, the Hermite transform is often limited to the first few terms, which introduces effects of filtering and aliasing. In order for the finite Hermite transform to describe the signal adequately, σ must be properly selected. On the one hand, σ should be as large as possible since integrating over large areas improves the output signal to noise ratio and the efficiency of our signal representation. On the other hand, σ cannot be too large because then the signal cannot be described accurately by the first few terms in the Hermite expansion. The important problem of selecting the right value of σ is the main topic of Martens (1990c). Remark 5.1 The Hermite transform provides the connection between the derivatives of Gaussians and the Hermite functions [see also Eq. (16)]. In the Hermite transform, the analysis functions Dσn are the derivatives of Gaussians, whereas the reconstruction functions Knσ are the Hermite functions. The difference between the Hermite function and the derivative of the Gaussian of order n is the scale of the Gaussian in relation to the scale of the Hermite polynomial. In case of the Hermite functions, the Hermite polynomials grow as fast as the exponential decays and hence the maxima of the Hermite functions all have approximately the same height, giving them the shape of a truncated sine/cosine wave. The Hermite functions have two interesting properties. First, they maximize the uncertainty principle (see subsection 5.3.3) and second, as for the Gaussian, their Fourier transform has the same functional form as the function itself [see Eq. (18)].

B. The Classical Multidimensional Hermite Transform The Hermite transform is generalized to higher dimension in a tensorial manner (for the 2D and 3D case, see Martens, 1990b). Let us start from the Gaussian window function  *σ (x) = V

1 √ πσ

m

 exp

x2 +···+x2 − 1 2σ 2 m

 .

*σ (x) = V *σ (x1 ) . . . V *σ (xm ). This window function is separable; that is, V Naturally, the polynomials

1 Giσ1 −i2 ,i2 −i3 ,...,im (x) =  2i1 (i1 − i2 )!(i2 − i3 )! . . . im ! x x x m 1 2 × Hi1 −i2 Hi2 −i3 . . . Him σ σ σ = Giσ1 −i2 (x1 ) Giσ2 −i3 (x2 ) . . . Giσm (xm )

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*σ (x))2 , that is, are orthonormal with respect to (V

 δi1 ,1 . . . δim ,m =

Rm

σ *σ (x))2 Gσ (V i1 −i2 ,...,im (x) G1 −2 ,...,m (x) dV(x)

for i1 , 1 = 0, . . . , ∞; i2 = 0, . . . , i1 ; 2 = 0, . . . , 1 ; . . . ; im = 0, . . . , im−1 ; m = 0, . . . , m−1 . In a manner similar to the 1D case, we obtain the following decomposition of a signal f (x) into the pattern functions Qσi1 −i2 ,...,im (x):

f (x) =

i1 ∞   i1 =0 i2 =0

im−1

...





im =0 (px1 ,...,pxm )∈P

ciσ1 −i2 ,...,im (px1 , . . . , pxm )

× Qσi1 −i2 ,...,im (x1 − px1 , . . . , xm − pxm ), where (px1 , . . . , pxm ) ranges over all coordinates in a square sampling grid P. Thus, the reconstruction of the signal consists again of interpolating the Hermite coefficients ciσ1 −i2 ,...,im (px1 , . . . , pxm ) with the pattern functions

Qσi1 −i2 ,...,im (x) =

*σ (x) Gσ V i1 −i2 ,...,im (x) = Qσi1 −i2 (x1 ) . . . Qσim (xm ), * σ (x) W

where

* σ (x) = W



*σ (x1 − px , . . . , xm − px ) V m 1

(px1 ,...,pxm )∈P

is the positive weight function. The Hermite coefficients ciσ1 −i2 ,...,im (px1 , . . . , pxm ) are again obtained by convolving the original signal f (x) with the filter functions

*σ (−x))2 Gσ Dσi1 −i2 ,...,im (x) = (V i1 −i2 ,...,im (−x)  m x x 1 m 1 . . . Him Hi1 −i2 = √ i σ σ πσ 2 1 (i1 − i2 )! . . . im !   x2 +···+x2 exp − 1 σ 2 m (−1)i1

= Dσi1 −i2 (x1 ) . . . Dσim (xm )

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followed by a downsampling along the grid P. These filter functions can be written as derivatives of a Gaussian:

Dσi1 −i2 ,...,im (x)



σ i1

= 2i1 (i1 − i2 )! . . . im ! ∂i1 −i2 ∂i2 −i3 i −i2

∂x11

i −i3

∂x22

...

∂im im ∂xm

m 1 √ πσ    2 x12 +···+xm exp − σ 2 .

Now we aim to construct a generating function of these filter functions. Putting

* Fiσ1 (u; x)

 i1   σ 2  1 ∂ ∂ ∂ * (x) V u1 + u2 + · · · + um i1 ! ∂(x1 /σ) ∂(x2 /σ) ∂(xm /σ) m   km−1 i1  k2 i −k k −k  u11 2 u22 3 . . . ukmm ∂i1 −k2 1 ... = √ (i1 − k2 )!(k2 − k3 )! . . . km ! ∂(x1 /σ)i1 −k2 πσ =

k2 =0 k3 =0

×

∂k2 −k3

∂ km ... ∂(xm /σ)km

∂(x2 /σ)k2 −k3 i1  k2 

i1 /2

=2

i −k2

...

 exp

x2 +···+x2 − 1 σ2 m





km

k −k3

u22



km−1

k2 =0 k3 =0

× u11

km =0

1  (i1 − k2 )!(k2 − k3 )! . . . km ! =0

. . . ukmm Dσi1 −k2 ,...,km (x),

we obtain

1

Dσi1 −i2 ,...,im (x) =

2i1 /2

√

∂im *σ 1 ∂i1 −i2 . . . [F (u; x)]. im i1 (i1 − i2 )! . . . im ! ∂ui1 −i2 ∂u m 1

Hence, the function

* Fσ (u; x) =

∞  i1 =0

=

1 2i1 /2

i1 ∞   i1 =0 k2 =0

1 i1 !



∂ ∂ u1 + · · · + um ∂(x1 /σ) ∂(xm /σ)

km−1

...



km

i1  

2 

*σ (x) V

1 i −k u11 2 . . . ukmm Dσi1 −k2 ,...,km (x)  (i1 − k2 )! · · · km ! =0

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generates the filter functions, since

1 Dσi1 −i2 ,...,im (x) = √ (i1 − i2 )! . . . im !



∂i1 −i2 i −i2

∂u11



∂im σ . . . i [* F (u; x)] ∂umm

. u=0

Moreover, we have that

ciσ1 −i2 ,...,im (t) = ( f ∗ Dσi1 −i2 ,...,im )(t)   ∂ im ∂i1 −i2 1 σ . . . i [( f (x) ∗ * F (u; x))(t)] =√ (i1 − i2 )! . . . im ! ∂ui1 −i2 ∂umm 1

.

u=0

So, if we define

cσ (u; t) = ( f (x) ∗ * Fσ (u; x))(t), we have obtained the generating function of the Hermite coefficients as follows:

* cσ (u; t) =

i1 ∞  

im−1

...

i1 =0 i2 =0



×

=

im =0

∂i1 −i2 i −i2

∂u11

i1 ∞   i1 =0 i2 =0



...

1 i −i u 1 2 . . . uimm (i1 − i2 )! . . . im ! 1 ∂im ∂uimm

im−1

...



im =0

√

 σ

[c (u; t)] u=0

1 i −i u 1 2 . . . uimm ciσ1 −i2 ,...,im (t) . (i1 − i2 ) . . . im ! 1

5.2.2. Expansion of the Classical Multidimensional Hermite Filters Into the Generalized Clifford–Hermite Polynomials In this subsection we show how the classical filter functions of the multidimensional Hermite transform can be expressed in terms of Clifford analysis. The starting point is the general term of the defining series of the generating function of the filter functions (see subsection 5.2.1), which, up to constants, may be rewritten as

Fk (u; x) =

1 < u, ∂x >k [V(x)], k!

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where we have put



2 V(x) = exp − |x|2 . Now the Fischer decomposition leads to the following expansion (see Sommen and Jancewicz, 1997):

 1 < u, x >k = us Zk,s (u, x) xs , k! k

s=0

where the functions Zk,s (u, x) are the so-called zonal monogenics. These zonal monogenics are homogeneous of degree k − s in u and x , left monogenic in u and right monogenic in x and have the form

1 Zk−s (u, x), Bs,k−s . . . B1,k−s

Zk,s (u, x) =

with B2s,k = −2s, B2s+1,k = −(2s + 2k + m) and

Zk (u, x) =

( m2 − 1)

(|u||x|)k 2k+1 (k + m2 )   u ∧ x m/2 m/2−1 Ck−1 (t) . × (k + m − 2)Ck (t) + (m − 2) |u||x|

Here Ck denotes the classical Gegenbauer polynomial with variable t = |u||x| . Hence, we obtain that

Fk (u; x) =

k 

s

u

Zk,s (u, ∂x ) ∂xs [V(x)]

s=0

=

k 

  us Zk,s (u, ∂x ) V(x) ∂xs .

s=0

Next, Proposition 2.4 leads to

Fk (u; x) =

k 

(−1)k−s us [Zk,s (u, x) V(x)]∂xs .

s=0

Moreover, as Zk,s (u, x) is a homogeneous right monogenic polynomial of degree k − s in x, the Rodrigues formula in Eq. (10) for the generalized

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Clifford–Hermite polynomials yields

[Zk,s (u, x) V(x)]∂xs = Zk,s (u, x) H s,k−s (x) V(x). Hence, we finally obtain the decomposition

 1 < u, ∂x >k [V(x)] = Fk (u; x) = (−1)k−s us Zk,s (u, x) H s,k−s (x) V(x). k! k

s=0

Substituting

x for x and putting σ

V σ (x) = V

x σ



|x|2 = exp − 2σ 2 ,

we obtain

Fk

k x 1  k 1 k σ u; =σ < u, ∂x > [V (x)] = k (−1)k−s σ s us Zk,s (u, x) σ k! σ s=0

× H s,k−s

x σ

V σ (x).

As

1 πm/2 σ m−i1 √2σ < u, ∂x >i1 [V σ (x)] = (i −m)/2 * Fi1 (u; x), i1 ! 2 1 we have finally obtained the following decomposition of the classical filter functions into the generalized Clifford–Hermite polynomials: √

2σ Di1 −i (x) = 2 ,...,im

1 2i1 /2

∂im *√2σ 1 ∂i1 −i2 . . . [Fi1 (u; x)] √ im (i1 − i2 )! . . . im ! ∂ui1 −i2 ∂u m 1

i1  1 1 1 (−1)i1 −s σ s = √ (2π)m/2 σ m+i1 (i1 − i2 )! . . . im ! s=0

∂i1 −i2 i −i2

∂u11

...

