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An orthogonal system of monogenic polynomials over prolate spheroids in R3
Mathematical and Computer Modelling 57 (2013) 425–434
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An orthogonal system of monogenic polynomials over prolate spheroids in R3 J. Morais Technical University of Mining, Freiberg, Germany
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Article history: Received 13 May 2011 Received in revised form 14 September 2011 Accepted 15 June 2012 Keywords: Quaternion analysis Moisil–Théodoresco system Monogenic functions Ferrer’s associated Legendre functions Chebyshev polynomials Hyperbolic functions
abstract The object of this paper is to construct a complete orthogonal system of monogenic polynomials as solutions of the Moisil–Théodoresco system over prolate spheroids in R3 . This will be done in the spaces of square integrable functions over H. A big breakthrough is that the orthogonality of the polynomials in question does not depend on the shape of the spheroids, but only on the location of the foci of the ellipse generating the spheroid. The representations of these polynomials are given explicitly, ready to be implemented on a computer. In addition, we show a corresponding orthogonality of the same polynomials over the surface of the spheroids with respect to a suitable weight function. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Consider the pair u = (u0 , u∗ ), where u0 is a real-valued continuously differentiable function defined on an open domain Ω of R3 , and u∗ = (u1 , u2 , u3 ) is a vector-valued function in Ω whose components ui = ui (x0 , x1 , x2 ) are real functions of real variables x0 , x1 , x2 . Following [1] (see also [2]) the Moisil–Théodoresco system is represented by ∂ u + ∂x1 u2 + ∂x2 u3 = 0 x0 1 div u∗ = 0 ∂x0 u0 − ∂x2 u2 + ∂x1 u3 = 0 (MT) ⇐⇒ grad u0 + rot u∗ = 0 ∂x1 u0 + ∂x2 u1 − ∂x0 u3 = 0 ∂x2 u0 − ∂x1 u1 + ∂x0 u2 = 0. The (MT)-system clearly generalizes the classical Cauchy–Riemann system for holomorphic functions in the complex plane. To start with, the (MT)-system may be obtained consistently by working with the quaternion algebra. Let H := {z = a0 + a1 i + a2 j + a3 k : ai ∈ R, i = 0, 1, 2, 3} be the Hamiltonian skew field, where the imaginary units i, j, and k are subject to the multiplication rules i2 = j2 = k2 = −1; ij = k = −ji,
jk = i = −kj,
ki = j = −ik.
At this stage we may introduce some terms which will be frequently used. The scalar and vector parts of z, Sc(z) and Vec(z), are defined as the a0 and a1 i + a2 j + a3 k terms, respectively. Like in the complex case, the conjugate of z is the quaternion z = a0 − a1 i − a2 j − a3 k, and the norm |z| of z is defined by |z|2 = zz = zz = a20 + a21 + a22 + a23 that coincides with its corresponding Euclidean norm as a vector in R4 . Now, let Ω be an open subset of R3 with a piecewise smooth boundary,
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J. Morais / Mathematical and Computer Modelling 57 (2013) 425–434
and x := (x0 , x1 , x2 ) ∈ Ω . The standard form of a quaternion-valued function or, briefly, an H-valued function, will be taken to be f : Ω −→ H,
f(x) = [f(x)]0 + [f(x)]1 i + [f(x)]2 j + [f(x)]3 k
where [f]i (i = 0, 1, 2, 3) are real-valued functions defined in Ω . Properties such as continuity, differentiability, integrability, and so on, which are ascribed to f have to be fulfilled by all components [f]i . We further introduce the right H-linear Hilbert space of square integrable H-valued functions defined in Ω , that we denote by L2 (Ω ; H; H). In this assignment, the H-valued inner product is defined by
⟨f, g⟩L2 (Ω ;H;H) =
f g dV
(1)
Ω
where dV denotes the Lebesgue measure on Ω . To simplify matters further we shall remark that using the embedding of R in H the inner product of two real-valued functions f , g : Ω −→ R can also be written by using the inner product (1), and it will be denoted simply by ⟨f , g ⟩L2 (Ω ) . The classical Dirac operator (also known as Moisil–Théodoresco operator [3,1])
∂=i
∂ ∂ ∂ +j +k ∂ x0 ∂ x1 ∂ x2
arises commonly in quaternion analysis and is very important. Nullsolutions to this operator provide us with the class of the so-called monogenic functions. Definition 1.1 (Monogenicity). An H-valued function f is called left (resp. right) monogenic in Ω if f is in C 1 (Ω ; H) and satisfies ∂ f = 0 (resp. f∂ = 0) in Ω . For, throughout this text, and except otherwise stated, we only use left H-valued monogenic functions that, for simplicity, we call monogenic. Nevertheless, all results accomplished to left H-valued monogenic functions can easily be adapted to right H-valued monogenic functions. Ultimately, consider a continuously differentiable H-valued function f, and the associated pair u = (u0 , u∗ ). Putting u0 = [f ]0 , u1 = [f ]3 , u2 = [f ]2 , and u3 = −[f ]1 , the (MT)-system can be rewritten as (MT) ⇐⇒ ∂ f = 0. From now on, the solutions of the (MT)-system will be called (MT)-solutions. The subspace of polynomial (MT)-solutions of degree n will be denoted by M + (Ω ; H; n). In [4], Sudbery proved that dim M + (Ω ; H; n) = n + 1. We also denote by M + (Ω ; H) := L2 (Ω ; H; H) ∩ ker ∂ the space of square integrable H-valued monogenic functions defined in Ω . 2. An orthogonal system of monogenic polynomials over the interior of 3D prolate spheroids In the sequel we shall make use of rectangular coordinates, and cylindrical coordinates: x0 = z , x1 = ρ cos φ, x2 = ρ sin φ , where z ∈ (−∞, +∞), ρ ∈ [0, +∞) and φ ∈ [0, 2π ), emphasizing that φ = arccos( xρ1 ) if ρ > 0 and φ = 0 if ρ = 0. In the present section we seek to assemble a special orthogonal system of H-valued monogenic polynomials in the interior of the surface of revolution
E :
z2 cosh2 α
+
ρ2 sinh2 α
= 1,
(2)
where α is a nonnegative real number. Representation (2) shows that surfaces of constant α do indeed form confocal prolate spheroids, since they are ellipses rotated about the axis joining their foci. At this stage it is convenient to introduce coordinates, θ and µ, defined by the relations: z = cos θ cosh µ and ρ = sin θ sinh µ, where θ ∈ [0, π] and µ ∈ [0, a]. In this case, the boundary of the above spheroid has the equation µ = α . In [5] the author found it necessary to focus the discussion, temporarily at least, on spaces of square integrable functions over R. With this outcome in mind, a complete orthogonal set
{E n,l , F n,m : l = 0, . . . , n + 1, m = 1, . . . , n + 1} of polynomial nullsolutions of the well known Riesz system has been developed over prolate spheroids in R3 . The mentioned prolate spheroidal monogenics are explicitly given by
E n,l (µ, θ , φ) :=
(n + l + 1) 2
+
1 4
An,l (µ, θ ) Tl (cos φ) +
1 4 (n − l + 1)
An,l+1 (µ, θ ) [Tl+1 (cos φ)i + sin φ Ul (cos φ)j]
(n + 1 + l)(n + l)(n − l + 2)An,l−1 (µ, θ ) [−Tl−1 (cos φ)i + sin φ Ul−2 (cos φ)j] ,
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427
and
(n + m + 1)
F n,m (µ, θ , φ) :=
2
An,m (µ, θ ) sin φ Um−1 (cos φ)
1 An,m+1 (µ, θ ) [sin φ Um (cos φ)i − Tm+1 (cos φ)j] 4 (n − m + 1) 1 − (n + 1 + m)(n + m)(n − m + 2)An,m−1 (µ, θ ) [sin φ Um−2 (cos φ)i + Tm−1 (cos φ)j] , 4 for l = 0, . . . , n + 1 and m = 1, . . . , n + 1, with the subscript coefficient function
+
n−l
2 (2n + 1 − 2k) (n + l)2k l Pn−2k (cos θ ) Pnl −2k (cosh µ) An,l (µ, θ ) := ( n + 1 − l ) 2k + 1 k=0
(3)
such that An,−1 (µ, θ ) = − n(n+1)12 (n+2) An,1 (µ, θ ). We shall use (a)r to stand the Pochhammer symbol, i.e. (a)r := a(a + 1) · · · (a + r − 1) = ΓΓ(a(+r )r ) for any integer r > 1, and (a)0 := 1. As usual, ⌈s⌉ denotes the smallest integer not less than s ∈ R. In addition, Pnl stands for the Ferrer’s associated Legendre functions of degree n and order l of the first kind, Tl and Ul are the Chebyshev polynomials of the first and second kinds, respectively. For the sake of completeness we set l
