A Construction of C1-Wavelets on the Two-Dimensional Sphere

Applied and Computational Harmonic Analysis 10, 1–26 (2001) doi:10.1006/acha.2000.0330, available online at http://www.idealibrary.com on

A Construction of C 1-Wavelets on the Two-Dimensional Sphere Ilona Weinreich Department of Mathematics and Technology, Rhein Ahr Campus Remagen, University of Applied Sciences Koblenz, Südallee 2, D-53424 Remagen, Germany E-mail: [email protected]

Communicated by Wim Sweldens Received June 17, 1998; revised September 4, 2000

In this paper a construction of C 1 -wavelets on the two-dimensional sphere is presented. First, we focus on the construction of a multiresolution analysis leading to C 1 -functions on S 2 . We show refinability of the constructed tensor product generators. Second, for the wavelet construction we employ a factorization of the refinement matrices which leads to refinement matrices characterizing complement spaces. With this method we achieve an initial stable completion. A desired stable completion can be gained by lifting the initial stable completion. The result is a biorthogonal wavelet basis leading to C 1 -functions on the sphere.  2001 Academic Press

Key Words: multiresolution analysis; wavelets; exponential splines; tensor product splines; wavelets on surfaces.

1. INTRODUCTION

During the past several years there has been great interest in wavelet constructions on closed manifolds. The two-dimensional sphere has to be considered as an example of central importance for many applications. Some of the common applications are the representation and manipulation of geographical data, the illumination algorithms in computer graphics, and the solution of integral or more general operator equations as well as medical problems because of the sphere-like structure of organs. The S 2 is a simple manifold from the mathematical point of view. Nevertheless there are remarkable difficulties to transfer the global concepts of classical wavelet theory, e.g., in R2 , to general manifolds. The first approach for the construction of wavelets on the sphere has been presented by Dahlke et al. [10]; here a characterization of C 1 -functions on S 2 (and S 3 ) with a wavelet representation is performed. The basic idea goes back to an approach using splines proposed by Schumaker and Traas [25] where the sphere is parameterized by polar coordinates. They employ a tensor product of trigonometric and polynomial splines for 1 1063-5203/01 $35.00 Copyright  2001 by Academic Press All rights of reproduction in any form reserved.

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ILONA WEINREICH

scattered data approximation. In [10] tensor products are employed, too. There, E-splines (exponential splines) [12, 21] are connected with interval wavelets [2]. A characteristic feature of an approach which yields C 1 -functions is the reproduction of trigonometric functions. This property is ensured with the help of appropriate E-splines. Before summarizing the contents of the present work we sketch some other approaches which also treat the sphere S 2 . Schröder and Sweldens [22] start with a triangulation of the sphere. Simple wavelets, e.g., Haar wavelets, are constructed on the triangles. With the lifting scheme one obtains smoother wavelets on the sphere. This approach is very useful in practical areas, e.g., for the compression of topographic data as well as in computer graphics. This approach is of great interest especially in connection to new results from Cohen et al. [6]. They establish stability of these wavelets. In [6] finite element wavelets with compactly supported duals are obtained by the lifting scheme on triangulations starting with an initial triangulation by uniform refinements. With a special treatment of the exceptional points a minimal Sobolev smoothness of the dual system can be shown. The results can be applied to the sphere by using a triangulation from an initial icosahedron or dodecahedron. With tensor products of interpolatory trigonometric wavelets and polynomial wavelets Potts and Tasche [19] construct continuous wavelets on S 2 . Here the C 1 -conditions are satisfied approximately. A similar idea with spherical harmonics is performed by Potts et al. [20]. They achieve smooth but as in [19] globally supported wavelets. In [18] Lyche and Schumaker present a construction of L-spline wavelets with compact support. An explicit multiresolution analysis for the sphere is not treated. Constructions of Freeden, Schreiner, and Windheuser are based on spherical harmonics and are applied to geodetical problems; see, e.g., [15, 27]. Another approach coming from CAGD, by Dahmen and Schneider [14], is based on a representation of the sphere (or more general manifolds) as a union of parametric patches which join at least continuously. On the single patches biorthogonal wavelets which satisfy certain transfer conditions are constructed. This approach is applied successfully for the treatment of integral equations. A further recent construction from Göttelmann [17] is based on polar coordinates. On a reduced grid with appropriately adapted splines a stable wavelet basis is obtained. An advantage of that approach in comparison with the present paper is the stability in a scale of Sobolev spaces H s (), 0 ≤ s < 1. Nevertheless, our approach seems to be simpler; it is not necessary to modify the grid. The central concern of the present paper is the construction of a multiresolution analysis as well as a wavelet basis of locally supported C 2 -functions on the sphere. Beginning with a parameterization of the sphere as already employed in [10] we reduce the problem to considering appropriate wavelets on the square [0, 1]2 for the moment. The fulfillment of side conditions (i.e., the reproduction of constant and trigonometric functions) must be guaranteed for obtaining C 1 -functions. For that reason tensor products are considered which are similar to those in [10]. The first component function is again an E-spline and the second component an interval wavelet. We show that the so-constructed functions are refinable. We obtain a multiresolution of L2 ([0, 1]2) and therefore a correspondence to C 1 -functions on S 2 by coordinate transformation. This approach allows a construction of a biorthogonal basis, the existence of which is crucial for stability considerations. The wavelet construction is based on a generalization of a matrix factorization method from Dahmen and Micchelli [13] to nonstationary splines. This method yields a so-called