∂im ∂uimm

[us Zi1 ,s (u, x)] H s,i1 −s

x σ

V σ (x).

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5.2.3. The Generalized Clifford–Hermite Transform In this subsection we develop a new multidimensional Hermite transform directly in Clifford analysis; we call it the generalized Clifford–Hermite transform. First, using the Gaussian window function

|x|2 V σ (x) = exp − 2σ 2 , we get the following decomposition of the original real-valued signal f (x):

f (x) =

 1 f (x) V σ (x − p) W σ (x)

(51)

p∈P

with

W σ (x) =



V σ (x − p)

p∈P

the positive weight function and P a sampling grid in R m .  For the decomposition of the filtered localized signal f (x) V σ (x − p) Fil , we use the generalized Clifford–Hermite polynomials Hn,k (x), which satisfy the orthogonality relation [see Eq. (13)]

 Rm

Pk† (x)Hn,k

√ 2x σ

√ 2x σ

Hn ,k

Pk (x)(V σ (x))2 dV(x) =

σ 2k+m 2(2k+m)/2 × γn,k δn,n δk,k .

The above orthogonality relation leads to the following decomposition of the filtered localized signal into the orthogonal generalized Clifford– √ σ (x) = V σ (x) H Hermite functions Kn,k n,k





σ

V (x − p) f (x)

Fil

=

∞  ∞ 

2x σ

Pk (x) (see also Section 3.3):



† σ σ cj, (p) Kj, (x − p) ,

(52)

j=0 =0

where we have put σ cn,k (p) =

=

2(2k+m)/2 σ 2k+m γn,k 2(2k+m)/2 σ 2k+m γn,k

√

 Rm



Rm

f (x)Hn,k

2(x−p) σ



Pk (x − p)(V σ (x − p))2 dV(x)

σ f (x) Kn,k (x − p) V σ (x − p) dV(x).

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σ (p) the generalized Clifford–Hermite coefficients. We call cn,k Combining formulas (51) and (52), we obtain the following decomposition of the filtered signal:

fFil (x) =

∞  ∞  

σ cj, (p) Qσj, (x − p),

j=0 =0 p∈P

where we have introduced the pattern functions

†

Qσj, (x)

=

σ (x) Kj,

W σ (x)

=

P† (x) H j,

√ 2x σ

V σ (x)

W σ (x)

.

Note that the generalized Clifford–Hermite coefficients may be expressed as the convolution of the original signal f (x) with the generalized Clifford–Hermite filter functions Dσn,k :

  σ cn,k (p) = f ∗ Dσn,k (p) with

Dσn,k (x) =

√ 2(2k+m)/2 H − σ2x Pk (−x) (V σ (−x))2 n,k 2k+m σ γn,k

=

√

(−1)n+k 2(2k+m)/2 2x |x|2 P H (x) exp − n,k k σ σ2 σ 2k+m γn,k

=



 2 2(2k+m−n)/2 σ n−m−2k P (−1)k ∂xn exp − |x| (x) . k σ2 γn,k

(53)

The last expression in Eq. (53) is obtained by using the Rodrigues formula [Eq. (10)] of the generalized Clifford–Hermite polynomials. Note that the parameters of the generalized Clifford–Hermite filters are the scale σ of the Gaussian, the derivative order n and the order k of the left solid inner spherical monogenic. Moreover, their Fourier transform in spherical coordinates ξ = ρη, ρ = |ξ|, η ∈ Sm−1 , is given by

F [Dσn,k ](ξ) =

2 2 ik+n σ n n n+k η P (η) ρ exp − σ 4ρ . k γn,k 2n/2

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Hence, the generalized Clifford–Hermite filters are polar separable, this means that their Fourier transform is expressed as the product of a spatial frequency tuning function and an orientation tuning function. Daugman (1983) has demonstrated the importance of polar separable filters.

5.2.4. Connection With the Clifford–Hermite Continuous Wavelet Transform The continuous wavelet transform (CWT) is a signal analysis technique suitable for nonstationary, inhomogeneous signals for which Fourier analysis is inadequate (see Chui, 1992, and Daubechies, 1992). In the 1D case, it is given by the integral transform

 f (x)  −→ F(a, b) =

+∞ 

−∞

c

ψa,b (x)

f (x) dx.

The kernel function of this integral transform is the dilate translate of a mother wavelet ψ:

  1 x−b , a > 0, b ∈ R, ψa,b (x) = √ ψ a a where the parameter b indicates the position of the wavelet, whereas the parameter a governs its frequency. The analyzing wavelet function ψ is a quite arbitrary L2 -function that is well localized in both the time domain and the frequency domain. Moreover, it must satisfy the admissibility condition:

 Cψ ≡ 2π

+∞

−∞

|F [ψ](ξ)|2 dξ < ∞. |ξ|

The constant Cψ is called the admissibility constant. In the case where ψ is also in L1 , this admissibility condition implies that ψ has “zero momentum,” that is,



+∞

−∞

ψ(x) dx = 0,

which can be fulfilled only if ψ is an oscillating function, explaining the terminology “wavelet.” A wavelet is a function that oscillates like a wave in a limited portion of time or space and vanishes outside of it:—it is a wavelike but localized function. Higher-dimensional CWTs, still enjoying the same properties as in the 1D case, traditionally originate as tensor products of 1D phenomena, an

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exception being the 2D CWT incorporating rotation as well as translation and dilation (see Antoine, Murenzi, and Vandergheynst , 1996). Brackx and Sommen (2001), introduced the so-called generalized Clifford–Hermite CWT. The building blocks in the construction of this multidimensional CWT are the generalized Clifford–Hermite polynomials. The mother wavelets of this transform take the form





2 2 ψn,k (x) = exp − |x|2 Hn,k (x) Pk (x) = (−1)n ∂xn exp − |x|2 Pk (x) , n = 1, 2, . . . Introducing the continuous family of wavelets a,b,s ψn,k (x)

=



1 am/2

s ψn,k

s(x − b)s a

 s,

the generalized Clifford–Hermite CWT is defined by

Fn,k (a, b, s) 

† a,b,s = ψn,k (x) f (x) dV(x) Rm

=



1 am/2



†

2 s(x−b)s s(x−b)s P s exp − |x−b| H s f (x) dV(x), k n,k a a 2a2

Rm

(54)   with f ∈ L2 Rm , dV(x) the signal to be analyzed, a ∈ R+ the dilation parameter, b ∈ Rm the translation parameter, and s ∈ SpinR (m) the spinorrotation parameter. The original signal may be reconstructed from its transform Fn,k (a, b, s) by the inverse transformation,

1 f (x) = Cn,k





SpinR (m) Rm

 0

+∞

a,b,s

ψn,k (x) Fn,k (a, b, s)

da am+1

dV(b) ds, (55)

with m

Cn,k = (2π)

(n + k − 1)! 2

the admissibility constant.

 Sm−1

|Pk (η)|2 dS(η) < +∞

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Fred Brackx et al.

√ Putting σ = 2μ in the generalized Clifford–Hermite coefficients of Subsection 5.2.3 σ cn,k (p) =

=

 Rm

Dσn,k (p − x) f (x) dV(x)

2(2k+m)/2 σ 2k+m γn,k

 Rm

  √

|x−p|2 Hn,k σ2 (x − p) exp − σ 2

  × Pk x − p f (x) dV(x), we obtain √

2μ cn,k (p)

=



1 μk+m γn,k



|x−p|2

exp − 2μ2 Rm

 Hn,k

x−p x−p μ Pk μ f (x) dV(x). (56)

Leaving the spinor-rotations in the generalized Clifford–Hermite CWT out of consideration, by comparing Eqs. (54) and (56), the connection between the generalized Clifford–Hermite transform and the ditto CWT is clear: σ plays the role of the dilation parameter a, p plays the role of the translation parameter b, and the translated filter functions Dσn,k (p − x) play the role of a,b,s the wavelets ψn,k . However, differences exist between the two transforms. The wavelet translation parameter b is continuous, whereas the parameter p in the Hermite transform is discrete. Furthermore, the reconstruction of the filtered signal in the Hermite transform

fFil (x) =

∞  ∞  

σ cj, (p) Qσj, (x − p)

(57)

j=0 =0 p∈P

depends on σ, which is not the case for the CWT. As mentioned previously, σ is an important parameter for practical applications. In the CWT, the quality of the signal reconstruction [Eq. (55)] depends on the mother wavelet ψn,k (x); in other words, it depends on the order of the derivative of the Gaussian modulated by Pk and of the order k of the left inner spherical monogenic. In the inverse Hermite transform [Eq. (57)], we sum over all these orders.

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5.3. The Two-Dimensional Clifford–Gabor Filters 5.3.1. The Classical One-Dimensional Gabor Filters The Gabor filter is one of the most prominent tools for local spectral image processing and analysis. Gabor first introduced the filters in the field of 1D signal processing in 1946 for a joint time-frequency analysis. Gabor filters have the primary advantage of being simultaneously optimally localized in the spatial and in the frequency domain. Hence, spatial and frequency properties are optimally analyzed at the same time by Gabor filters (see also Subsection 5.3.3). Gabor filters also give access to the local phase of a signal. A close correspondence has been shown between the local structure of a signal and its local phase. Furthermore, certain regions in the human visual cortex can be modeled as Gabor filters (see Section 5.1). Hence, Gabor filters conform well to the human visual system’s capabilities. Gabor filters have been successfully applied to different image processing and analysis tasks such as texture segmentation (see Bülow, 1999; Weldon, Higgins, and Dunn, 1996), edge detection, and local phase and frequency estimation for image matching.

a. One-dimensional complex Gabor filters. In this subsection we consider the classical Fourier transform with the angular frequency in the kernel function 1 F [ f ](u) = √ 2π



+∞

−∞

f (x) exp (−i2πux) dx.

Complex Gabor filters are closely related to Fourier analysis in the following way. They are linear shift-invariant (LSI) filters; hence, they can be applied by simply convolving the signal with the impulse response of the filter. The impulse response h of a complex Gabor filter is the complex conjugated integral kernel of the classical Fourier transform F of some frequency u∗ multiplied by a Gaussian g centered at the origin; that is,

h(x) = g(x) exp (i2πu∗ x) with



x2 g(x) = N exp − 2σ 2 .

(58)

Whereas Gabor filters analyze the local spectral properties of a signal, the Fourier transform decomposes a signal into its global spectral components. Hence, the Fourier transform can be considered the basis on which Gabor filters were introduced. The parameters of the complex Gabor filter are the normalization constant, N, the center frequency, u∗ , and the variance or standard

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deviation, σ, of the Gaussian. Normally, N is chosen such that the Gaussian is amplitude normalized

  g(x) = 1



+∞ −∞

g(x) dx = 1,

which implies N = √ 1 . 2πσ Sometimes other parameterizations of the complex Gabor filter than the one given above are used, viz.

h(x) = g(x) exp (iξ ∗ x) = g(x) exp



icx σ

.