Pn (cosh µ) = Pn0 (cosh µ) and Pnl (cosh µ) = (−1)l (sinh µ)l dtd l [Pn (t )]|t =cosh µ . Remark 2.1. One may show by elementary reasoning that the well-known recurrence properties of the Legendre polynomials Pn (cosh µ) and their associated Legendre functions Pnl (cosh µ)(l > 0) are different from the ones we are familiar with. We shall now be able to extend the results in that connection to a quaternionic Hilbert subspace; in particular, we construct a complete orthogonal system of polynomial nullsolutions of the Moisil–Théodoresco system over 3D prolate spheroids, whose study will require us to explore a new situation. We emphasize that the used methods are quite general and can be either extended to oblate spheroids or even to arbitrary ellipsoids in the 3D space. But such a procedure would make the further study of the underlying polynomials somewhat laborious, and for this reason we shall prefer not to discuss these cases here. For, in continuation of [5] (cf. [6,7]) we designate the new n + 1 (prolate) spheroidal monogenics by
S n,l := E n,l+1 i + F n,l+1 j,
l = 0 , . . . , n,
(4)
namely functions with respect to the variables µ, θ , and the azimuthal angle φ of the quaternion form:
S n,l (µ, θ , φ) :=
1 2
+
(n + 2 + l) (n + 1 + l) (n − l + 1) An,l (µ, θ ) Tl (cos φ) + 1 2 1
1 2
(n + 2 + l) An,l+1 (µ, θ ) Tl+1 (cos φ) i
(n + 2 + l) An,l+1 (µ, θ ) sin φ Ul (cos φ) j
(n + 2 + l) (n + 1 + l) (n − l + 1) An,l (µ, θ ) sin φ Ul−1 (cos φ) k 2 with the subscript coefficient function An,l (µ, θ ) given by (3). It is convenient to regard (4) as a new special system of monogenic polynomials despite its similarity with other systems in literature. −
Remark 2.2. Although the monogenic polynomials S n,0 are built in terms of the Legendre polynomials while S n,l (l > 0) are built in terms of the Ferrer’s associated Legendre functions, we still include the treatment of the first into the general case, whenever this does not raise any confusion and the treatment remains the same. Of course, the results can be extended to the case of the region outside a spheroid as well. One has merely to replace the Ferrer’s associated Legendre functions by the Legendre functions of second kind [8]. In the considerations to follow we will often omit the argument and write simply S n,l instead of S n,l (µ, θ , φ). In particular, these n + 1 system constituents satisfy the first order partial differential equation 0 = ∂ S n ,l
∂ S n,l sin θ cosh µ ∂ S n,l − sin2 θ + sinh2 µ ∂µ sin2 θ + sinh2 µ ∂θ sin θ cosh µ cos φ ∂ S n,l cos θ sinh µ cos φ ∂ S n,l sin φ ∂ S n,l +j + − sin θ sinh µ ∂φ sin2 θ + sinh2 µ ∂µ sin2 θ + sinh2 µ ∂θ sin θ cosh µ sin φ ∂ S n,l cos θ sinh µ sin φ ∂ S n,l cos φ ∂ S n ,l +k + + . sin θ sinh µ ∂φ sin2 θ + sinh2 µ ∂µ sin2 θ + sinh2 µ ∂θ
=i
cos θ sinh µ
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We further assume the reader to be familiar with the fact that ∂ is a square root of the Laplace operator in R3 in the sense that
∆3 S n,l = −∂ 2 S n,l
1
=
sin2 θ + sinh2 µ
∂ 2 S n ,l ∂ S n ,l ∂ S n ,l ∂ 2 S n,l + + coth µ + cot θ ∂µ2 ∂θ 2 ∂µ ∂θ
+
∂ 2 S n ,l . sin2 θ sinh2 µ ∂φ 2 1
For, the monogenic polynomials S n,l can be seen as a refinement of the harmonic polynomials exploited by Garabedian in [9]. We formulate a first result to our initial calculation. Lemma 2.1. The monogenic polynomials S n,l are the zero functions for l ≥ n + 1. We may now deliver the orthogonality of the above-mentioned polynomials over the interior of the prolate spheroid (2) in the sense of the quaternion product (1), which is the main objective of the present section. A big step forward is that the orthogonality in question does not depend on the shape of the spheroids, but only on their size. More specifically, the polynomials S n,l (l ≥ 0) depend on the location of the foci of the ellipse generating the spheroid only, and not on its eccentricity. Theorem 2.1. For each n ∈ N0 , the set {S n,l : l = 0, . . . , n} is orthogonal over the interior of the prolate spheroid (2) in the sense of the quaternion product (1), and their norms are given by
(n + 2 + l)! ∥S n,l ∥2L2 (E ;H;H) = π (n + 2 + l) (n − l)! × (n + 1 + l) (n − l + 1)
a
Pnl (cosh µ) sinh µ cosh µ Pnl +1 (cosh µ) dµ 0
−
1 2n + 3
+
(n + 1 + l)2 (n − l + 1)
a
[Pnl (cosh µ)]2 sinh µ dµ 0
a 1 Pnl+1 (cosh µ) sinh µ cosh µ Pnl+ +1 (cosh µ) dµ
0
(n + 2 + l) − 2n + 3
a
[Pnl+1 (cosh µ)]2 sinh µ dµ .
0
Proof. We begin by proving the orthogonality of the polynomials. By definition of the quaternion product (1) for a fixed n ∈ N0 we have
⟨S n,l1 , S n,l2 ⟩L2 (E ;H;H) =
S n,l1 S n,l2 dV E
= (I) + (II)i + (III)j + (IV)k where
(I) =
Sc(S n,l1 ) Sc(S n,l2 ) + [S n,l1 ]1 [S n,l2 ]1 + [S n,l1 ]2 [S n,l2 ]2 + [S n,l1 ]3 [S n,l2 ]3 dV
E
and, moreover
(II) = (III) =
(IV) =
Sc(S n,l1 )[S n,l2 ]1 − [S n,l1 ]1 Sc(S n,l2 ) − [S n,l1 ]2 [S n,l2 ]3 + [S n,l1 ]3 [S n,l2 ]2 dV
E
Sc(S n,l1 )[S n,l2 ]2 + [S n,l1 ]1 [S n,l2 ]3 − [S n,l1 ]2 Sc(S n,l2 ) − [S n,l1 ]3 [S n,l2 ]1 dV
E
Sc(S n,l1 )[S n,l2 ]3 − [S n,l1 ]1 [S n,l2 ]2 + [S n,l1 ]2 [S n,l2 ]1 − [S n,l1 ]3 Sc(S n,l2 ) dV .
E
Assume we have the change of variables z = z (µ, θ ) = cos θ cosh µ and ρ = ρ(µ, θ ) = sin θ sinh µ. A first straightforward computation shows
(I) =
=
∂(µ, θ ) dφ dρ dz ∂(z , ρ) E ∂(µ, θ ) + [S n,l1 ]1 [S n,l2 ]1 + [S n,l1 ]2 [S n,l2 ]2 ρ dφ dρ dz ∂(z , ρ) E
1 4
Sc(S n,l1 )Sc(S n,l2 ) + [S n,l1 ]3 [S n,l2 ]3 ρ
(n + 2 + l1 ) (n + 1 + l1 ) (n − l1 + 1) (n + 2 + l2 ) (n + 1 + l2 ) (n − l2 + 1)
J. Morais / Mathematical and Computer Modelling 57 (2013) 425–434
π
429
2π ∂(µ, θ ) Tl1 −l2 (cos φ) dφ dθ dµ ∂(z , ρ) 0 0 0 a π 1 ∂(µ, θ ) + An,l1 +1 (µ, θ ) An,l2 +1 (µ, θ ) sin θ sinh µ dθ dµ 4 0 0 ∂(z , ρ) 2π Tl1 −l2 (cos φ) dφ × (n + 2 + l1 ) (n + 2 + l2 ) a
×
An,l1 (µ, θ ) An,l2 (µ, θ ) sin θ sinh µ
0
= 0,
l1 ̸= l2 .
We will proceed in such a manner to compute the remaining integrals
∂(µ, θ ) dφ dρ dz ∂(z , ρ) E ∂(µ, θ ) [S n,l1 ]1 Sc(S n,l2 ) + [S n,l1 ]2 [S n,l2 ]3 ρ dφ dρ dz − ∂(z , ρ) a E π ∂(µ, θ ) = An,l1 (µ, θ ) An,l2 +1 (µ, θ ) sin θ sinh µ dθ dµ ∂(z , ρ) 0 0 2π 1 Tl1 +1+l2 (cos φ) dφ × (n + 2 + l1 ) (n + 1 + l1 ) (n − l1 + 1) (n + 2 + l2 ) 4 0 a π ∂(µ, θ ) − An,l1 +1 (µ, θ ) An,l2 (µ, θ ) sin θ sinh µ dθ dµ ∂(z , ρ) 0 0 2π 1 × (n + 2 + l1 ) (n + 2 + l2 ) (n + 1 + l2 ) (n − l2 + 1) Tl1 +1+l2 (cos φ) dφ
(II) =
Sc(S n,l1 )[S n,l2 ]1 + [S n,l1 ]3 [S n,l2 ]2 ρ
4
= 0,
0
l1 ̸= l2
and moreover,
∂(µ, θ ) dφ dρ dz ∂(z , ρ) E ∂(µ, θ ) + [S n,l1 ]1 [S n,l2 ]3 − [S n,l1 ]2 Sc(S n,l2 ) ρ dφ dρ dz ∂(z , ρ) a E π ∂(µ, θ ) = An,l1 (µ, θ ) An,l2 +1 (µ, θ ) sin θ sinh µ dθ dµ ∂(z , ρ) 0 0
(III) =
Sc(S n,l1 )[S n,l2 ]2 − [S n,l1 ]3 [S n,l2 ]1 ρ
2π
1
(n + 2 + l1 ) (n + 1 + l1 ) (n − l1 + 1) (n + 2 + l2 ) sin φ Ul1 +l2 (cos φ) dφ 4 0 a π ∂(µ, θ ) dθ dµ − An,l1 +1 (µ, θ ) An,l2 (µ, θ ) sin θ sinh µ ∂(z , ρ) 0 0 2π 1 × (n + 2 + l1 ) (n + 2 + l2 ) (n + 1 + l2 ) (n − l2 + 1) sin φ Ul1 +l2 (cos φ) dφ ×
4
0
= 0, l1 ̸= l2 . ∂(µ, θ ) dφ dρ dz (IV) = Sc(S n,l1 )[S n,l2 ]3 − [S n,l1 ]3 Sc(S n,l2 ) ρ ∂(z , ρ) E ∂(µ, θ ) − [S n,l1 ]1 [S n,l2 ]2 − [S n,l1 ]2 [S n,l2 ]1 ρ dφ dρ dz ∂(z , ρ) E =
1
(n + 2 + l1 ) (n + 1 + l1 ) (n − l1 + 1) (n + 2 + l2 ) (n + 1 + l2 ) (n − l2 + 1) 4 2π a π
∂(µ, θ ) dθ dµ sin φ Ul1 −1−l2 (cos φ) dφ ∂(z , ρ) 0 0 a 0 π ∂(µ, θ ) + An,l1 +1 (µ, θ ) An,l2 +1 (µ, θ ) sin θ sinh µ dθ dµ ∂(z , ρ) 0 0 An,l1 (µ, θ ) An,l2 (µ, θ ) sin θ sinh µ
×
× = 0,
1
4
2π
(n + 2 + l1 ) (n + 2 + l2 )
l1 ̸= l2 .
sin φ Ul1 −1−l2 (cos φ) dφ
0
Thus, for a fixed n ∈ N0 the set {S n,l : l = 0, . . . , n} is orthogonal in the sense of the quaternion product (1). We proceed to prove the orthogonality of the system for each degree n. The proof is mainly based on the expressions of the
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aforementioned spheroidal monogenics combining several recurrence properties of the Legendre polynomials and Ferrer’s associated Legendre functions, and some equations between the Chebyshev polynomials of the first and second kinds. Therefore, it requires extensive calculations. Here, we only give the main steps of the proof. Once again, by definition of the quaternion product (1) it follows that
⟨S n1 ,l , S n2 ,l ⟩L2 (E ;H;H) =
S n1 ,l S n2 ,l dV E
= (V) + (VI)i + (VII)j + (VIII)k where
(V) =
Sc(S n1 ,l ) Sc(S n2 ,l ) + [S n1 ,l ]1 [S n2 ,l ]1 + [S n1 ,l ]2 [S n2 ,l ]2 + [S n1 ,l ]3 [S n2 ,l ]3 dV
E
and
(VI) = (VII) =
E
(VIII) =
Sc(S n1 ,l )[S n2 ,l ]1 − [S n1 ,l ]1 Sc(S n2 ,l ) − [S n1 ,l ]2 [S n2 ,l ]3 + [S n1 ,l ]3 [S n2 ,l ]2 dV
Sc(S n1 ,l )[S n2 ,l ]2 + [S n1 ,l ]1 [S n2 ,l ]3 − [S n1 ,l ]2 Sc(S n2 ,l ) − [S n1 ,l ]3 [S n2 ,l ]1 dV
E
Sc(S n1 ,l )[S n2 ,l ]3 − [S n1 ,l ]1 [S n2 ,l ]2 + [S n1 ,l ]2 [S n2 ,l ]1 − [S n1 ,l ]3 Sc(S n2 ,l ) dV .
E
In the following we only address the calculations for the integrals (V) and (VI). The calculations for (VII) and (VIII) follow the same principle and are therefore omitted. Note that
(V) =
Sc(S n1 ,l ) Sc(S n2 ,l ) dV +
E
[S n1 ,l ]1 [S n2 ,l ]1 dV +
[S n1 ,l ]2 [S n2 ,l ]2 dV +
E
E
[S n1 ,l ]3 [S n2 ,l ]3 dV E
= (A) + (B) + (C) + (D). For simplicity we only present the calculations for (A)-integral. The proofs for the remaining integrals follow the same procedure and are therefore straightforward. Setting h as a harmonic polynomial of variables θ , µ, and φ . For each n ∈ N0 a direct computation shows that
⟨Sc(S n,l ), h⟩L2 (E ) = =
Sc(S n,l ) h ρ E
∂(µ, θ ) dφ dρ dz ∂(z , ρ)
(n + 2 + l) (n + 1 + l) (n − l + 1) 2
An,l (µ, θ )Tl (cos φ) h ρ E
∂(µ, θ ) dφ dρ dz . ∂(z , ρ)
(5)
Our attack is based rather upon the following closed-form representation proved in [5]: An,l (µ, θ ) =
1
cosh µ Pnl (cos θ ) Pnl +1 (cosh µ) − cos θ Pnl +1 (cos θ ) Pnl (cosh µ) .
sin θ + sinh µ 2
2
Hence, substituting in (5) we finally obtain
⟨Sc(S n,l ), h⟩L2 (E ) =
1 2
a
0
π
0
× cosh µ
2π
0 Pnl
h Tl (cos φ) sin θ sinh µ(n + 2 + l) (n + 1 + l) (n − l + 1)
(cos θ ) Pnl +1 (cosh µ) − cos θ Pnl +1 (cos θ ) Pnl (cosh µ) dφ dθ dµ.
As we may see the last integral vanishes when h is a harmonic polynomial of the form Psl (cos θ ) Psl (cosh µ) Tl (cos φ) with s < n, since obviously π
0
π
Pnl (cos θ ) sin θ Psl (cos θ ) dθ = 0 Pnl +1 (cos θ ) cos θ Psl (cos θ ) sin θ dθ = 0.
0
Hence, ⟨Sc(S n1 ,l ), Sc(S n2 ,l )⟩L2 (E ) = 0 for n1 ̸= n2 . It is significant to note in this connection that the remaining integrals (B), (C), and (D) also vanish. Thus, for n1 ̸= n2 the integral (V) vanishes.
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431
The same reasoning can be repeated to the integral
(VI) =
Sc(S n1 ,l )[S n2 ,l ]1 dV − E
[S n1 ,l ]1 Sc(S n2 ,l ) dV − E
[S n1 ,l ]2 [S n2 ,l ]3 dV + E
[S n1 ,l ]3 [S n2 ,l ]2 dV E
= (E) − (F) − (G) + (H). For each n ∈ N0 , a few straightforward computations show that
(E) + (H) − (F) − (G) =
2π T2l+1 (cos φ) dφ (n1 + 2 + l) (n1 + 1 + l) (n1 − l + 1) (n2 + 2 + l) 4 0 a π An2 ,l+1 sin θ sinh µ cosh µPnl 1 (cos θ ) Pnl 1 +1 (cosh µ) × 0 0 − cos θ Pnl 1 +1 (cos θ ) Pnl 1 (cosh µ) dθ dµ 2π 1 − (n1 + 2 + l) (n2 + 2 + l) (n2 + 1 + l) (n2 − l + 1) T2l+1 (cos φ) dφ 4 0 a π 1 × An2 ,l sin θ sinh µ cosh µ Pnl+ (cos θ ) Pnl+1 +1 1 (cosh µ) 1 0 0 − cos θ Pnl+1 +1 1 (cos θ ) Pnl+1 1 (cosh µ) dθ dµ 1
= 0,
l ≥ 0.
In summary, the set {S n,l : l = 0, . . . , n; n = 0, 1, . . .} is orthogonal in the sense of the quaternion product (1). Now we calculate the L2 -norms of the polynomials. We call attention to the fact that for s = n, we have the representations h := Sc(S n,l )
=
1 2
(n + 2 + l) (n + 1 + l) (n − l + 1)
(2n + 1) l P (cos θ ) Pnl (cosh µ) Tl (cos φ) (n + 1 − l) n
n−l
2 (2n + 1 − 2k) (n + l)2k l 1 Pn−2k (cos θ )Pnl −2k (cosh µ) Tl (cos φ) + (n + 2 + l) (n + 1 + l) (n − l + 1) 2 (n + 1 − l)2k+1 k=1
and h := [S n,l ]3 1
= − (n + 2 + l) (n + 1 + l) (n − l + 1) 2
(2n + 1) l P (cos θ ) Pnl (cosh µ) sin φ Ul−1 (cos φ) (n + 1 − l) n
n−l
2 1 (2n + 1 − 2k) (n + l)2k l − (n + 2 + l) (n + 1 + l) (n − l + 1) Pn−2k (cos θ ) 2 (n + 1 − l)2k+1 k=1
× Pnl −2k (cosh µ) sin φ Ul−1 (cos φ) where the last summands in each expression indicate harmonic polynomials of lower degree, which are orthogonal to Sc(S n,l ) and [S n,l ]3 , respectively. Thus, by direct inspection of previous expressions a first straightforward computation shows that
⟨Sc(S n1 ,l1 ), Sc(S n2 ,l2 )⟩L2 (E ) + ⟨[S n1 ,l1 ]3 , [S n2 ,l2 ]3 ⟩L2 (E ) =
1
a
π
2π
(n1 + 2 + l1 ) (n1 + 1 + l1 ) (n1 − l1 + 1) δn1 ,n2 δl1 ,l2 An1 ,l1 (µ, θ ) [Tl1 (cos φ)]2 sin θ sinh µ 4 0 0 0 l l × cosh µ Pnl11 (cos θ ) Pn11 +1 (cosh µ) − cos θ Pn11 +1 (cos θ ) Pnl11 (cosh µ) dφ dθ dµ 2
1
2
2
(n1 + 2 + l1 )2 (n1 + 1 + l1 )2 (n1 − l1 + 1)2 δn1 ,n2 δl1 ,l2 a π 2π × An1 ,l1 (µ, θ ) [sin φ Ul1 −1 (cos φ)]2 sin θ sinh µ 0 0 0 l l × cosh µ Pnl11 (cos θ ) Pn11 +1 (cosh µ) − cos θ Pn11 +1 (cos θ ) Pnl11 (cosh µ) dφ dθ dµ
+
4
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(n1 + l1 )! = π (n1 + 2 + l1 )2 (n1 + 1 + l1 )2 (n1 − l1 + 1)2 δn ,n δl ,l (n1 + 1 − l1 )! 1 2 1 2 a (n1 + 1 + l1 ) a l1 l Pnl11 (cosh µ) sinh µ cosh µ Pn11 +1 (cosh µ) dµ − × [Pn1 (cosh µ)]2 sinh µ dµ . 2n1 + 3 0 0 In a similar way, for s = n the following representations hold h := [S n,l ]1
=
1 2
(n + 2 + l)
(2n + 1) l+1 P (cos θ ) Pnl+1 (cosh µ) Tl+1 (cos φ) (n − l) n
n−1−l 2
(2n + 1 − 2k) (n + 1 + l)2k 1 1 l+1 Pnl+ + (n + 2 + l) −2k (cos θ )Pn−2k (cosh µ) Tl (cos φ) 2 (n − l)2k+1 k=1 and h := [S n,l ]2
=
1 2
(n + 2 + l)
(2n + 1) l+1 P (cos θ ) Pnl+1 (cosh µ) sin φ Ul (cos φ) (n − l) n
n−1−l 2
(2n + 1 − 2k) (n + 1 + l)2k 1 1 l+1 Pnl+ + (n + 2 + l) −2k (cos θ )Pn−2k (cosh µ) sin φ Ul (cos φ) 2 (n − l)2k+1 k=1 where the last summands in each expression indicate harmonic polynomials of lower degree, which are orthogonal to [S n,l ]1 and [S n,l ]2 , respectively. Thus, by direct inspection of previous expressions a direct computation shows that
⟨[S n1 ,l1 ]1 , [S n2 ,l2 ]1 ⟩L2 (E ) + ⟨[S n1 ,l1 ]2 , [S n2 ,l2 ]2 ⟩L2 (E ) a π 2π 1 An1 ,l1 +1 (µ, θ ) [Tl1 +1 (cos φ)]2 sin θ sinh µ = (n1 + 2 + l1 )2 δn1 ,n2 δl1 ,l2 4 0 0 0 l +1 l +1 × cosh µ Pnl11+1 (cos θ ) Pn11 +1 (cosh µ) − cos θ Pn11 +1 (cos θ ) Pnl11+1 (cosh µ) dφ dθ dµ a π 2π 1 + (n1 + 2 + l1 )2 δn1 ,n2 δl1 ,l2 An1 ,l1 +1 (µ, θ ) [sin φ Ul1 (cos φ)]2 sin θ sinh µ 4 0 0 0 l +1 l +1 × cosh µ Pnl11+1 (cos θ ) Pn11 +1 (cosh µ) − cos θ Pn11 +1 (cos θ ) Pnl11+1 (cosh µ) dφ dθ dµ a (n1 + 1 + l1 )! l +1 δn1 ,n2 δl1 ,l2 Pnl11+1 (cosh µ) sinh µ cosh µ Pn11 +1 (cosh µ) dµ = π (n1 + 2 + l1 )2 (n1 − l1 )! 0 (n1 + 2 + l1 ) a l1 +1 2 [Pn1 (cosh µ)] sinh µ dµ . − 2n1 + 3 0 In closing this section, note that for each degree n ∈ N0 , the system {S n,l : l = 0, . . . , n} is formed by n + 1 = dim M+ (E ; H; n) monogenicpolynomials, and therefore, it is complete in M+ (E ; H; n). Based on the orthogonal ∞ decomposition M + (E ; H) = ⊕ n=0 M + (E ; H; n), and the completeness of the system in each subspace M + (E ; H; n), the result follows. Theorem 2.2. For each n, the set of n + 1 linearly independent monogenic polynomials {S n,l : l = 0, . . . , n} forms an orthogonal basis in the subspace M + (E ; H; n) in the sense of the quaternion product (1). Consequently, S n,l : l = 0, . . . , n; n = 0, 1, . . . is an orthogonal basis in M + (E ; H). 3. Concluding remarks In this section we prove the orthogonality of the system {S n,l : l = 0, . . . , n; n = 0, 1, . . .} over the interior of the ellipse (2) for all values of α to obtain a corresponding orthogonality of the same system over the surface S of the spheroid E with respect to a suitable weight function. Next we formulate the result. Theorem 3.1. For each n ∈ N0 , the set {S n,l : l = 0, . . . , n} forms a complete orthogonal system over the surface of the spheroid (2) in the sense of the quaternion product