Here ξ ∗ = 2πu∗ is the angular frequency and c = ξ ∗ σ is the oscillation parameter. The transfer function of a complex Gabor filter is a shifted Gaussian:

  1 H(u) := F [h](u) = √ exp − 2π2 σ 2 (u − u∗ )2 . 2π In terms of the angular frequency, ξ = 2πu, this becomes

2 1 H(ξ) = √ exp − σ2 (ξ − ξ ∗ )2 . 2π Hence, Gabor filters are bandpass filters. The majority of energy of the Gabor filter is centered around the frequency, u∗ , in the positive half of the frequency domain. Analogously, the definition of the 2D complex Gabor filters is based on the classical 2D Fourier transform.

b. One-dimensional real Gabor filters. In addition to complex Gabor filters, real Gabor filters also appear in the literature (see Rivero-Moreno and Bres, 2003a, b). Naturally they are obtained as the real and imaginary part of the complex Gabor filters introduced above. Hence, the impulse responses of these real Gabor filters take the form cx hc (x) = g(x) cos (2πu∗ x) = g(x) cos (ξ ∗ x) = g(x) cos σ and

hs (x) = g(x) sin (2πu∗ x) = g(x) sin (ξ ∗ x) = g(x) sin with g the Gaussian given by Eq. (58).

cx σ

,

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Their associated transfer functions are

 σ exp (−σ 2 2π2 (u + u∗ )2 ) Hc/s (u) := F [hc/s ](u) = N 2  ± exp (−σ 2 2π2 (u − u∗ )2 ) , where the plus sign and the minus sign, correspond, respectively, with the cosine and the sine, Gabor filter. The above expression can be rewritten in terms of the angular frequency as





 σ (ξ+ξ ∗ )2 (ξ−ξ ∗ )2 2 2 Hc/s (ξ) = N ± exp −σ . exp −σ 2 2 2

5.3.2. Different Types of Two-Dimensional Gabor Filters a. Quaternionic Gabor filters. In Bülow (1998, 1999), quaternionic Gabor filters were constructed and applied to the problems of disparity estimation and texture segmentation. The impulse response hq of a quaternionic Gabor filter is a Gaussian windowed kernel function of the quaternionic Fourier transform mentioned in Section 4.1: hq (x) = g(x) exp (i2πu∗1 x1 ) exp ( j2πu∗2 x2 ) with

 g(x) = N exp

x2 +(x )2 − 1 2σ 2 2

 .

(59)

The parameter  is the aspect ratio. In the quaternionic frequency domain, these Gabor filters are shifted Gaussians:



 q

q

q

H (u) := F [h ](u) = exp

−2π2 σ 2

(u1 − u∗1 )2

+

(u2 −u∗2 )2 2

 .

b. Gabor filters of Ebling and Scheuermann. Ebling and Scheuermann, (2004) introduce 2D and 3D Gabor filters based on their Clifford–Fourier transforms (see Subsection 4.5.3) and use them to describe local patterns in flow fields. The impulse responses he of their 2D Gabor filters take the

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form

he (x) = g(x ) exp (e12 (ξ1∗ x1 + ξ2∗ x2 )), with x a rotated version of x and g the Gaussian given by Eq. (59).

5.3.3. The Two-Dimensional Clifford–Gabor Filters a. Definition. As we dispose of a closed form for the integral kernel of our 2D Clifford–Fourier transform (see Subsection 4.5.1), we are now able to define a new type of 2D Gabor filters (see Brackx, De Schepper, and Sommen, 2006a). Definition 5.1 The 2D Clifford–Gabor filters G ± are linear shift-invariant filters with impulse response given by



h± (x) = g(x) exp ±(x ∧ ξ ∗ ) = g(x) cos (x1 ξ2∗ − x2 ξ1∗ ) ± e12 g(x) × sin (x1 ξ2∗ − x2 ξ1∗ ), where g is the Gaussian given by

g(x) =

1 |x|2 . exp − 2 2σ 2πσ 2

The parameters of the Clifford–Gabor filters are the angular frequency ξ ∗ and the variance σ, which determines the scale of the Gaussian envelope. It turns out that both types of Clifford–Gabor filters G ± have the same transfer function. Proposition 5.1 The transfer function of the Clifford–Gabor filter G ± is given by

H ± (ξ) := FH± [h± ](ξ) =

2 1 exp − σ2 |ξ − ξ ∗ |2 . 2π

Proof. By means of the modulation theorem of the 2D Clifford–Fourier transform (see Proposition 4.9 (iv)), we have



 H ± (ξ) = FH± exp x ∧ (±ξ ∗ ) g(x) (ξ) = FH± [ g(x)](ξ − ξ ∗ ) = F [ g(x)](ξ − ξ ∗ ), since the Gaussian g(x) is a radial function.

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163

The desired result now follows from

  1 |ξ|2 2 exp −σ 2 . F [ g(x)](ξ) = 2π Note that, similar to the classical case, the transfer functions H ± (ξ) are shifted Gaussians, which implies that the Clifford–Gabor filters G ± are bandpass filters. Remark 5.2 In a similar way, one obtains

FH± [h∓ ](ξ) =

2 1 exp − σ2 |ξ + ξ ∗ |2 . 2π

b. Relationship with other Gabor filters. Using the properties of the Clifford numbers P± introduced in Subsection 4.5.1, we can derive an explicit connection between the 2D Clifford–Gabor filters G ± and the classical complex Gabor filters (see Subsection 5.3.1). By a straightforward computation, we obtain h± (x) = P+ h(∓e12 x) + P− h(±e12 x), where

  h(x1 , x2 ) = g(x) exp i(ξ1∗ x1 + ξ2∗ x2 ) is the classical 2D Gabor filter in case of a symmetric Gaussian. Naturally the Clifford–Gabor filters G ± can also be expressed in terms of classical 1D Gabor filters which, for the sake of clarity, are now denoted with a superscript specifying the angular frequency: ∗

hξ (x) = g(x) exp (iξ ∗ x). It is easily seen that ∗

∗

∗

∗

h+ (x) = P+ h−ξ2 (x1 ) hξ1 (x2 ) + P− hξ2 (x1 ) h−ξ1 (x2 ). A similar result holds for the impulse response of G − . Finally, let us look for a relationship between the Clifford–Gabor filters G ± and the Gabor filters of Ebling and Scheuermann in case of a symmetric Gaussian (see Subsection 5.3.2).

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We have

  he (x) = g(x) exp e12 (ξ1∗ x1 + ξ2∗ x2 ) = h± (∓e12 x). c. Localization in the spatial and in the frequency domain. An often-cited property of Gabor filters is their optimal simultaneous localization in the spatial and the frequency domain. This makes them suitable for local frequency analysis. The notion “optimal simultaneous localization” is formalized by the uncertainty principle, which in its most cited form states that a nonzero function and its Fourier transform cannot both be sharply localized. The uncertainty principle appeared in 1927 under the name Heisenberg inequality in the field of quantum mechanics in Heisenberg’s paper (1927). However, it also has a useful interpretation in classical physics; namely, it expresses a limitation on the extent to which a signal can be both timelimited and band-limited. This aspect of the uncertainty principle was already expounded by Wiener in a lecture in Göttingen in 1925. Unfortunately, no written record of this lecture seems to have survived, apart from the nontechnical account in Wiener’s autobiography (1956). The uncertainty principle became really fundamental in the field of signal processing after the publication of Gabor’s famous 1946 article. For a 1D complex-valued signal f , the uncertainty principle takes the form x ξ ≥

1 . 2

(60)

Here x denotes the width or spatial uncertainty of f , defined as the square root of the variance of the energy distribution of f

+ +∞

x2 f (x) f c (x) dx . (x) = −∞ + +∞ c (x) dx f (x) f −∞ 2

Analogously, the bandwidth ξ is given by

+ +∞ 2

(ξ) =

 c 2 −∞ ξ F [ f ](ξ) F [ f ](ξ) dξ . + +∞  c −∞ F [ f ](ξ) F [ f ](ξ) dξ

The functions that minimize the inequality (60) are the complex Gabor filters. Hence, depending on the parameter σ, the Gabor filters are better localized in position or frequency space but they always exhibit the best possible joint localization.

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Daugman extended the uncertainty principle to 2D complex-valued filters or signals (see Daugman, 1985):

x1 x2 ξ1 ξ2 ≥

1 , 4

(61)

where x1 is defined by

+

x12 f (x1 , x2 ) f c (x1 , x2 ) dV(x) . c R2 f (x1 , x2 ) f (x1 , x2 ) dV(x)

(x1 ) = R+

2

2

The uncertainties x2 , ξ1 , and ξ2 are defined analogously. It can be shown that 2D complex Gabor filters achieve the minimum product of uncertainties; that is,

x1 x2 ξ1 ξ2 =

1 . 4

Let us now consider 2D Clifford algebra–valued functions:

f : R2 −→ C2 x = (x1 , x2 ) −→ f (x) = f (x1 , x2 ) = f0 (x) + f1 (x) e1 + f2 (x) e2 + f12 (x) e12 , with fα : R2 −→ C, α = 0, 1, 2, 12. First, we extend the definition of the uncertainties to these Clifford algebra–valued functions as follows:

+ 2

(x1 ) = and

+ (ξ1 )2 =

x12 [ f (x) f † (x)]0 dV(x) † R2 [ f (x) f (x)]0 dV(x)

R+2



† ξ12 [FH± [ f ](ξ) FH± [ f ](ξ) ]0 dV(ξ) .

† + R2 [FH± [ f ](ξ) FH± [ f ](ξ) ]0 dV(ξ)

R2

Analogous definitions hold for x2 and ξ2 . For complex-valued signals, the real-valued energy distribution is given by |f |2 = ff c . For Clifford algebra–valued signals, this is given by

| f (x)|2 = [ f (x) f † (x)]0 = | f0 (x)|2 + | f1 (x)|2 + | f2 (x)|2 + | f12 (x)|2 .

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Hence, the uncertainty relation for 2D Clifford algebra–valued signals is identical to Daugman’s relation [Eq. (61)]. In case of the Clifford–Gabor filters G ± , we have



±

∗

2

∗



|h (x)| = g(x) exp (±(x ∧ ξ )) g(x) exp (∓(x ∧ ξ )) = [(g(x))2 ]0 =

0

2



1 , exp − |x| σ2 4π2 σ 4

where we have used the fact that



exp (±(x ∧ ξ ∗ ))

†

= exp (∓(x ∧ ξ ∗ )).