⟨f, g⟩L2 (S ;H;H) =
S
f g |1 − (z + i ρ)2 |1/2 dσ ,
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433
with weight function |1 − (z + i ρ)2 |1/2 equal to the square root of the product of the distances from (z , ρ, φ) to the north and south poles. Here, dσ denotes the Lebesgue measure on S . Proof. Let α be a nonnegative real number. By the same reasoning as in [9] and based on the previous theorem, it follows that d dα
[⟨Sc(S n1 ,l1 ), Sc(S n2 ,l2 )⟩L2 (E ) + ⟨[S n1 ,l1 ]3 , [S n2 ,l2 ]3 ⟩L2 (E ) ]
(n1 + l1 )! = π (n1 + 2 + l1 )2 (n1 + 1 + l1 )2 (n1 − l1 + 1)2 δ n ,n δ l ,l (n1 + 1 − l1 )! 1 2 1 2 a a (n1 + 1 + l1 ) d d l × Pnl11 (cosh µ) sinh µ cosh µ Pn11 +1 (cosh µ) dµ − [Pnl11 (cosh µ)]2 sinh µ dµ , dα 0 2n1 + 3 dα 0
and d
⟨[S n1 ,l1 ]1 , [S n1 ,l1 ]1 ⟩L2 (E ) + ⟨[S n1 ,l1 ]2 , [S n2 ,l2 ]2 ⟩L2 (E ) a l +1 2 (n1 + 1 + l1 )! Pnl11+1 (cosh µ) sinh µ cosh µ Pn11 +1 (cosh µ) dµ = π (n1 + 2 + l1 ) δn1 ,n2 δl1 ,l2 (n1 − l1 )! 0 (n1 + 2 + l1 ) a l1 +1 2 [Pn1 (cosh µ)] sinh µ dµ , − 2n1 + 3 0
dα
whence
⟨S n1 ,l1 , S n2 ,l2 ⟩L2 (S ;H;H) =
S n1 ,l1 S n2 ,l2 |1 − (z + i ρ)2 |1/2 dσ S
(n1 + 2 + l1 )! δn1 ,n2 δl1 ,l2 = π (n1 + 2 + l1 ) (n1 − l1 )! l × (n1 + 1 + l1 ) (n1 − l1 + 1) Pnl11 (cosh α) sinh α cosh α Pn11 +1 (cosh α) −
1 2n1 + 3
(n1 + 1 + l1 )2 (n1 − l1 + 1)[Pnl11 (cosh α)]2 sinh α l +1
+ Pnl11+1 (cosh α) sinh α cosh α Pn11 +1 (cosh α) −
(n1 + 2 + l1 ) l1 +1 [Pn1 (cosh α)]2 sinh α . 2n1 + 3
Here the weight function |1 − (z + i ρ)2 |1/2 equals the square root of the product of the distances from (z , ρ, φ) to the points (1, 0, 0) and (−1, 0, 0). 4. Perspectives An explicit complete orthogonal system of polynomial (MT)-solutions has been presented, and we hope it constitutes a new class of practical and flexible basis within the context of quaternion analysis. In our point of view the system can be seen as a refinement of the three-dimensional homogeneous monogenic polynomials recently exploited by several authors. For a general orientation the reader is suggested to check some of the existing pioneering works [10–15] and [11,16,14, 17–19], of which the first six deal entirely with the R3 −→ R4 case and the last six the R3 −→ R3 case. It should be noticed that the application of homotopy methods in such polynomials will not preserve their monogenicity, and therefore it will not allow the generation of the polynomials presented here. The constructed system is composed by piecewise monogenic polynomials, and is capable of describing exactly the general information expressed by the commonly used Ferrer’s associated Legendre functions, which seems to be suitable for the treatment of monogenic functions by the use of power series expansions. In addition, we give explicit representations of the L2 -norms of these polynomials. This makes it possible to construct a complete orthonormal system, and consequently to define the Fourier expansion of a square integrable H-valued monogenic function in M + (E ; H). Further investigations on this topic will be reported in a forthcoming paper. Acknowledgments This work was supported by FEDER funds through COMPETE–Operational Programme Factors of Competitiveness (‘‘Programa Operacional Factores de Competitividade’’) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (‘‘FCT–Fundação para a Ciência e a Tecnologia’’), within project PEst-C/MAT/UI4106/2011 with COMPETE
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number FCOMP-01-0124-FEDER-022690. Partial support from the Foundation for Science and Technology (FCT) via the post-doctoral grant SFRH/BPD/66342/2009 is also acknowledged by the author. References [1] G. Moisil, N. Théodoresco, Fonctions holomorphes dans l’espace, Matematica (Cluj) 5 (1931) 142–159. [2] V. Kravchenko, M. Shapiro, Integral Representations for Spatial Models of Mathematical Physics, in: Research Notes in Mathematics, Pitman Advanced Publishing Program, London, 1996. [3] R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Commentarii Mathematici Helvetici 4 (1932) 9–20. [4] A. Sudbery, Quaternionic analysis, Mathematical Proceedings Cambridge Philosophical Society 85 (1979) 199–225. [5] J. Morais, A complete orthogonal system of spheroidal monogenics, Journal of Numerical Analysis, Industrial and Applied Mathematics 6 (3–4) (2011) 105–119. [6] S. Georgiev, J. Morais, On convergence aspects of spheroidal monogenics, AIP Conference Proceedings 1389 (2011) 276–279. [7] J. Morais, S. Georgiev, Complete orthogonal systems of 3D spheroidal monogenics, in: K. Gürlebeck, T. Lahmer, F. Werner (Eds.), Proceedings of the 19th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar. [8] E. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge, 1931. [9] P. Garabedian, Orthogonal harmonic polynomials, Pacific Journal of Mathematics 3 (3) (1953) 585–603. [10] S. Bock, K. Gürlebeck, On a generalized Appell system and monogenic power series, Mathematical Methods in the Applied Sciences 33 (4) (2009) 394–411. [11] I. Cação, Constructive approximation by monogenic polynomials, Ph.D. Thesis, Universidade de Aveiro, 2004. [12] I. Cação, K. Gürlebeck, S. Bock, Complete orthonormal systems of spherical monogenics—a constructive approach, in: L.H. Son, et al. (Eds.), Methods of Complex and Clifford Analysis, SAS International Publications, Delhi, 2005, pp. 241–260. [13] I. Cação, K. Gürlebeck, S. Bock, On derivatives of spherical monogenics, Complex Variables and Elliptic Equations 51 (8–11) (2006) 847–869. [14] I. Cação, Complete orthonormal sets of polynomial solutions of the Riesz and Moisil–Théodoresco systems in R3 , Numerical Algorithms 55 (2–3) (2010) 191–203. [15] R. Delanghe, On homogeneous polynomial solutions of the Moisil–Théodoresco system in R3 , Computational Methods and Function Theory 9 (1) (2009) 199–212. [16] I. Cação, H. Malonek, Remarks on some properties of monogenic polynomials, in: T.E. Simos, G. Psihoyios, Ch. Tsitouras (Eds.), ICNAAM, Special Volume of Wiley-VCH, 2006, pp. 596–599. [17] R. Delanghe, On homogeneous polynomial solutions of the Riesz system and their harmonic potentials, Complex Variables and Elliptic Equations 52 (10–11) (2007) 1047–1062. [18] J. Morais, K. Gürlebeck, Real-part estimates for solutions of the Riesz system in R3 , Complex Variables and Elliptic Equations 57 (5) (2012) 505–522. [19] J. Morais, Approximation by homogeneous polynomial solutions of the Riesz system in R3 , Ph.D. Thesis, Bauhaus-Universität Weimar, 2009.
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An orthogonal system of monogenic polynomials over prolate spheroids in R3 J. Morais Technical University of Mining, Freiberg, Germany
article
info
Article history: Received 13 May 2011 Received in revised form 14 September 2011 Accepted 15 June 2012 Keywords: Quaternion analysis Moisil–Théodoresco system Monogenic functions Ferrer’s associated Legendre functions Chebyshev polynomials Hyperbolic functions
abstract The object of this paper is to construct a complete orthogonal system of monogenic polynomials as solutions of the Moisil–Théodoresco system over prolate spheroids in R3 . This will be done in the spaces of square integrable functions over H. A big breakthrough is that the orthogonality of the polynomials in question does not depend on the shape of the spheroids, but only on the location of the foci of the ellipse generating the spheroid. The representations of these polynomials are given explicitly, ready to be implemented on a computer. In addition, we show a corresponding orthogonality of the same polynomials over the surface of the spheroids with respect to a suitable weight function. © 2012 Elsevier Ltd. All rights reserved.
1. Introduction Consider the pair u = (u0 , u∗ ), where u0 is a real-valued continuously differentiable function defined on an open domain Ω of R3 , and u∗ = (u1 , u2 , u3 ) is a vector-valued function in Ω whose components ui = ui (x0 , x1 , x2 ) are real functions of real variables x0 , x1 , x2 . Following [1] (see also [2]) the Moisil–Théodoresco system is represented by ∂ u + ∂x1 u2 + ∂x2 u3 = 0 x0 1 div u∗ = 0 ∂x0 u0 − ∂x2 u2 + ∂x1 u3 = 0 (MT) ⇐⇒ grad u0 + rot u∗ = 0 ∂x1 u0 + ∂x2 u1 − ∂x0 u3 = 0 ∂x2 u0 − ∂x1 u1 + ∂x0 u2 = 0. The (MT)-system clearly generalizes the classical Cauchy–Riemann system for holomorphic functions in the complex plane. To start with, the (MT)-system may be obtained consistently by working with the quaternion algebra. Let H := {z = a0 + a1 i + a2 j + a3 k : ai ∈ R, i = 0, 1, 2, 3} be the Hamiltonian skew field, where the imaginary units i, j, and k are subject to the multiplication rules i2 = j2 = k2 = −1; ij = k = −ji,
jk = i = −kj,
ki = j = −ik.
At this stage we may introduce some terms which will be frequently used. The scalar and vector parts of z, Sc(z) and Vec(z), are defined as the a0 and a1 i + a2 j + a3 k terms, respectively. Like in the complex case, the conjugate of z is the quaternion z = a0 − a1 i − a2 j − a3 k, and the norm |z| of z is defined by |z|2 = zz = zz = a20 + a21 + a22 + a23 that coincides with its corresponding Euclidean norm as a vector in R4 . Now, let Ω be an open subset of R3 with a piecewise smooth boundary,
E-mail address: [email protected]. 0895-7177/$ – see front matter © 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.mcm.2012.06.020
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and x := (x0 , x1 , x2 ) ∈ Ω . The standard form of a quaternion-valued function or, briefly, an H-valued function, will be taken to be f : Ω −→ H,
f(x) = [f(x)]0 + [f(x)]1 i + [f(x)]2 j + [f(x)]3 k
where [f]i (i = 0, 1, 2, 3) are real-valued functions defined in Ω . Properties such as continuity, differentiability, integrability, and so on, which are ascribed to f have to be fulfilled by all components [f]i . We further introduce the right H-linear Hilbert space of square integrable H-valued functions defined in Ω , that we denote by L2 (Ω ; H; H). In this assignment, the H-valued inner product is defined by
⟨f, g⟩L2 (Ω ;H;H) =
f g dV
(1)
Ω
where dV denotes the Lebesgue measure on Ω . To simplify matters further we shall remark that using the embedding of R in H the inner product of two real-valued functions f , g : Ω −→ R can also be written by using the inner product (1), and it will be denoted simply by ⟨f , g ⟩L2 (Ω ) . The classical Dirac operator (also known as Moisil–Théodoresco operator [3,1])
∂=i
∂ ∂ ∂ +j +k ∂ x0 ∂ x1 ∂ x2
arises commonly in quaternion analysis and is very important. Nullsolutions to this operator provide us with the class of the so-called monogenic functions. Definition 1.1 (Monogenicity). An H-valued function f is called left (resp. right) monogenic in Ω if f is in C 1 (Ω ; H) and satisfies ∂ f = 0 (resp. f∂ = 0) in Ω . For, throughout this text, and except otherwise stated, we only use left H-valued monogenic functions that, for simplicity, we call monogenic. Nevertheless, all results accomplished to left H-valued monogenic functions can easily be adapted to right H-valued monogenic functions. Ultimately, consider a continuously differentiable H-valued function f, and the associated pair u = (u0 , u∗ ). Putting u0 = [f ]0 , u1 = [f ]3 , u2 = [f ]2 , and u3 = −[f ]1 , the (MT)-system can be rewritten as (MT) ⇐⇒ ∂ f = 0. From now on, the solutions of the (MT)-system will be called (MT)-solutions. The subspace of polynomial (MT)-solutions of degree n will be denoted by M + (Ω ; H; n). In [4], Sudbery proved that dim M + (Ω ; H; n) = n + 1. We also denote by M + (Ω ; H) := L2 (Ω ; H; H) ∩ ker ∂ the space of square integrable H-valued monogenic functions defined in Ω . 2. An orthogonal system of monogenic polynomials over the interior of 3D prolate spheroids In the sequel we shall make use of rectangular coordinates, and cylindrical coordinates: x0 = z , x1 = ρ cos φ, x2 = ρ sin φ , where z ∈ (−∞, +∞), ρ ∈ [0, +∞) and φ ∈ [0, 2π ), emphasizing that φ = arccos( xρ1 ) if ρ > 0 and φ = 0 if ρ = 0. In the present section we seek to assemble a special orthogonal system of H-valued monogenic polynomials in the interior of the surface of revolution
E :
z2 cosh2 α
+
ρ2 sinh2 α
= 1,
(2)
where α is a nonnegative real number. Representation (2) shows that surfaces of constant α do indeed form confocal prolate spheroids, since they are ellipses rotated about the axis joining their foci. At this stage it is convenient to introduce coordinates, θ and µ, defined by the relations: z = cos θ cosh µ and ρ = sin θ sinh µ, where θ ∈ [0, π] and µ ∈ [0, a]. In this case, the boundary of the above spheroid has the equation µ = α . In [5] the author found it necessary to focus the discussion, temporarily at least, on spaces of square integrable functions over R. With this outcome in mind, a complete orthogonal set
{E n,l , F n,m : l = 0, . . . , n + 1, m = 1, . . . , n + 1} of polynomial nullsolutions of the well known Riesz system has been developed over prolate spheroids in R3 . The mentioned prolate spheroidal monogenics are explicitly given by
E n,l (µ, θ , φ) :=
(n + l + 1) 2
+
1 4
An,l (µ, θ ) Tl (cos φ) +
1 4 (n − l + 1)
An,l+1 (µ, θ ) [Tl+1 (cos φ)i + sin φ Ul (cos φ)j]
(n + 1 + l)(n + l)(n − l + 2)An,l−1 (µ, θ ) [−Tl−1 (cos φ)i + sin φ Ul−2 (cos φ)j] ,
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427
and
(n + m + 1)
F n,m (µ, θ , φ) :=
2
An,m (µ, θ ) sin φ Um−1 (cos φ)
1 An,m+1 (µ, θ ) [sin φ Um (cos φ)i − Tm+1 (cos φ)j] 4 (n − m + 1) 1 − (n + 1 + m)(n + m)(n − m + 2)An,m−1 (µ, θ ) [sin φ Um−2 (cos φ)i + Tm−1 (cos φ)j] , 4 for l = 0, . . . , n + 1 and m = 1, . . . , n + 1, with the subscript coefficient function
+
n−l
2 (2n + 1 − 2k) (n + l)2k l Pn−2k (cos θ ) Pnl −2k (cosh µ) An,l (µ, θ ) := ( n + 1 − l ) 2k + 1 k=0
(3)
such that An,−1 (µ, θ ) = − n(n+1)12 (n+2) An,1 (µ, θ ). We shall use (a)r to stand the Pochhammer symbol, i.e. (a)r := a(a + 1) · · · (a + r − 1) = ΓΓ(a(+r )r ) for any integer r > 1, and (a)0 := 1. As usual, ⌈s⌉ denotes the smallest integer not less than s ∈ R. In addition, Pnl stands for the Ferrer’s associated Legendre functions of degree n and order l of the first kind, Tl and Ul are the Chebyshev polynomials of the first and second kinds, respectively. For the sake of completeness we set l