Hence, for G ± we obtain

2 dV(x) x12 exp − |x| σ2 σ2

= . (x1 )2 = + |x|2 2 exp − dV(x) 2 2 R σ +

R2

Furthermore, we have

  FH± [h± ](ξ)2 = 1 exp (−σ 2 |ξ − ξ ∗ |2 ), 4π2 which yields

+

(ξ1 )2 =

2 R2 ξ 1

+

R2

exp (−σ 2 |ξ − ξ ∗ |2 ) dV(ξ)

exp (−σ 2 |ξ

− ξ ∗ |2 )

dV(ξ)

=

1 . 2σ 2

Summarizing, the uncertainties of the Clifford–Gabor filters G ± are given by

σ x1 = x2 = √ 2

and

1 ξ1 = ξ2 = √ , 2σ

which implies

x1 x2 ξ1 ξ2 =

1 . 4

Hence, as in the classical setting, the Clifford–Gabor filters G ± are jointly optimally localized in the spatial and in the frequency domain.

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167

6. THE CYLINDRICAL FOURIER TRANSFORM In Section 4 we defined the Clifford–Fourier transform as an operator exponential. In only the 2D case, we were able to write it as a standard integral transform with a closed form for the integral kernel function— namely, the exponential of the wedge product of the old and new vector variable. Now taking a generalization of this specific 2D Clifford–Fourier kernel as a new integral kernel, we are able to introduce a new multidimensional Fourier transform. As the phase of the new integral kernel takes constant values on coaxial cylinders, we call this new Fourier transform the cylindrical Fourier transform.

6.1. Definition The cylindrical Fourier transform is obtained by substituting for the standard inner product in the classical exponential Fourier kernel a wedge product of the old and new vector variable as argument. Definition 6.1 The cylindrical Fourier transform of a function f is given by

1 Fcyl [ f ](ξ) = (2π)m/2 with exp(x ∧ ξ) =

∞

r=0

 Rm

exp (x ∧ ξ) f (x) dV(x)

(x∧ξ)r r! .

Remark 6.1 As is to be expected, the Clifford–Fourier transform and the cylindrical Fourier transform reduce to the same integral transform in the 2D case as follows:

FH± [ f ](ξ) = Fcyl [ f ](∓ξ). In the sequel, we often appeal to the following basic formulas. Lemma 6.1 For all x, t ∈ Rm , one has

(x ∧ t)2 = −|x ∧ t|2 = (< x, t >)2 − |x|2 |t|2 . Proof. First, by definition of the Clifford norm, we have

|x ∧ t|2 = [(x ∧ t)† (x ∧ t)]0 = −[(x ∧ t)2 ]0 . Next, let us decompose t as t = t + t⊥ , where t , respectively t⊥ , denotes the component of t which is parallel, respectively perpendicular, to x. This

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implies that

x ∧ t = x ∧ t⊥ = xt⊥ = −t⊥ ∧ x = −t⊥ x , which in turn leads to

(x ∧ t)2 = xt⊥ xt⊥ = −t⊥ x2 t⊥ = |x|2 t2⊥ = −|x|2 |t⊥ |2 . We thus have proved that (x ∧ t)2 is scalar valued and consequently we obtain that |x ∧ t|2 = −(x ∧ t)2 . We now prove the second part of the statement, namely,

|x ∧ t|2 = |x|2 |t|2 − (< x, t >)2 . Using spherical coordinates t = |t| η with η ∈ Sm−1 , we find |x ∧ t|2 = |t|2 |x ∧ η|2 . Now, we decompose x into its components parallel and perpendicular to η:

x = x + x⊥ = < x, η > η + x⊥ . By a similar reasoning as above, this yields

(x ∧ η)2 = x⊥ η x⊥ η = −x2⊥ η2 = x2⊥ = −|x⊥ |2 . Hence, we obtain

|x ∧ η|2 = −[(x ∧ η)2 ]0 = −(x⊥ )2 . Moreover, we have consecutively

(x⊥ )2 = (x− < x, η > η)(x− < x, η > η) = −|x|2 − < x, η > (ηx + xη) − (< x, η >)2 = −|x|2 − < x, η > (−2 < x, η >) − (< x, η >)2 = −|x|2 + (< x, η >)2 . So, finally we indeed find

  |x ∧ t|2 = |t|2 |x|2 − (< x, η >)2 = |x|2 |t|2 − (< x, t >)2 .

The Fourier Transform in Clifford Analysis

169

This result enables us to rewrite the integral kernel of our newly introduced Fourier transform in terms of the sine and cosine function. Proposition 6.1 The kernel of the cylindrical Fourier transform can be rewritten as

exp (x ∧ ξ) = cos (|x ∧ ξ|) + I sin (|x ∧ ξ|) = cos (|x ∧ ξ|) + x ∧ ξ sinc(|x ∧ ξ|), where I =

x∧ξ |x∧ξ|

and sinc(x) :=

sin (x) x

the unnormalized sinc function.

Proof. Splitting the defining series expansion of exp (x ∧ ξ) into its even and odd part yields

exp (x ∧ ξ) =

∞  (x ∧ ξ)r r=0

r!

=

∞  (x ∧ ξ)2 =0

(2)!

+

∞  (x ∧ ξ)2+1 =0

(2 + 1)!

∞ ∞  |x ∧ ξ|2 (x ∧ ξ)  |x ∧ ξ|2+1  = + (−1) (−1) (2)! |x ∧ ξ| (2 + 1)! =0

=0

= cos (|x ∧ ξ|) + I sin (|x ∧ ξ|). Remark 6.2 1. From Proposition 6.1 it is clear that the cylindrical Fourier kernel is parabivector valued. 2. As I satisfies I 2 = −1, it can be viewed as a kind of imaginary unit. Let us end this subsection by explaining why we have chosen the name cylindrical for our new Fourier transform. To that end, we rewrite Lemma 6.1 as follows:

  ,ξ)2 |x ∧ ξ|2 = |x|2 |ξ|2 − (< x, ξ >)2 = |x|2 |ξ|2 1 − cos (x, ,ξ) . = |x|2 |ξ|2 sin (x, 2

,ξ) is Hence, for ξ fixed, the “phase” |x ∧ ξ| is constant if and only if |x| sin (x, constant. In other words, for ξ fixed, the phase |x ∧ ξ| is constant on coaxial cylinders w.r.t. ξ (Figure 2). For comparison, in the case of the classical Fourier transform, for ξ fixed ,ξ) is constant if and only if |x| cos (x, ,ξ) the phase < x, ξ > = |x||ξ| cos (x,

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x x sin(x,␰)

␰

FIGURE 2 to ξ.

For fixed ξ, the phase |x ∧ ξ| is constant on coaxial cylinders with regard

x

x cos(x, ␰)

␰

FIGURE 3 For fixed ξ, the phase < x, ξ > is constant on planes perpendicular to ξ.

is constant. Hence, for ξ fixed, the level surfaces of the traditional Fourier kernel are planes perpendicular to that fixed vector (Figure 3).

6.2. Properties Let us start by showing that the cylindrical Fourier transform Fcyl [ f ] is   well defined for each integrable function f ∈ L1 Rm , dV(x) .

The Fourier Transform in Clifford Analysis

  Theorem 6.1 Let f ∈ L1 Rm , dV(x) .   m C0 R , dV(x) and moreover

  Fcyl [ f ]

∞

Then

171

  Fcyl [ f ] ∈ L∞ Rm , dV(x) ∩

 m/2   2 f . ≤2 1 π

Proof. Taking into account Proposition 6.1, the proof is similar to the one of Theorem 4.2. Next, we collect some operational formulas satisfied by the cylindrical Fourier transform.   Proposition 6.2 Let f , g ∈ L1 Rm , dV(x) . The cylindrical Fourier transform satisfies (i) the linearity property

Fcyl [ f λ + gμ] = Fcyl [ f ] λ + Fcyl [ g] μ

λ, μ ∈ Cm

(ii) the reflection property

Fcyl [ f (−x)](ξ) = Fcyl [ f (x)](−ξ) (iii) Hermitean conjugation



†

1 = (2π)m/2

Fcyl [ f ](ξ)

 Rm

f † (x) exp (ξ ∧ x) dV(x)

(iv) the change of scale property

  ξ 1 for a ∈ R+ Fcyl [ f (ax)](ξ) = m Fcyl [ f (x)] a a (v) the differentiation rule

Fcyl [∂x [ f (x)]](ξ) = −ξ Fcyl [ f (x)](−ξ) + (2 − m) ξ  × with sinc(x) :=

sin (x) x

Rm

sinc(|x ∧ ξ|) f (x) dV(x)

the unnormalized sinc function.

1 (2π)m/2

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(vi) the multiplication rule

Fcyl [x f (x)](ξ) = −∂ξ [Fcyl [ f (x)](−ξ)] + (2 − m)

1 (2π)m/2

 ×

Rm

sinc(|x ∧ ξ|) x f (x) dV(x)

(vii) the transfer formula





Rm

†

Fcyl [ f ](ξ)

 g(ξ) dV(ξ) =

Rm

f † (ξ) Fcyl [ g](ξ) dV(ξ)

(viii) the rotation rule

Fcyl [ f (sxs)](ξ) = s Fcyl [sf (x)](sξs) with s ∈ SpinR (m). Proof. (i) Trivial. (ii) Straightforward. (iii) This result can be proved taking into account that x ∧ ξ = ξ ∧ x, which implies that

exp (x ∧ ξ) = cos (|x ∧ ξ|) +

ξ∧x |x ∧ ξ|

sin (|x ∧ ξ|) = exp (ξ ∧ x).

(iv) By means of the substitution u = ax, we have

Fcyl [ f (ax)](ξ) =

1 (2π)m/2

 Rm

1 1 = m a (2π)m/2

exp (x ∧ ξ) f (ax) dV(x) 

 Rm

exp u ∧

  ξ 1 . = m Fcyl [ f (x)] a a

ξ a

 f (u) dV(u)

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The Fourier Transform in Clifford Analysis

(v) First, by means of the Clifford–Stokes theorem (see Theorem 2.1), we obtain

 1 Fcyl [∂x [ f (x)]](ξ) = exp (x ∧ ξ) ∂x [ f (x)] dV(x) (2π)m/2 Rm  1 = exp (x ∧ ξ) dσx f (x) (2π)m/2 ∂Rm  1 − [exp (x ∧ ξ)]∂x f (x) dV(x) (2π)m/2 Rm  1 =− [exp (x ∧ ξ)]∂x f (x) dV(x). (2π)m/2 Rm Next, a straightforward computation yields

  x∧ξ sin (|x ∧ ξ|) ∂x [exp (ξ∧x)] = ∂x cos (|x ∧ ξ|) − |x ∧ ξ| = − sin (|x∧ξ|) ∂x [|x∧ξ|]− +

sin (|x∧ξ|) |x∧ξ|2

cos (|x∧ξ|) |x∧ξ|

∂x [|x∧ξ|](x∧ξ) −

∂x [|x∧ξ|](x∧ξ)

sin (|x ∧ ξ|) |x ∧ ξ|

∂x [x ∧ ξ] (62)

and

∂x [x ∧ ξ] = ∂x [xξ+ < x, ξ >] = −mξ + ξ = (1 − m) ξ.

(63)

Furthermore, in view of Lemma 6.1 we also have

∂x [|x ∧ ξ|2 ] = ∂x [|x|2 |ξ|2 − (< x, ξ >)2 ] = 2x |ξ|2 − 2 < x, ξ > ξ = −2ξ (ξx+ < x, ξ >) = −2ξ (ξ ∧ x) = 2ξ (x ∧ ξ). Combining the above result with

∂x [|x ∧ ξ|2 ] = 2 |x ∧ ξ| ∂x [|x ∧ ξ|], we obtain

∂x [|x ∧ ξ|] = ξ

(x ∧ ξ) |x ∧ ξ|

.

(64)

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Fred Brackx et al.

Inserting Eqs. (63) and (64) in (62) and using (x ∧ ξ)2 = −|x ∧ ξ|2 yields

∂x [exp (ξ ∧ x)] = ξ cos (|x ∧ ξ|) − ξ ×ξ

(x ∧ ξ)

sin (|x ∧ ξ|) − (2 − m)

|x ∧ ξ|

sin (|x ∧ ξ|) |x ∧ ξ|

= ξ exp (ξ ∧ x) − (2 − m) ξ

sin (|x ∧ ξ|) |x ∧ ξ|

.

Taking the Hermitean conjugate of the above result, we obtain

[exp (x ∧ ξ)]∂x = exp (x ∧ ξ) ξ + (m − 2) ξ = ξ exp (ξ ∧ x) + (m − 2) ξ

sin (|x ∧ ξ|) |x ∧ ξ| sin (|x ∧ ξ|) |x ∧ ξ|

.

Hence, we finally have

1 Fcyl [∂x [ f (x)]](ξ) = −ξ (2π)m/2 + (2 − m) ξ

 Rm

exp (ξ ∧ x) f (x) dV(x)

1 (2π)m/2

 sinc(|x ∧ ξ|) f (x) dV(x)

Rm

1 = −ξ Fcyl [ f (x)](−ξ) + (2 − m) ξ (2π)m/2  × sinc(|x ∧ ξ|) f (x) dV(x). Rm

(vi) As

∂ξ [exp (ξ ∧ x)] = − exp (x ∧ ξ) x − (m − 2) x

sin (|ξ ∧ x|) |ξ ∧ x|

we indeed obtain

 1 Fcyl [xf (x)](ξ) = exp (x ∧ ξ) x f (x) dV(x) (2π)m/2 Rm  1 =− ∂ξ [exp (ξ ∧ x)] f (x) dV(x) (2π)m/2 Rm

,

The Fourier Transform in Clifford Analysis

1 + (2 − m) (2π)m/2

175

 Rm

sinc(|ξ ∧ x|) x f (x) dV(x)

1 = −∂ξ [Fcyl [ f (x)](−ξ)] + (2 − m) (2π)m/2  × sinc(|x ∧ ξ|) x f (x) dV(x). Rm

(vii) Similar to the proof of the transfer formula of the 2D Clifford–Fourier transform (see Proposition 4.9). (viii) See the proof of the rotation rule for the 2D Clifford–Fourier transform (Proposition 4.9). Remark 6.3 1. The differentiation and multiplication rules, once more, indicate that the 2D case is special. 2. Note that the shift theorem, the modulation theorem, and the convolution theorem (see Proposition 4.9 for the 2D case) do not hold in the general m-dimensional case, since

(u ∧ ξ) (x ∧ ξ) = ±(x ∧ ξ) (u ∧ ξ) for m ≥ 3.

6.3. Cylindrical Fourier Spectrum of the L2 -Basis Consisting of Generalized Clifford–Hermite Functions In this final subsection, we aim at calculating the cylindrical Fourier spectrum of the L2 -basis

φs,k,j (x) =

√ 2m/4 (j) √ |x|2 H ( 2x) P ( 2x) exp − s,k 2 ; k (γs,k )1/2  + s, k ∈ N, j ≤ dim(M (k))

consisting of generalized Clifford–Hermite functions.Note that these basis elements belong to the space S(Rm ) ⊂ L1 Rm , dV(x) . Hence, their cylindrical Fourier image should be a bounded and continuous function (see Theorem 6.1). The calculation method is based on the Funk-Hecke theorem in space (see Theorem 2.4), which needs dimension m > 2. Moreover, introducing spherical coordinates

x = rω,

ξ = ρη,

r = |x|,

ρ = |ξ|,

ω, η ∈ Sm−1 ,

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denoting < ω, η > = tη and taking into account Lemma 6.1 and Proposition 6.1, we can rewrite the cylindrical Fourier kernel as follows:

exp (x ∧ ξ) = cos rρ 1 − tη2 − ξ ∧ x sinc rρ 1 − tη2

= cos rρ 1 − tη2 − rρ tη sinc rρ 1 − tη2

− rρ η ω sinc rρ 1 − tη2 .

(65)

Let us denote these three terms in the decomposition of the kernel by A, B, and C—we put

A = cos rρ 1 − tη2 ,

B = rρ tη sinc rρ 1 − tη2 ,

C = rρ η ω sinc rρ 1 − tη2 .

Note that A and C are even functions in tη , while B is an odd function in tη .

6.3.1. The Cylindrical Fourier Spectrum of the Gaussian (= φ0,0,1 ) By means of the above decomposition of the integral kernel, we have that



Fcyl exp



2 − |x|2



1 (ξ) = (2π)m/2 − −



1 (2π)m/2 1 (2π)m/2

2 A exp − |x|2 dV(x)

Rm

 Rm



Rm

2 B exp − |x|2 dV(x)

2 C exp − |x|2 dV(x).

In view of Corollary 2.1 to the Funk-Hecke theorem, the integrals containing the B- and C-term of the kernel decomposition reduce to zero. Furthermore, applying the Funk-Hecke theorem in space (Theorem 2.4), we obtain



 2 Fcyl exp − |x|2 (ξ) =

1 (2π)m/2

 Rm

2 cos rρ 1 − tη2 exp − r2 dV(x)

The Fourier Transform in Clifford Analysis

=

177

 +∞  2 Am−1 m−1 r exp − dr r 2 (2π)m/2 0  1 

 2 (m−3)/2 2 × cos rρ 1 − t (1 − t ) dt . −1

Next, taking into account the series expansion of the cosine function

cos (u) =

∞  (−1) =0

(2)!

u2 ,

(66)

this becomes



 2 (ξ) Fcyl exp − |x|2  +∞  ∞ 2 Am−1  (−1) 2 2+m−1 r ρ exp − 2 r dr = (2)! (2π)m/2 0 =0   1 2 (2+m−3)/2 (1 − t ) dt × −1

=

Am−1 (2π)m/2

∞  (−1) =0

(2)!

 √π  2+m−1 

2 m

2+m/2−1   + 2+m 2  2

 ρ2



  ∞  2+m−1 (−1) ρ2 2

=  (2)! 2  m−1 1

2

=0

  m−1 1 |ξ|2 = 1 F1 ; ;− . 2 2 2

Recall that 1 F1 (a; c; z) denotes Kummer’s function, also called the confluent hypergeometric function, which is defined by the following infinite series, and all its analytic continuations: ∞  (a) z =0

(c) !

=

∞  (a + ) (c) z =0

(a) (c + ) !

with (α) = α(α + 1) . . . (α +  − 1) = also Subsection 2.4.1).

,

(α+) (α)

c  = 0, −1, −2, . . . Pochhammer’s symbol (see

Conclusion 6.1 The cylindrical Fourier image of the Gaussian is the radial function given by (see also Figure 4):

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Fred Brackx et al.

1

m 3 m 4 m 5 m 6

0.8 0.6 0.4 0.2 0 2

4

6

0.2

|␰|

8

10

0.4 0.6

2 FIGURE 4 The cylindrical Fourier spectrum of the Gaussian exp − |x|2 for m = 3, m = 4, m = 5, and m = 6.



Fcyl exp



2 − |x|2





 m − 1 1 |ξ|2 (ξ) = 1 F1 ; ;− 2 2 2     m 1 |ξ|2 |ξ|2 exp − 2 . = 1 F1 1 − ; ; 2 2 2

Note that in the case where the dimension m is even and hence 1 − m2 ∈ −N, the   2 m 1 |ξ| Kummer function 1 F1 1 − 2 ; 2 ; 2 reduces to the classical Hermite polyno  mials [Eq. (17)] of even degree associated with the weight function exp −u2 , since

  1 2 (−1)n (2n)! 1 F1 −n; ; u and H2n (u) = n! 2   3 (−1)n (2n + 1)! 2u 1 F1 −n; ; u2 . H2n+1 (u) = n! 2

6.3.2. The Cylindrical Fourier Spectrum of the Gaussian Multiplied With the Clifford Vector (= φ1,0,1 ) Let us now calculate the cylindrical Fourier transform of the basis function φ1,0,1 (x) which is given, up to constants, by x exp − |x|2 . Now, Corollary 2.1 implies that the integral containing the A-term of the kernel 2

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The Fourier Transform in Clifford Analysis

decomposition in Eq. (65) reduces to zero. By means of the Funk-Hecke theorem in space, we thus obtain the following:



 2 Fcyl exp − |x|2 x (ξ) 

1 1 |x|2 =− x dV(x) − B exp − 2 m/2 m (2π) (2π)m/2 R 

2 × C exp − |x|2 x dV(x) Rm

 2 1 2 r 2 ω dV(x) ρ r exp − sinc rρ 1 − t =− t η η 2 (2π)m/2 Rm  2 1 2 r 2 dV(x) sinc rρ ρ η r exp − 1 − t + η 2 (2π)m/2 Rm  +∞ 2  Am−1 m+1 r exp − r2 dr = −ρ (2π)m/2 0   1

 2 2 (m−3)/2 2 t sinc rρ 1 − t (1 − t ) dt η × −1

 +∞ 2  Am−1 m+1 +ρη r exp − r2 dr (2π)m/2 0   1

 2 (m−3)/2 2 sinc rρ 1 − t (1 − t ) dt × −1

 +∞ 2  Am−1 m+1 =ξ r exp − r2 dr (2π)m/2 0   1 

sinc rρ 1 − t2 (1 − t2 )(m−1)/2 dt , × −1

where we have used the fact that P1,m (t) = t. Moreover, in view of the series expansion of the sinc function

sinc(u) =

∞  (−1) u2 , (2 + 1)!