Pn (cosh µ) = Pn0 (cosh µ) and Pnl (cosh µ) = (−1)l (sinh µ)l dtd l [Pn (t )]|t =cosh µ . Remark 2.1. One may show by elementary reasoning that the well-known recurrence properties of the Legendre polynomials Pn (cosh µ) and their associated Legendre functions Pnl (cosh µ)(l > 0) are different from the ones we are familiar with. We shall now be able to extend the results in that connection to a quaternionic Hilbert subspace; in particular, we construct a complete orthogonal system of polynomial nullsolutions of the Moisil–Théodoresco system over 3D prolate spheroids, whose study will require us to explore a new situation. We emphasize that the used methods are quite general and can be either extended to oblate spheroids or even to arbitrary ellipsoids in the 3D space. But such a procedure would make the further study of the underlying polynomials somewhat laborious, and for this reason we shall prefer not to discuss these cases here. For, in continuation of [5] (cf. [6,7]) we designate the new n + 1 (prolate) spheroidal monogenics by
S n,l := E n,l+1 i + F n,l+1 j,
l = 0 , . . . , n,
(4)
namely functions with respect to the variables µ, θ , and the azimuthal angle φ of the quaternion form:
S n,l (µ, θ , φ) :=
1 2
+
(n + 2 + l) (n + 1 + l) (n − l + 1) An,l (µ, θ ) Tl (cos φ) + 1 2 1
1 2
(n + 2 + l) An,l+1 (µ, θ ) Tl+1 (cos φ) i
(n + 2 + l) An,l+1 (µ, θ ) sin φ Ul (cos φ) j
(n + 2 + l) (n + 1 + l) (n − l + 1) An,l (µ, θ ) sin φ Ul−1 (cos φ) k 2 with the subscript coefficient function An,l (µ, θ ) given by (3). It is convenient to regard (4) as a new special system of monogenic polynomials despite its similarity with other systems in literature. −
Remark 2.2. Although the monogenic polynomials S n,0 are built in terms of the Legendre polynomials while S n,l (l > 0) are built in terms of the Ferrer’s associated Legendre functions, we still include the treatment of the first into the general case, whenever this does not raise any confusion and the treatment remains the same. Of course, the results can be extended to the case of the region outside a spheroid as well. One has merely to replace the Ferrer’s associated Legendre functions by the Legendre functions of second kind [8]. In the considerations to follow we will often omit the argument and write simply S n,l instead of S n,l (µ, θ , φ). In particular, these n + 1 system constituents satisfy the first order partial differential equation 0 = ∂ S n ,l
∂ S n,l sin θ cosh µ ∂ S n,l − sin2 θ + sinh2 µ ∂µ sin2 θ + sinh2 µ ∂θ sin θ cosh µ cos φ ∂ S n,l cos θ sinh µ cos φ ∂ S n,l sin φ ∂ S n,l +j + − sin θ sinh µ ∂φ sin2 θ + sinh2 µ ∂µ sin2 θ + sinh2 µ ∂θ sin θ cosh µ sin φ ∂ S n,l cos θ sinh µ sin φ ∂ S n,l cos φ ∂ S n ,l +k + + . sin θ sinh µ ∂φ sin2 θ + sinh2 µ ∂µ sin2 θ + sinh2 µ ∂θ
=i
cos θ sinh µ
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We further assume the reader to be familiar with the fact that ∂ is a square root of the Laplace operator in R3 in the sense that
∆3 S n,l = −∂ 2 S n,l
1
=
sin2 θ + sinh2 µ
∂ 2 S n ,l ∂ S n ,l ∂ S n ,l ∂ 2 S n,l + + coth µ + cot θ ∂µ2 ∂θ 2 ∂µ ∂θ
+
∂ 2 S n ,l . sin2 θ sinh2 µ ∂φ 2 1
For, the monogenic polynomials S n,l can be seen as a refinement of the harmonic polynomials exploited by Garabedian in [9]. We formulate a first result to our initial calculation. Lemma 2.1. The monogenic polynomials S n,l are the zero functions for l ≥ n + 1. We may now deliver the orthogonality of the above-mentioned polynomials over the interior of the prolate spheroid (2) in the sense of the quaternion product (1), which is the main objective of the present section. A big step forward is that the orthogonality in question does not depend on the shape of the spheroids, but only on their size. More specifically, the polynomials S n,l (l ≥ 0) depend on the location of the foci of the ellipse generating the spheroid only, and not on its eccentricity. Theorem 2.1. For each n ∈ N0 , the set {S n,l : l = 0, . . . , n} is orthogonal over the interior of the prolate spheroid (2) in the sense of the quaternion product (1), and their norms are given by
(n + 2 + l)! ∥S n,l ∥2L2 (E ;H;H) = π (n + 2 + l) (n − l)! × (n + 1 + l) (n − l + 1)
a
Pnl (cosh µ) sinh µ cosh µ Pnl +1 (cosh µ) dµ 0
−
1 2n + 3
+
(n + 1 + l)2 (n − l + 1)
a
[Pnl (cosh µ)]2 sinh µ dµ 0
a 1 Pnl+1 (cosh µ) sinh µ cosh µ Pnl+ +1 (cosh µ) dµ
0
(n + 2 + l) − 2n + 3
a
[Pnl+1 (cosh µ)]2 sinh µ dµ .
0
Proof. We begin by proving the orthogonality of the polynomials. By definition of the quaternion product (1) for a fixed n ∈ N0 we have
⟨S n,l1 , S n,l2 ⟩L2 (E ;H;H) =
S n,l1 S n,l2 dV E
= (I) + (II)i + (III)j + (IV)k where
(I) =
Sc(S n,l1 ) Sc(S n,l2 ) + [S n,l1 ]1 [S n,l2 ]1 + [S n,l1 ]2 [S n,l2 ]2 + [S n,l1 ]3 [S n,l2 ]3 dV
E
and, moreover
(II) = (III) =
(IV) =
Sc(S n,l1 )[S n,l2 ]1 − [S n,l1 ]1 Sc(S n,l2 ) − [S n,l1 ]2 [S n,l2 ]3 + [S n,l1 ]3 [S n,l2 ]2 dV
E
Sc(S n,l1 )[S n,l2 ]2 + [S n,l1 ]1 [S n,l2 ]3 − [S n,l1 ]2 Sc(S n,l2 ) − [S n,l1 ]3 [S n,l2 ]1 dV
E
Sc(S n,l1 )[S n,l2 ]3 − [S n,l1 ]1 [S n,l2 ]2 + [S n,l1 ]2 [S n,l2 ]1 − [S n,l1 ]3 Sc(S n,l2 ) dV .
E
Assume we have the change of variables z = z (µ, θ ) = cos θ cosh µ and ρ = ρ(µ, θ ) = sin θ sinh µ. A first straightforward computation shows
(I) =
=
∂(µ, θ ) dφ dρ dz ∂(z , ρ) E ∂(µ, θ ) + [S n,l1 ]1 [S n,l2 ]1 + [S n,l1 ]2 [S n,l2 ]2 ρ dφ dρ dz ∂(z , ρ) E
1 4
Sc(S n,l1 )Sc(S n,l2 ) + [S n,l1 ]3 [S n,l2 ]3 ρ
(n + 2 + l1 ) (n + 1 + l1 ) (n − l1 + 1) (n + 2 + l2 ) (n + 1 + l2 ) (n − l2 + 1)
J. Morais / Mathematical and Computer Modelling 57 (2013) 425–434
π
429
2π ∂(µ, θ ) Tl1 −l2 (cos φ) dφ dθ dµ ∂(z , ρ) 0 0 0 a π 1 ∂(µ, θ ) + An,l1 +1 (µ, θ ) An,l2 +1 (µ, θ ) sin θ sinh µ dθ dµ 4 0 0 ∂(z , ρ) 2π Tl1 −l2 (cos φ) dφ × (n + 2 + l1 ) (n + 2 + l2 ) a
×
An,l1 (µ, θ ) An,l2 (µ, θ ) sin θ sinh µ
0
= 0,
l1 ̸= l2 .
We will proceed in such a manner to compute the remaining integrals
∂(µ, θ ) dφ dρ dz ∂(z , ρ) E ∂(µ, θ ) [S n,l1 ]1 Sc(S n,l2 ) + [S n,l1 ]2 [S n,l2 ]3 ρ dφ dρ dz − ∂(z , ρ) a E π ∂(µ, θ ) = An,l1 (µ, θ ) An,l2 +1 (µ, θ ) sin θ sinh µ dθ dµ ∂(z , ρ) 0 0 2π 1 Tl1 +1+l2 (cos φ) dφ × (n + 2 + l1 ) (n + 1 + l1 ) (n − l1 + 1) (n + 2 + l2 ) 4 0 a π ∂(µ, θ ) − An,l1 +1 (µ, θ ) An,l2 (µ, θ ) sin θ sinh µ dθ dµ ∂(z , ρ) 0 0 2π 1 × (n + 2 + l1 ) (n + 2 + l2 ) (n + 1 + l2 ) (n − l2 + 1) Tl1 +1+l2 (cos φ) dφ
(II) =
Sc(S n,l1 )[S n,l2 ]1 + [S n,l1 ]3 [S n,l2 ]2 ρ
4
= 0,
0
l1 ̸= l2
and moreover,
∂(µ, θ ) dφ dρ dz ∂(z , ρ) E ∂(µ, θ ) + [S n,l1 ]1 [S n,l2 ]3 − [S n,l1 ]2 Sc(S n,l2 ) ρ dφ dρ dz ∂(z , ρ) a E π ∂(µ, θ ) = An,l1 (µ, θ ) An,l2 +1 (µ, θ ) sin θ sinh µ dθ dµ ∂(z , ρ) 0 0
(III) =
Sc(S n,l1 )[S n,l2 ]2 − [S n,l1 ]3 [S n,l2 ]1 ρ
2π
1
(n + 2 + l1 ) (n + 1 + l1 ) (n − l1 + 1) (n + 2 + l2 ) sin φ Ul1 +l2 (cos φ) dφ 4 0 a π ∂(µ, θ ) dθ dµ − An,l1 +1 (µ, θ ) An,l2 (µ, θ ) sin θ sinh µ ∂(z , ρ) 0 0 2π 1 × (n + 2 + l1 ) (n + 2 + l2 ) (n + 1 + l2 ) (n − l2 + 1) sin φ Ul1 +l2 (cos φ) dφ ×
4
0
= 0, l1 ̸= l2 . ∂(µ, θ ) dφ dρ dz (IV) = Sc(S n,l1 )[S n,l2 ]3 − [S n,l1 ]3 Sc(S n,l2 ) ρ ∂(z , ρ) E ∂(µ, θ ) − [S n,l1 ]1 [S n,l2 ]2 − [S n,l1 ]2 [S n,l2 ]1 ρ dφ dρ dz ∂(z , ρ) E =
1
(n + 2 + l1 ) (n + 1 + l1 ) (n − l1 + 1) (n + 2 + l2 ) (n + 1 + l2 ) (n − l2 + 1) 4 2π a π
∂(µ, θ ) dθ dµ sin φ Ul1 −1−l2 (cos φ) dφ ∂(z , ρ) 0 0 a 0 π ∂(µ, θ ) + An,l1 +1 (µ, θ ) An,l2 +1 (µ, θ ) sin θ sinh µ dθ dµ ∂(z , ρ) 0 0 An,l1 (µ, θ ) An,l2 (µ, θ ) sin θ sinh µ
×
× = 0,
1
4
2π
(n + 2 + l1 ) (n + 2 + l2 )
l1 ̸= l2 .
sin φ Ul1 −1−l2 (cos φ) dφ
0
Thus, for a fixed n ∈ N0 the set {S n,l : l = 0, . . . , n} is orthogonal in the sense of the quaternion product (1). We proceed to prove the orthogonality of the system for each degree n. The proof is mainly based on the expressions of the
430
J. Morais / Mathematical and Computer Modelling 57 (2013) 425–434
aforementioned spheroidal monogenics combining several recurrence properties of the Legendre polynomials and Ferrer’s associated Legendre functions, and some equations between the Chebyshev polynomials of the first and second kinds. Therefore, it requires extensive calculations. Here, we only give the main steps of the proof. Once again, by definition of the quaternion product (1) it follows that
⟨S n1 ,l , S n2 ,l ⟩L2 (E ;H;H) =
S n1 ,l S n2 ,l dV E
= (V) + (VI)i + (VII)j + (VIII)k where
(V) =
Sc(S n1 ,l ) Sc(S n2 ,l ) + [S n1 ,l ]1 [S n2 ,l ]1 + [S n1 ,l ]2 [S n2 ,l ]2 + [S n1 ,l ]3 [S n2 ,l ]3 dV
E
and
(VI) = (VII) =
E
(VIII) =
Sc(S n1 ,l )[S n2 ,l ]1 − [S n1 ,l ]1 Sc(S n2 ,l ) − [S n1 ,l ]2 [S n2 ,l ]3 + [S n1 ,l ]3 [S n2 ,l ]2 dV
Sc(S n1 ,l )[S n2 ,l ]2 + [S n1 ,l ]1 [S n2 ,l ]3 − [S n1 ,l ]2 Sc(S n2 ,l ) − [S n1 ,l ]3 [S n2 ,l ]1 dV
E
Sc(S n1 ,l )[S n2 ,l ]3 − [S n1 ,l ]1 [S n2 ,l ]2 + [S n1 ,l ]2 [S n2 ,l ]1 − [S n1 ,l ]3 Sc(S n2 ,l ) dV .
E
In the following we only address the calculations for the integrals (V) and (VI). The calculations for (VII) and (VIII) follow the same principle and are therefore omitted. Note that
(V) =
Sc(S n1 ,l ) Sc(S n2 ,l ) dV +
E
[S n1 ,l ]1 [S n2 ,l ]1 dV +
[S n1 ,l ]2 [S n2 ,l ]2 dV +
E
E
[S n1 ,l ]3 [S n2 ,l ]3 dV E
= (A) + (B) + (C) + (D). For simplicity we only present the calculations for (A)-integral. The proofs for the remaining integrals follow the same procedure and are therefore straightforward. Setting h as a harmonic polynomial of variables θ , µ, and φ . For each n ∈ N0 a direct computation shows that
⟨Sc(S n,l ), h⟩L2 (E ) = =
Sc(S n,l ) h ρ E
∂(µ, θ ) dφ dρ dz ∂(z , ρ)
(n + 2 + l) (n + 1 + l) (n − l + 1) 2
An,l (µ, θ )Tl (cos φ) h ρ E
∂(µ, θ ) dφ dρ dz . ∂(z , ρ)
(5)
Our attack is based rather upon the following closed-form representation proved in [5]: An,l (µ, θ ) =
1
cosh µ Pnl (cos θ ) Pnl +1 (cosh µ) − cos θ Pnl +1 (cos θ ) Pnl (cosh µ) .
sin θ + sinh µ 2
2
Hence, substituting in (5) we finally obtain
⟨Sc(S n,l ), h⟩L2 (E ) =
1 2
a
0
π
0
× cosh µ
2π
0 Pnl
h Tl (cos φ) sin θ sinh µ(n + 2 + l) (n + 1 + l) (n − l + 1)
(cos θ ) Pnl +1 (cosh µ) − cos θ Pnl +1 (cos θ ) Pnl (cosh µ) dφ dθ dµ.
As we may see the last integral vanishes when h is a harmonic polynomial of the form Psl (cos θ ) Psl (cosh µ) Tl (cos φ) with s < n, since obviously π
0
π
Pnl (cos θ ) sin θ Psl (cos θ ) dθ = 0 Pnl +1 (cos θ ) cos θ Psl (cos θ ) sin θ dθ = 0.
0
Hence, ⟨Sc(S n1 ,l ), Sc(S n2 ,l )⟩L2 (E ) = 0 for n1 ̸= n2 . It is significant to note in this connection that the remaining integrals (B), (C), and (D) also vanish. Thus, for n1 ̸= n2 the integral (V) vanishes.
J. Morais / Mathematical and Computer Modelling 57 (2013) 425–434
431
The same reasoning can be repeated to the integral
(VI) =
Sc(S n1 ,l )[S n2 ,l ]1 dV − E
[S n1 ,l ]1 Sc(S n2 ,l ) dV − E
[S n1 ,l ]2 [S n2 ,l ]3 dV + E
[S n1 ,l ]3 [S n2 ,l ]2 dV E
= (E) − (F) − (G) + (H). For each n ∈ N0 , a few straightforward computations show that
(E) + (H) − (F) − (G) =
2π T2l+1 (cos φ) dφ (n1 + 2 + l) (n1 + 1 + l) (n1 − l + 1) (n2 + 2 + l) 4 0 a π An2 ,l+1 sin θ sinh µ cosh µPnl 1 (cos θ ) Pnl 1 +1 (cosh µ) × 0 0 − cos θ Pnl 1 +1 (cos θ ) Pnl 1 (cosh µ) dθ dµ 2π 1 − (n1 + 2 + l) (n2 + 2 + l) (n2 + 1 + l) (n2 − l + 1) T2l+1 (cos φ) dφ 4 0 a π 1 × An2 ,l sin θ sinh µ cosh µ Pnl+ (cos θ ) Pnl+1 +1 1 (cosh µ) 1 0 0 − cos θ Pnl+1 +1 1 (cos θ ) Pnl+1 1 (cosh µ) dθ dµ 1
= 0,
l ≥ 0.
In summary, the set {S n,l : l = 0, . . . , n; n = 0, 1, . . .} is orthogonal in the sense of the quaternion product (1). Now we calculate the L2 -norms of the polynomials. We call attention to the fact that for s = n, we have the representations h := Sc(S n,l )
=
1 2
(n + 2 + l) (n + 1 + l) (n − l + 1)
(2n + 1) l P (cos θ ) Pnl (cosh µ) Tl (cos φ) (n + 1 − l) n
n−l
2 (2n + 1 − 2k) (n + l)2k l 1 Pn−2k (cos θ )Pnl −2k (cosh µ) Tl (cos φ) + (n + 2 + l) (n + 1 + l) (n − l + 1) 2 (n + 1 − l)2k+1 k=1
and h := [S n,l ]3 1
= − (n + 2 + l) (n + 1 + l) (n − l + 1) 2
(2n + 1) l P (cos θ ) Pnl (cosh µ) sin φ Ul−1 (cos φ) (n + 1 − l) n
n−l
2 1 (2n + 1 − 2k) (n + l)2k l − (n + 2 + l) (n + 1 + l) (n − l + 1) Pn−2k (cos θ ) 2 (n + 1 − l)2k+1 k=1
× Pnl −2k (cosh µ) sin φ Ul−1 (cos φ) where the last summands in each expression indicate harmonic polynomials of lower degree, which are orthogonal to Sc(S n,l ) and [S n,l ]3 , respectively. Thus, by direct inspection of previous expressions a first straightforward computation shows that
⟨Sc(S n1 ,l1 ), Sc(S n2 ,l2 )⟩L2 (E ) + ⟨[S n1 ,l1 ]3 , [S n2 ,l2 ]3 ⟩L2 (E ) =
1
a
π
2π
(n1 + 2 + l1 ) (n1 + 1 + l1 ) (n1 − l1 + 1) δn1 ,n2 δl1 ,l2 An1 ,l1 (µ, θ ) [Tl1 (cos φ)]2 sin θ sinh µ 4 0 0 0 l l × cosh µ Pnl11 (cos θ ) Pn11 +1 (cosh µ) − cos θ Pn11 +1 (cos θ ) Pnl11 (cosh µ) dφ dθ dµ 2
1
2
2
(n1 + 2 + l1 )2 (n1 + 1 + l1 )2 (n1 − l1 + 1)2 δn1 ,n2 δl1 ,l2 a π 2π × An1 ,l1 (µ, θ ) [sin φ Ul1 −1 (cos φ)]2 sin θ sinh µ 0 0 0 l l × cosh µ Pnl11 (cos θ ) Pn11 +1 (cosh µ) − cos θ Pn11 +1 (cos θ ) Pnl11 (cosh µ) dφ dθ dµ
+
4
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J. Morais / Mathematical and Computer Modelling 57 (2013) 425–434
(n1 + l1 )! = π (n1 + 2 + l1 )2 (n1 + 1 + l1 )2 (n1 − l1 + 1)2 δn ,n δl ,l (n1 + 1 − l1 )! 1 2 1 2 a (n1 + 1 + l1 ) a l1 l Pnl11 (cosh µ) sinh µ cosh µ Pn11 +1 (cosh µ) dµ − × [Pn1 (cosh µ)]2 sinh µ dµ . 2n1 + 3 0 0 In a similar way, for s = n the following representations hold h := [S n,l ]1
=
1 2
(n + 2 + l)
(2n + 1) l+1 P (cos θ ) Pnl+1 (cosh µ) Tl+1 (cos φ) (n − l) n
n−1−l 2
(2n + 1 − 2k) (n + 1 + l)2k 1 1 l+1 Pnl+ + (n + 2 + l) −2k (cos θ )Pn−2k (cosh µ) Tl (cos φ) 2 (n − l)2k+1 k=1 and h := [S n,l ]2
=
1 2
(n + 2 + l)
(2n + 1) l+1 P (cos θ ) Pnl+1 (cosh µ) sin φ Ul (cos φ) (n − l) n
n−1−l 2
(2n + 1 − 2k) (n + 1 + l)2k 1 1 l+1 Pnl+ + (n + 2 + l) −2k (cos θ )Pn−2k (cosh µ) sin φ Ul (cos φ) 2 (n − l)2k+1 k=1 where the last summands in each expression indicate harmonic polynomials of lower degree, which are orthogonal to [S n,l ]1 and [S n,l ]2 , respectively. Thus, by direct inspection of previous expressions a direct computation shows that
⟨[S n1 ,l1 ]1 , [S n2 ,l2 ]1 ⟩L2 (E ) + ⟨[S n1 ,l1 ]2 , [S n2 ,l2 ]2 ⟩L2 (E ) a π 2π 1 An1 ,l1 +1 (µ, θ ) [Tl1 +1 (cos φ)]2 sin θ sinh µ = (n1 + 2 + l1 )2 δn1 ,n2 δl1 ,l2 4 0 0 0 l +1 l +1 × cosh µ Pnl11+1 (cos θ ) Pn11 +1 (cosh µ) − cos θ Pn11 +1 (cos θ ) Pnl11+1 (cosh µ) dφ dθ dµ a π 2π 1 + (n1 + 2 + l1 )2 δn1 ,n2 δl1 ,l2 An1 ,l1 +1 (µ, θ ) [sin φ Ul1 (cos φ)]2 sin θ sinh µ 4 0 0 0 l +1 l +1 × cosh µ Pnl11+1 (cos θ ) Pn11 +1 (cosh µ) − cos θ Pn11 +1 (cos θ ) Pnl11+1 (cosh µ) dφ dθ dµ a (n1 + 1 + l1 )! l +1 δn1 ,n2 δl1 ,l2 Pnl11+1 (cosh µ) sinh µ cosh µ Pn11 +1 (cosh µ) dµ = π (n1 + 2 + l1 )2 (n1 − l1 )! 0 (n1 + 2 + l1 ) a l1 +1 2 [Pn1 (cosh µ)] sinh µ dµ . − 2n1 + 3 0 In closing this section, note that for each degree n ∈ N0 , the system {S n,l : l = 0, . . . , n} is formed by n + 1 = dim M+ (E ; H; n) monogenicpolynomials, and therefore, it is complete in M+ (E ; H; n). Based on the orthogonal ∞ decomposition M + (E ; H) = ⊕ n=0 M + (E ; H; n), and the completeness of the system in each subspace M + (E ; H; n), the result follows. Theorem 2.2. For each n, the set of n + 1 linearly independent monogenic polynomials {S n,l : l = 0, . . . , n} forms an orthogonal basis in the subspace M + (E ; H; n) in the sense of the quaternion product (1). Consequently, S n,l : l = 0, . . . , n; n = 0, 1, . . . is an orthogonal basis in M + (E ; H). 3. Concluding remarks In this section we prove the orthogonality of the system {S n,l : l = 0, . . . , n; n = 0, 1, . . .} over the interior of the ellipse (2) for all values of α to obtain a corresponding orthogonality of the same system over the surface S of the spheroid E with respect to a suitable weight function. Next we formulate the result. Theorem 3.1. For each n ∈ N0 , the set {S n,l : l = 0, . . . , n} forms a complete orthogonal system over the surface of the spheroid (2) in the sense of the quaternion product