(67)

=0

this becomes



 2 Fcyl exp − |x|2 x (ξ)  +∞ ∞ 2  Am−1  (−1) 2 2+m+1 = ξ ρ r exp − r2 dr (2 + 1)! (2π)m/2 0 =0

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Fred Brackx et al.

 ×

1 −1

2 (2+m−1)/2

(1 − t )

 dt

 ∞

 m Am−1  (−1) 2 +m/2 ρ 2  + +1 = ξ (2 + 1)! 2 (2π)m/2 =0

 √π  2+m+1  2

× 2+m+2  2   ∞  2 + m + 1 (−1) 2 

ξ ρ 2  = (2 + 1)! 2  m−1 2

=0

2



= (m − 1) 1 F1

m + 1 3 |ξ|2 ; ;− 2 2 2



ξ.



2 Conclusion 6.2 The cylindrical Fourier spectrum of exp − |x|2 x is the vectorvalued function given by (see also Figure 5):

Fcyl





 2 exp − |x|2 x (ξ) = (m − 1) 1 F1



m + 1 3 |ξ|2 ; ;− 2 2 2



m 3 |ξ|2 1− ; ; 2 2 2

= (m − 1) 1 F1

 In the even-dimensional case, Kummer’s function 1 F1 1 −



 ξ   |ξ|2 exp − 2 ξ.

2 m 3 |ξ| 2 ; 2; 2

 now yields

classical Hermite polynomials of odd degree. Remark 6.4 We can now easily check the multiplication rule (see Proposition 6.2

(vi)) on the Gaussian exp − |x|2

(2 − m) (2π)m/2

 Rm

2

:



2 sinc(|x ∧ ξ|) x exp − |x|2 dV(x)



  

 2 2 (ξ) + ∂ξ Fcyl exp − |x|2 (−ξ) = Fcyl x exp − |x|2 

 = (m − 1) 1 F1

m + 1 3 |ξ|2 ; ;− 2 2 2



. ξ + ∂ξ

1 F1



m − 1 1 |ξ|2 ; ;− 2 2 2

/ .

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The Fourier Transform in Clifford Analysis

m 53 m 54 m 55 m 56

1.5

1

0.5

0

2

4

6

8

10

|␰ | 20.5

2 FIGURE 5 The norm of the cylindrical Fourier spectrum of x exp − |x|2 for m = 3, m = 4, m = 5, and m = 6.

As

d dz [1 F1 (a; c; z)]

(2 − m) (2π)m/2

 Rm

=

a c 1 F1 (a + 1; c

+ 1; z) (see Magnus et al., 1966), we arrive at

2 sinc(|x ∧ ξ|) x exp − |x|2 dV(x)

    m+1 3 |ξ|2 m+1 3 |ξ|2 = (m−1)1 F1 ; ;− ξ − (m−1)1 F1 ; ;− ξ = 0, 2 2 2 2 2 2

2 which was to be expected, since sinc(|x ∧ ξ|) x exp − |x|2 is an odd function in x.

6.3.3. The Cylindrical Fourier Spectrum of the Gaussian Multiplied With Pk (= φ0,k,j )

2 The L2 -basis function φ0,k,j (x) is given, up to constants, by Pk (x) exp − |x|2 with Pk a left solid inner spherical monogenic of order k. From Corollary 2.1 it is clear that we must make a distinction between k even and odd.

a. k even (B- and C-term of kernel decomposition yield zero). In the case where k is even, the integrals containing the B- and C-term of the kernel

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decomposition are zero. Applying the Funk-Hecke theorem and the cosine expansion (66), we arrive at

 

2 Fcyl exp − |x|2 Pk (x) (ξ) =

1 (2π)m/2

=

Am−1 (2π)m/2 

1 −1

 Rm



2 exp − r2 rk cos rρ 1 − tη2 Pk (ω) dV(x)

+∞

0

 2 exp − r2 rk+m−1 dr



 2 (m−3)/2 2 cos rρ 1 − t (1 − t ) Pk,m (t) dt Pk (η) ∞

=

k!(m−3)! Am−1  (−1) 2 ρ (k+m − 3)! (2π)m/2 (2)! =0



1 −1

(m−2)/2

(1 − t2 )(2+m−3)/2 Ck

 0

 2 exp − r2 r2+k+m−1 dr

+∞

 (t) dt Pk (η),

(68)

where we have also used the expression in Eq. (7) of the Legendre polynomials in Rm in terms of the Gegenbauer, also called ultraspherical, polynomials Ckλ (t). As these Gegenbauer polynomials Ckλ are orthogonal

on ] − 1, 1[ w.r.t. the weight function (1 − t2 )λ−1/2 λ > − 12 , it is easily seen that for  ≤



1

−1

k 2

− 1 holds (m−2)/2

(1 − t2 ) (1 − t2 )(m−3)/2 Ck

(t) dt = 0.

Moreover, combining the integral formula (see Gradshteyn and Ryzhik, 1980, p. 826, formula 4 with α = β)



1

−1

2 α

(1 − t )

Ckλ (t)

 2 22α+1 (α + 1) (k + 2λ) dt = 3 F2 k! (2λ) (2α + 2)   1 × −k, k + 2λ, α + 1; λ + , 2α + 2; 1 , 2

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The Fourier Transform in Clifford Analysis

valid for Re(α) > −1, with Watson’s theorem (see Erdélyi et al., 1953a)







√ 1 a+b+1 1−a−b+2c π  c +   2 2 2 a+b+1







, 2c; 1 = a, b, c; b+1 1−a+2c 1−b+2c 2     a+1 2 2 2 2 

 3 F2

results in



1

−1

(1 − t2 )α Ckλ (t) dt





2 √ 2α+1  (α + 1) (k + 2λ) α + 32  λ + 12  2α−2λ+3 π2 2







. = k+2λ+1 2α+3+k 2α−2λ+3−k k!(2λ)(2α + 2) −k+1    2 2 2 2 (69) Hence, Eq. (68) becomes 

 2 Fcyl exp − |x|2 Pk (x) (ξ)   ∞    2π(m−1)/2  (−1) 2 (k+m+2−2)/2 1 k+m+2 k! (m−3)! 2 ρ  =   (k+m−3)! (2π)m/2  m−1 (2)! 2 2 =k/2

⎛

⎞  2+m   m−1  √ 2+m−2   2+m−1 2 π2 (k + m − 2)   ( + 1) 2 2 ⎝



2



⎠ Pk (η). k+m−1 2+m+k 2+2−k k (m − 2)(2 + m − 1) −k+1    2 2 2 2

Taking into account that (see Magnus, Oberhettinger, and Soni, 1966)

(2z) = π

−1/2

 1 , (z)  z + 2 

2

2z−1

the last result can be simplified to



 2 Fcyl exp − |x|2 Pk (x) (ξ)

√ ∞ (−1) 2 !  2+m−1  2 π



Pk (ξ)

= |ξ|2−k −k+1 k+m−1 2+2−k   (2)!  =k/2 2 2 2   2 k − 1 + m k + 1 |ξ| ; ;− . = Pk (ξ) 1 F1 2 2 2 2k/2

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b. k odd (A-term of kernel decomposition yields zero). For k odd, the FunkHecke theorem implies that 

 2 Fcyl exp − |x|2 Pk (x) (ξ) 

2 1 k+1 r 2 P (ω) dV(x) =− r ρ exp − t sinc rρ 1 − t η k η 2 (2π)m/2 Rm 

2 1 k+1 r 2 ω P (ω) dV(x) − r ρ η exp − sinc rρ 1 − t k η 2 (2π)m/2 Rm  +∞  2 Am−1 = −ρ exp − r2 rk+m dr (2π)m/2 0   1 

2 (m−3)/2 2 t sinc rρ 1 − t (1 − t ) Pk,m (t) dt Pk (η) −1

 +∞  2 Am−1 k+m r exp − 2 r dr −ρ η (2π)m/2 0   1 

2 (m−3)/2 2 sinc rρ 1 − t (1 − t ) Pk+1,m (t) dt η Pk (η) −1

 +∞  2 Am−1 k+m r Pk (η) exp − 2 r dr =ρ (2π)m/2 0  1 

   sinc rρ 1 − t2 (1 − t2 )(m−3)/2 Pk+1,m (t) − tPk,m (t) dt . −1

Now, taking into account the Gegenbauer recurrence relation (see Magnus, Oberhettinger, and Soni, 1966) λ+1 λ (k + 2λ) t Ckλ (t) − (k + 1) Ck+1 (t) = 2λ (1 − t2 ) Ck−1 (t),

we obtain

(k + 1)! (m − 3)! (m−2)/2 Ck+1 (t) (k + m − 2)! k! (m − 3)! (m−2)/2 −t C (t) (k + m − 3)! k k! (m − 3)!  (m−2)/2 = (k + 1) Ck+1 (t) (k + m − 2)! (m−2)/2  (t) − t (k + m − 2) Ck

Pk+1,m (t) − tPk,m (t) =

=−

k! (m − 2)! m/2 (1 − t2 ) Ck−1 (t), (k + m − 2)!

The Fourier Transform in Clifford Analysis

185

which in its turn yields



 2 k! (m − 2)! Am−1 Fcyl exp − |x|2 Pk (x) (ξ) = − ρ Pk (η) (k + m − 2)! (2π)m/2  +∞  2 k+m r × exp − 2 r dr 0



1

−1



 2 (m−1)/2 m/2 2 sinc rρ 1 − t (1 − t ) Ck−1 (t) dt .

Next, applying the series expansion in Eq. (67) the orthogonality of the Gegenbauer polynomials, and expression (69) we find consecutively



 2 Fcyl exp − |x|2 Pk (x) (ξ)  +∞  ∞

k! (m−2)! Am−1 Pk (ξ)  (−1) ρ2 2+k+m r2 exp − 2 r dr =− (k+m−2)! (2π)m/2 |ξ|k−1 (2 + 1)! 0 =0



1 −1

=−

2 (2+m−1)/2

(1 − t )

m/2 Ck−1 (t)

⎛

 dt

2π(m−1)/2

⎞

Pk (ξ) 1 k! (m − 2)! ⎝

⎠ m/2 m−1 (k + m − 2)! (2π) |ξ|k−1  2 ∞ 

=(k−1)/2

(−1) ρ2 (2 + 1)!