⟨f, g⟩L2 (S ;H;H) =
S
f g |1 − (z + i ρ)2 |1/2 dσ ,
J. Morais / Mathematical and Computer Modelling 57 (2013) 425–434
433
with weight function |1 − (z + i ρ)2 |1/2 equal to the square root of the product of the distances from (z , ρ, φ) to the north and south poles. Here, dσ denotes the Lebesgue measure on S . Proof. Let α be a nonnegative real number. By the same reasoning as in [9] and based on the previous theorem, it follows that d dα
[⟨Sc(S n1 ,l1 ), Sc(S n2 ,l2 )⟩L2 (E ) + ⟨[S n1 ,l1 ]3 , [S n2 ,l2 ]3 ⟩L2 (E ) ]
(n1 + l1 )! = π (n1 + 2 + l1 )2 (n1 + 1 + l1 )2 (n1 − l1 + 1)2 δ n ,n δ l ,l (n1 + 1 − l1 )! 1 2 1 2 a a (n1 + 1 + l1 ) d d l × Pnl11 (cosh µ) sinh µ cosh µ Pn11 +1 (cosh µ) dµ − [Pnl11 (cosh µ)]2 sinh µ dµ , dα 0 2n1 + 3 dα 0
and d
⟨[S n1 ,l1 ]1 , [S n1 ,l1 ]1 ⟩L2 (E ) + ⟨[S n1 ,l1 ]2 , [S n2 ,l2 ]2 ⟩L2 (E ) a l +1 2 (n1 + 1 + l1 )! Pnl11+1 (cosh µ) sinh µ cosh µ Pn11 +1 (cosh µ) dµ = π (n1 + 2 + l1 ) δn1 ,n2 δl1 ,l2 (n1 − l1 )! 0 (n1 + 2 + l1 ) a l1 +1 2 [Pn1 (cosh µ)] sinh µ dµ , − 2n1 + 3 0
dα
whence
⟨S n1 ,l1 , S n2 ,l2 ⟩L2 (S ;H;H) =
S n1 ,l1 S n2 ,l2 |1 − (z + i ρ)2 |1/2 dσ S
(n1 + 2 + l1 )! δn1 ,n2 δl1 ,l2 = π (n1 + 2 + l1 ) (n1 − l1 )! l × (n1 + 1 + l1 ) (n1 − l1 + 1) Pnl11 (cosh α) sinh α cosh α Pn11 +1 (cosh α) −
1 2n1 + 3
(n1 + 1 + l1 )2 (n1 − l1 + 1)[Pnl11 (cosh α)]2 sinh α l +1
+ Pnl11+1 (cosh α) sinh α cosh α Pn11 +1 (cosh α) −
(n1 + 2 + l1 ) l1 +1 [Pn1 (cosh α)]2 sinh α . 2n1 + 3
Here the weight function |1 − (z + i ρ)2 |1/2 equals the square root of the product of the distances from (z , ρ, φ) to the points (1, 0, 0) and (−1, 0, 0). 4. Perspectives An explicit complete orthogonal system of polynomial (MT)-solutions has been presented, and we hope it constitutes a new class of practical and flexible basis within the context of quaternion analysis. In our point of view the system can be seen as a refinement of the three-dimensional homogeneous monogenic polynomials recently exploited by several authors. For a general orientation the reader is suggested to check some of the existing pioneering works [10–15] and [11,16,14, 17–19], of which the first six deal entirely with the R3 −→ R4 case and the last six the R3 −→ R3 case. It should be noticed that the application of homotopy methods in such polynomials will not preserve their monogenicity, and therefore it will not allow the generation of the polynomials presented here. The constructed system is composed by piecewise monogenic polynomials, and is capable of describing exactly the general information expressed by the commonly used Ferrer’s associated Legendre functions, which seems to be suitable for the treatment of monogenic functions by the use of power series expansions. In addition, we give explicit representations of the L2 -norms of these polynomials. This makes it possible to construct a complete orthonormal system, and consequently to define the Fourier expansion of a square integrable H-valued monogenic function in M + (E ; H). Further investigations on this topic will be reported in a forthcoming paper. Acknowledgments This work was supported by FEDER funds through COMPETE–Operational Programme Factors of Competitiveness (‘‘Programa Operacional Factores de Competitividade’’) and by Portuguese funds through the Center for Research and Development in Mathematics and Applications (University of Aveiro) and the Portuguese Foundation for Science and Technology (‘‘FCT–Fundação para a Ciência e a Tecnologia’’), within project PEst-C/MAT/UI4106/2011 with COMPETE
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number FCOMP-01-0124-FEDER-022690. Partial support from the Foundation for Science and Technology (FCT) via the post-doctoral grant SFRH/BPD/66342/2009 is also acknowledged by the author. References [1] G. Moisil, N. Théodoresco, Fonctions holomorphes dans l’espace, Matematica (Cluj) 5 (1931) 142–159. [2] V. Kravchenko, M. Shapiro, Integral Representations for Spatial Models of Mathematical Physics, in: Research Notes in Mathematics, Pitman Advanced Publishing Program, London, 1996. [3] R. Fueter, Analytische Funktionen einer Quaternionenvariablen, Commentarii Mathematici Helvetici 4 (1932) 9–20. [4] A. Sudbery, Quaternionic analysis, Mathematical Proceedings Cambridge Philosophical Society 85 (1979) 199–225. [5] J. Morais, A complete orthogonal system of spheroidal monogenics, Journal of Numerical Analysis, Industrial and Applied Mathematics 6 (3–4) (2011) 105–119. [6] S. Georgiev, J. Morais, On convergence aspects of spheroidal monogenics, AIP Conference Proceedings 1389 (2011) 276–279. [7] J. Morais, S. Georgiev, Complete orthogonal systems of 3D spheroidal monogenics, in: K. Gürlebeck, T. Lahmer, F. Werner (Eds.), Proceedings of the 19th International Conference on the Application of Computer Science and Mathematics in Architecture and Civil Engineering, Weimar. [8] E. Hobson, The Theory of Spherical and Ellipsoidal Harmonics, Cambridge, 1931. [9] P. Garabedian, Orthogonal harmonic polynomials, Pacific Journal of Mathematics 3 (3) (1953) 585–603. [10] S. Bock, K. Gürlebeck, On a generalized Appell system and monogenic power series, Mathematical Methods in the Applied Sciences 33 (4) (2009) 394–411. [11] I. Cação, Constructive approximation by monogenic polynomials, Ph.D. Thesis, Universidade de Aveiro, 2004. [12] I. Cação, K. Gürlebeck, S. Bock, Complete orthonormal systems of spherical monogenics—a constructive approach, in: L.H. Son, et al. (Eds.), Methods of Complex and Clifford Analysis, SAS International Publications, Delhi, 2005, pp. 241–260. [13] I. Cação, K. Gürlebeck, S. Bock, On derivatives of spherical monogenics, Complex Variables and Elliptic Equations 51 (8–11) (2006) 847–869. [14] I. Cação, Complete orthonormal sets of polynomial solutions of the Riesz and Moisil–Théodoresco systems in R3 , Numerical Algorithms 55 (2–3) (2010) 191–203. [15] R. Delanghe, On homogeneous polynomial solutions of the Moisil–Théodoresco system in R3 , Computational Methods and Function Theory 9 (1) (2009) 199–212. [16] I. Cação, H. Malonek, Remarks on some properties of monogenic polynomials, in: T.E. Simos, G. Psihoyios, Ch. Tsitouras (Eds.), ICNAAM, Special Volume of Wiley-VCH, 2006, pp. 596–599. [17] R. Delanghe, On homogeneous polynomial solutions of the Riesz system and their harmonic potentials, Complex Variables and Elliptic Equations 52 (10–11) (2007) 1047–1062. [18] J. Morais, K. Gürlebeck, Real-part estimates for solutions of the Riesz system in R3 , Complex Variables and Elliptic Equations 57 (5) (2012) 505–522. [19] J. Morais, Approximation by homogeneous polynomial solutions of the Riesz system in R3 , Ph.D. Thesis, Bauhaus-Universität Weimar, 2009.