2(2+k+m−1)/2 



2 + k + m + 1 2



⎞

2



2+m+2 m+1  ! π 22+m  2+m+1 (k + m − 1) 2 2 2 ⎟ ⎜





⎠ ⎝ −k+2 k+m 2+m+1+k 2−k+3 (k−1)! (m)(2+m+1)  2  2   2 2 ⎛

√

√ 2π 2k/2

Pk (ξ) =  − 2k  k+m 2  = −Pk (ξ) 1 F1

(−1) 2 !  2+m+1 2

|ξ|2−k+1 2−k+3 =(k−1)/2 (2 + 1)!  2 ∞ 

 m + k k + 2 |ξ|2 ; ;− . 2 2 2

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Fred Brackx et al.

2 Conclusion 6.3 The cylindrical Fourier spectrum of exp − |x|2 Pk (x) is once more expressed in terms of Kummer’s function: a. k even



Fcyl exp



2 − |x|2





Pk (x) (ξ) = 1 F1

k − 1 + m k + 1 |ξ|2 ; ;− 2 2 2

 = 1 F1

m k+1 |ξ|2 ; 1− ; 2 2 2



 Pk (ξ)

  |ξ|2 exp − 2 Pk (ξ)

b. k odd



Fcyl exp



2 − |x|2





Pk (x) (ξ) = − 1 F1

 k + m k + 2 |ξ|2 ; ;− Pk (ξ) 2 2 2



m k+2 |ξ|2 ; =−1 F1 1− ; 2 2 2



 2 |ξ| exp − 2 Pk (ξ).

6.3.4. The Cylindrical Fourier Spectrum of the Gaussian Multiplied With the Clifford Vector and Pk (= φ1,k,j ) The calculation of the cylindrical Fourier transform

of the basis function |x|2 φ1,k,j , which is given, up to constants, by exp − 2 x Pk (x), runs along similar lines as in the previous subsection. Hence, we restrict ourselves to stating the results.

a. k even (A-term of kernel decomposition yields zero) 

 2 Fcyl exp − |x|2 x Pk (x) (ξ) (k + m − 1) = 1 F1 (k + 1)



k + m + 1 k + 3 |ξ|2 ; ;− 2 2 2

 ξ Pk (ξ)

    m k + 3 |ξ|2 (k + m − 1) |ξ|2 ; ; ξ Pk (ξ) exp − 2 . = 1 F1 1 − (k + 1) 2 2 2

The Fourier Transform in Clifford Analysis

187

b. k odd (B- and C-terms of kernel decomposition yield zero) 

Fcyl exp



2 − |x|2

 k + m k + 2 |ξ|2 x Pk (x) (ξ) = 1 F1 ; ;− ξ Pk (ξ) 2 2 2    2 m k+2 |ξ|2 |ξ| ; ξ Pk (ξ) exp − 2 . =1 F1 1− ; 2 2 2





6.3.5. The Cylindrical Fourier Spectrum of φ2p,k,j Let us now tackle the problem of calculating the cylindrical Fourier transform of the general basis element φ2p,k,j (x), which is, again up to constants, given by

√ 2 H2p,k ( 2x) Pk (x) exp − |x|2 . The starting point of the calculation is very similar to the one performed in subsection 6.3.3. Hence, we will skip the details.

a. k even (B- and C-terms of kernel decomposition yield zero) Expressing the generalized Clifford–Hermite polynomial of even degree in terms of the classical Laguerre polynomial on the real line [see Eq. (11)] and applying moreover the integral formula (see Magnus, Oberhettinger, and Soni, 1966, p. 245)  0

+∞

(α)

exp (−zu) uλ Ln (u) du =

(λ + 1) (α + n + 1) n! (α + 1) × z−λ−1 2 F1 (−n, λ + 1; α + 1; z−1 ) (70)

valid for Re(λ) > −1 and Re(z) > 0, we arrive at



 √ 2 (ξ) Fcyl H2p,k ( 2x) Pk (x) exp − |x|2 

 2 m/2+k−1 = 2p p! Fcyl Lp (|x|2 ) exp − |x|2 Pk (x) (ξ)   √ 2p+k/2 π  m2 + k + p



 =  Pk (ξ) k+m−1 m    −k+1 + k 2 2 2

  ∞ (−1) 2 !  2+m−1  2 k + 2 + m m

2 F1 −p; ; + k; 2 |ξ|2−k . 2+2−k 2 2 (2)!  =k/2

2

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Fred Brackx et al.

By means of the substitution  =  − 2k , the above result is turned into  √  

 √ 2p+k/2 π  m2 +k+p (−2)k/2 |x|2



 Fcyl H2p,k ( 2x) Pk (x) exp − 2 (ξ) =  Pk (ξ)  k+m−1  m2 +k  −k+1 2 2



 ∞ (−2)  + k !  2 +k+m−1  2 2 (2 +k)! !

 =0

  2 +2k+m m  F1 −p, ; +k; 2 |ξ|2 . 2 2 2 (71)

Moreover, when a or b are nonpositive integers, the hypergeometric function 2 F1 (a, b; c; z) represented by the infinite series, and all its analytic continuations 2 F1 (a, b; c; z)

=

∞  (a) (b) z , (c) ! =0

reduce to a polynomial. In fact, taking into account

(−s) =

0

if  > s

s! (−1) (s−)!

if  ≤ s

,

we find 2 F1 (−s, b; c; z) =

s  (−1) =0

s 

(c)

(b)

z =

  s  s (b + ) (c) z . (−1) (b) (c + )  =0



 Hence, the hypergeometric function 2 F1 −p, 2 +2k+m ; m2 + k; 2 takes the 2 following form:

  2 + 2k + m m ; + k; 2 −p, F 2 1 2 2 

  2 +2k+m+2u p     m2 + k  2 p u u 

 (−1) = 2 . 2 +2k+m m u  +k+u  u=0

2

2

The Fourier Transform in Clifford Analysis

189

Inserting the above result in Eq. (71) and exchanging the sum and series yields   √ 

 √ 2 2p+k/2 π  m2 +k+p (−2)k/2



Fcyl H2p,k ( 2x) Pk (x) exp − |x|2 Pk (ξ) (ξ) = k+m−1   −k+1 2 2





p   ∞ (−2)  + k !  2+k+m−1  2+2k+2u+m u   2 2 2 p (−2)  

|ξ|2 . 2+2k+m u  m2 + k + u (2 + k)! !  u=0 =0 2

Fortunately, the series over  can now be written in a closed form: ∞ (−2) 



+

k 2

!



2+k+m−1 2







2+2k+m 2

2+2k+2u+m 2



|ξ|2

(2 + k)! ! 



√  2k+m+2u 2−k π  m−1+k 2 2



= 2k+m   k+1 2 2   2k + m + 2u m − 1 + k 2k + m k + 1 |ξ|2 ×2 F2 , ; , ;− , 2 2 2 2 2

=0

where 2 F2 (a, b; c, d; z) is the generalized hypergeometric series given by 2 F2 (a, b; c, d; z) =

∞  (a) (b) z . (c) (d) ! =0





Finally, taking into account that  12 + z  12 − z = Oberhettinger, and Soni, 1966, p. 2), we obtain

π cos (πz)

(see Magnus,

 

 √ |x|2 p m + 2k (ξ) = 2 Fcyl H2p,k ( 2x) Pk (x) exp − 2 Pk (ξ) 2 p   p    p 2k + m + 2u m − 1 + k 2k + m k + 1 |ξ|2 u , ; , ;− . (−2) 2 F2 2 2 2 2 2 u 

u=0

(72) Now our aim is to express the above cylindrical Fourier image in terms of Kummer’s function. For that purpose, let us start with the following lemma.

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Fred Brackx et al.

Lemma 6.2 One has

n    n (b)u zu 1 F1 (b + u; c + u; z). u (a)u (c)u

2 F2 (a + n, b; a, c; z) =

(73)

u=0

Proof. We prove this result by induction on n. For n = 0, it is clear that 2 F2 (a, b; a, c; z)

= 1 F1 (b; c; z),

while for n = 1, a straightforward calculation yields 2 F2 (a + 1, b; a, c; z)

= 1 F1 (b; c; z) +

b z 1 F1 (b + 1; c + 1; z). ac

Hence, the proposed sum representation in Eq. (73) holds for the lower order cases n = 0, 1. Assuming that it holds for order n, we now prove the expression (73) in case of order n + 1. First, one can easily verify that 2 F2 (a + n + 1, b; a, c; z)

bz ac × 2 F2 (a + n + 1, b + 1; a + 1, c + 1; z).

= 2 F2 (a + n, b; a, c; z) +

Next, applying the induction hypothesis we arrive at 2 F2 (a + n + 1, b; a, c; z) =

n    n (b)u zu 1 F1 (b + u; c + u; z) u (a)u (c)u u=0

n   bz  n (b + 1)u zu + 1 F1 (b + 1 + u; c + 1 + u; z). u (a + 1)u (c + 1)u ac u=0

Executing in the second sum the substitution u = u + 1 and rearranging the terms indeed yields the desired summation as follows: 2 F2 (a + n + 1, b; a, c; z)

=

n    n (b)u zu 1 F1 (b + u; c + u; z) u (a)u (c)u u=0

+

  n+1  n (b + 1)u −1 zu −1 bz    1 F1 (b + u ; c + u ; z) u − 1 (a + 1)u −1 (c + 1)u −1 ac  u =1

The Fourier Transform in Clifford Analysis

191

   n    n (b)u b n (b+1)u−1 = 1 F1 (b; c; z) + zu + u (a)u (c)u ac u − 1 (a+1)u−1 (c+1)u−1 u=1

1 F1 (b + u; c

=

+ u; z) +

b (b + 1)n zn+1 1 F1 (b + n + 1; c + n + 1; z) ac (a + 1)n (c + 1)n

 n+1   n + 1 (b)u zu 1 F1 (b + u; c + u; z). u (a)u (c)u u=0

We can now easily prove the following proposition. Proposition 6.3 One has p    p n=0

n

(−2)n 2 F2 (a + n, b; a, c; z) p

= (−1)

p  u=0

p

= (−1)

p  u=0

  p (b)u zu 2 1 F1 (b + u; c + u; z) u (a)u (c)u u

  p (b)u zu 2 1 F1 (c − b; c + u; −z) exp (z). u (a)u (c)u u

Proof. Using the previous lemma, followed by an exchange of the two summations, we obtain p    p n=0

=

n

(−2)n 2 F2 (a + n, b; a, c; z)

p    p n=0

n    n (b)u zu (−2) 1 F1 (b + u; c + u; z) n u (a)u (c)u n

u=0

p

 (b)u zu = (a)u (c)u u=0

 p

    p n (−2) 1 F1 (b + u; c + u; z). n u n=u n

Using the symbolic software Maple (Maplesoft, Waterloo, Ontario, Canada), we find p  n=u

(−2)n

     p n p = 2u (−1)p , n u u

which yields the desired expression.

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Fred Brackx et al.

By means of the above proposition, we can rewrite the cylindrical Fourier transform in Eq. (72) in terms of Kummer’s function as follows:    2 

 √ 2 2k + m |ξ| (ξ) = (−2)p Fcyl H2p,k ( 2x)Pk (x)exp − |x|2 Pk (ξ) exp − 2 2 p p    p

u

u=0



m−1+k 2

u 2k+m k+1 2 2 u u

 2 u

(−|ξ| )

1 F1

 |ξ|2 m k+1 1− ; + u; . 2 2 2

Expressing also the generalized Clifford–Hermite polynomial in terms of Kummer’s function [see Eq. (12)], we derive Fcyl

 2

 m |ξ| |x|2 2 p −p; P (ξ) F (x) exp − = (−1) P (ξ) exp − 2 + k; |x| 1 1 k k 2 2



p    p u=0

u



m−1+k 2

u 2k+m k+1 2 2 u u

 |ξ|2 m k+1 + u; . 1− ; 2 2 2

 (−|ξ|2 )u 1 F1

When k = 0 we can also calculate the cylindrical Fourier spectrum using the integral formula (see Gradshteyn and Ryzhik, 1980, p. 427, formula 4 with u = 1):



1

x 0

2ν−1

2 μ−1

(1 − x )

  a2 1 1 (74) cos (ax) dx = B(μ, ν) 1 F2 ν; , ν + μ; − 2 2 4

valid for Re(μ) > 0 and Re(ν) > 0. Indeed, by means of the Funk-Hecke theorem we find, as before,



 √ 2 (ξ) Fcyl H2p,0 ( 2x)exp − |x|2 Am−1 = 2 p! (2π)m/2



p



1 −1

0

+∞

m/2−1 2 Lp (r )

 2 m−1 r exp − 2 r dr

  2 (m−3)/2 2 cos (rρ 1 − t ) (1 − t ) dt .

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The Fourier Transform in Clifford Analysis

Executing the substitution x = we have that



1 −1

 1 − t2 and taking into account Eq. (74),

  cos (rρ 1 − t2 ) (1 − t2 )(m−3)/2 dt = 2

0

√ =

1

dx cos (rρx) xm−2  1 − x2

  π  m−1 2 m − 1 1 m r 2 ρ2   ; , ;− , 1 F2 2 2 2 4  m2

which yields



 √ 2 2p−m/2+1 p!   Fcyl H2p,0 ( 2x) exp − |x|2 (ξ) =  m2 

+∞ 0

m/2−1 2 Lp (r )



2 − r2

exp



r

m−1

1 F2

m − 1 1 m r2 |ξ|2 ; , ;− 2 2 2 4



 dr .

Comparing the above with [see Eq. (72)]



 √ 2 (ξ) Fcyl H2p,0 ( 2x) exp − |x|2 = 2p



   p   2  p |ξ| + p m − 1 m 1 m + 2u 2  , ; , ;− , (−2)u 2 F2 2 2 2 2 2 u  m2 u=0

m

we have thus proved the following integral formula. Proposition 6.4 One has



+∞ 0

m/2−1 2 Lp (r )

exp

2 − r2



r

m−1

1 F2

m − 1 1 m r2 |ξ|2 ; , ;− 2 2 2 4

 dr

    p   2 |ξ| 2m/2−1  m2 + p  p m − 1 m 1 m + u, ; , ;− , (−2)u 2 F2 = p! 2 2 2 2 2 u u=0

where 1 F2 (a; b, c; z) and 2 F2 (a, b; c, d; z) denote generalized hypergeometric series (α) and L the generalized Laguerre polynomial on the real line.

194

Fred Brackx et al.

a. k odd (A-term of kernel decomposition yields zero). Taking into account formula (70), we find in a similar way as in Subsection 6.3.3, case b Fcyl



 √ 2 H2p,k ( 2x)Pk (x)exp − |x|2 (ξ) =

=(k−1)/2



  k + 2+m+1 m

2 F1 −p, ; + k; 2 |ξ|2+1−k . 2 2 (2 + 1)!  2+3−k 2

(−1) 2 ! 

∞ 

√   2π 2p+k/2  m2 + k + p

  Pk (ξ)  m2 + k  − 2k  k+m 2



2+m+1 2

Executing the substitution  =  − k−1 2 , writing the hypergeometric function as a summation, and exchanging the sum and series, we find Fcyl



 √ 2 H2p,k ( 2x) Pk (x) exp − |x|2 (ξ) =

√   2π(−1)(k−1)/2 2p+k−1/2  m2 +k+p

Pk (ξ)  − 2k  k+m 2







p   ∞ (−2)  + k − 1 !  2+k+m  2+2k+m+2u   2 2 2 2 p 1

 |ξ|2 , (−2)u  m u (2 + k)!!  2 +k+u  2+2k+m =0

u=0

2

which can further be simplified to 

√

Fcyl H2p,k ( 2x) Pk (x) exp p    p u=0

u

 (−2)

u

2 F2



2 − |x|2



(ξ) = −

2p 

m



+k+p

2  2k+m 2

Pk (ξ)

 2k + m k + m 2k + m k + 2 |ξ|2 + u, ; , ;− . 2 2 2 2 2

Applying Proposition 6.3 we can equivalently state     

 √ 2 2k + m |ξ|2 (ξ) = −(−2)p Fcyl H2p,k ( 2x)Pk (x)exp − |x|2 Pk (ξ)exp − 2 2 p

  k+m   (−|ξ|2 )u  p 2 m k + 2 + 2u |ξ|2 u



1 F1 1 − ; ; 2k+m k+2 u 2 2 2 p

u=0

2

u

2

u

The Fourier Transform in Clifford Analysis

195

or rewriting also the generalized Clifford–Hermite polynomial in terms of Kummer’s function  2 

 m |ξ| |x|2 2 p+1 (ξ) = (−1) Fcyl 1 F1 −p; + k; |x| Pk (x) exp − 2 Pk (ξ) exp − 2 2

  p   k+m (−|ξ|2 )u  2 p m k + 2 + 2u |ξ|2 u



1 F1 1 − ; ; . 2k+m k+2 u 2 2 2 u=0

2

u

2

u

6.3.6. The Cylindrical Fourier Spectrum of φ2p+1,k,j Since the calculations of the cylindrical Fourier spectrum of the basis function φ2p+1,k,j given, up to constants, by



√ 2 H2p+1,k ( 2x) Pk (x) exp − |x|2 are very similar to the ones of the previous subsection, we only give the results.

a. k even (A-term of kernel decomposition yields zero) 

 √ 2 Fcyl H2p+1,k ( 2x) Pk (x) exp − |x|2 (ξ)

m (k + m − 1) k + + 1 ξ Pk (ξ) k+1 2 p     p 2  |ξ| 2k+m+2+2u m+k+1 2k+m+2 3+k p (−2)u F2 , ; , ;− 2 2 2 2 2 u 2

= 2p+1/2

u=0

 

√ (k + m − 1) m |ξ|2 k + + 1 ξ exp − 2 Pk (ξ) 2 k+1 2 p

  p   m+k+1 (−|ξ|2 )u  |ξ|2 2 m 3+k p u



1 F1 1 − ; + u; 2k+m+2 3+k 2 2 2 u

= (−2)p

u=0

2

u

2

u

or in terms of Kummer’s function [see Eq. (12)],

Fcyl





 m |x|2 2 −p; x P + k + 1; |x| F exp − (x) (ξ) = 1 1 k 2 2

(75)

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  k+m−1 |ξ|2 ξ exp − 2 k+1

  p   m+k+1 (−|ξ|2 )u  p |ξ|2 2 m 3+k u



1 F1 1 − ; Pk (ξ) + u; . 3+k 2 2 2 u 2k+m+2

(−1)p

u=0

2

u

2

u

b. k odd (B- and C-terms of kernel decomposition yield zero) 

 √ 2 Fcyl H2p+1,k ( 2x) Pk (x) exp − |x|2 (ξ)

m = 2p+1/2 k + + 1 ξ Pk (ξ) 2 p     p  2k+m+2+2u m+k 2k+m+2 2+k |ξ|2 u p , ; , ;− (−2) 2 F2 2 2 2 2 2 u u=0  

√ m |ξ|2 = (−2)p 2 k + + 1 ξ exp − 2 Pk (ξ) 2 p

  m+k p   2 (−|ξ|2 )u  |ξ| 2 p m 2 + k u

1 F1 1 − ; + u; 2k+m+2 2+k u 2 2 2 u=0

2

2

u

u

or in terms of Kummer’s function   

m |ξ|2 |x|2 2 p Fcyl 1 F1 −p; +k+1; |x| x exp − 2 Pk (x) (ξ) = (−1) ξ exp − 2 Pk (ξ) 2

  m+k p   (−|ξ|2 )u  |ξ|2 2 p m 2+k u



1 F1 1 − ; + u; . 2k+m+2 2+k u 2 2 2 

u=0

2

u

2

u

Taking into account the integral formula (see Gradshteyn and Ryzhik, 1980, p. 426, formula 3 with u = 1):





1

x 0

2ν−1

a 1 (1 − x )μ−1 sin (ax) dx = B μ, ν + 2



2 

1 a2 1 3 × 1 F2 ν + ; , μ + ν + ; − 2 2 2 4

2



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valid for Re(μ) > 0 and Re(ν) > − 12 , we also find that 

 √ 2 (m − 1)

ξ (ξ) = 2p+1/2−m/2 p! Fcyl H2p+1,0 ( 2x) exp − |x|2  m+2 2     +∞ 2 m + 1 3 m + 2 r2 |ξ|2 m/2 2 m+1 r Lp (r ) exp − 2 r ; , ;− dr 1 F2 2 2 2 4 0 which combined with [see Eq. (75)] Fcyl



 √ 2  H2p+1,0 ( 2x) exp − |x|2 (ξ) = 2p+1/2 (m − 1)



  p  u p (−2) 2 F2 u u=0



m



2 + p +1  m2 + 1

ξ

m + 2 + 2u m + 1 m + 2 3 |ξ|2 , ; , ;− 2 2 2 2 2



yields the following result. Proposition 6.5 One has 

 m + 1 3 m + 2 r2 |ξ|2 ; , ;− dr r exp 1 F2 2 2 2 4 0     p   2m/2  m2 +p+1  m+2+2u m+1 m+2 3 |ξ|2 u p , ; , ;− , (−2) = 2 F2 u p! 2 2 2 2 2 

+∞

m/2 Lp (r2 )



2 − r2

m+1

u=0

where 1 F2 (a; b, c; z) and 2 F2 (a, b; c, d; z) denote generalized hypergeometric series (α) and L the generalized Laguerre polynomial on the real line.

ACKNOWLEDGMENTS We are indebted to Wouter Hamelinck for assistance in preparing the figures.

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