3

SMOOTH WAVELETS ON THE SPHERE

initial stable completion. A suitable stable completion corresponding to the biorthogonal system is obtained with a method from Carnicer et al. [4]. This method is very similar and algebraically equivalent to the lifting scheme invented by Sweldens [23] which has been developed independently. The lifting scheme provides a relationship between different multiresolution analyses with the same scaling function. It enables the construction of wavelets for a particular application using the degrees of freedom after fixing the biorthogonality relations. For details see [24]. We proceed as follows: In Section 2 we introduce nonstationary multiresolution analysis with E-splines. In Section 3 we present our approach for the sphere. We consider a parameterization of the sphere by polar coordinates. The claim for C 1 -continuity makes things more difficult. The basis functions must represent not only constant but also trigonometric functions. We develop a multiresolution analysis which satisfies the corresponding conditions. Refinability is explicitly shown. In Section 4 we construct wavelets as so-called initial stable completion which can be mapped to each desired stable completion, especially the one induced by the biorthogonal system, with the aid of the method introduced by Carnicer et al. [4]. For the constructed basis, L2 -stability can be shown, if a biorthogonal basis exists.

2. BASIC CONCEPTS: E-SPLINES AND INTERVAL WAVELETS

In this section we introduce two concepts of multiresolution analysis which we are going to employ for the construction of a spherical multiresolution analysis. On the one hand, this is a nonstationary multiresolution analysis with E-splines and, on the other hand, a construction for a bounded interval. First, we consider a nonstationary multiresolution analysis where the generator depends on the refinement level. A typical example arises in connection with exponential splines [12, 21]. D EFINITION 2.1. Let µ ∈ CN be fixed and X a (d ×N)-matrix where x 1 , . . . , x N ∈ Rd are the columns of X. The E-spline Cµ (·|X) of dimension d is defined by Z

Z Rd

f (x)Cµ (x|X) dx =

[0,1]N

e−µ·u f (X · u) du

(2.1)

for all f ∈ C(Rd ). If we choose f (x) = e−iξ x in Definition 2.1 we obtain the Fourier transform of Cµ (·|X) Cˆ µ (ξ |X) =

 l N  Y 1 − e−(iξ ·x +µl ) . iξ · x l + µl

(2.2)

l=1

Formula (2.2) implies N 1 Y 1 Cˆ µ (ξ |X) l = (1 + e−(iξ/2·x +µl /2) ) =: a µ (z), ξ N 2 Cˆ µ/2 ( |X) 2 2

l=1

z = e−iξ/2 .

(2.3)

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ILONA WEINREICH

This means that Cµ (·|X) is refinable with symbol a µ (z) = scale relation of the form Cµ (x|X) =

X

P

µ k k ak z

and satisfies a two-

µ

ak Cµ/2 (2x − k|X),

(2.4)

k∈Zd µ

where the mask {ak }k∈Z is given by the coefficients of the symbol a µ (z). Let the spaces Sj be defined by Sj := span{Cµ/2j (2j · −k|X) | k ∈ Zd };

(2.5)

then Formula (2.4) implies that {Sj }j ∈Z is a sequence of nested spaces; i.e., · · · ⊂ S−1 ⊂ S0 ⊂ S1 ⊂ · · · .

(2.6)

For Re µ = 0 the spaces {Sj }j ∈Z form a (nonstationary) multiresolution analysis. This has been shown by de Boor et al. [9]. In the following lemma we summarize some useful properties of E-splines. For the general theory we refer to [12, 21]. L EMMA 2.1. If we consider X as the indexed set of column vectors of the matrix X and |X| as the number of columns in X then the following holds: (i) The translates {Cµ (· − k|X)}k∈Zd are linearly independent if and only if X is / 2πiZ\{0} for all y, y 0 . unimodular and µy − µy 0 ∈ (ii) Let Y be a maximal linearly independent subset of X with |Y | = d. Then the following formula is valid: X

e−uy ·k Cµ (x − k|X) =

k∈L(Y )

cµ (iuy |X) C e−uy ·x | det Y |

(2.7)

with uy = Y −T µy , L(Y ) = {Y l | l ∈ Z}. For our application we employ several special properties of the E-splines Cµ (·|X), which can be formulated as conditions concerning the parameters µ = (µ1 , . . . , µN ). We consider the univariate E-splines matrix X = (1, . . . , 1). With the following special choice of µ we obtain the desired representation of constant and trigonometric functions with E-splines. P p Remark 2.1. Let φ j = 2j/2 Cµ/2j (2j ·) = 2j/2 l∈Z Cµ/2j (2j · −2j l) be a periodic E-spline with parameter vector µ = (0, iν, −iν). Then for ν ∈ R and j > log2 |(2/π)ν| j j there exist coefficients Kj , Lk , Mk such that j −1 2X

Kj φ j (2j x − k) = 1,

(2.8)

k=0 j −1 2X

j

(2.9)

j

(2.10)

Lk φ j (2j x − k) = cos(νx),

k=0 j −1 2X

k=0

Mk φ j (2j x − k) = sin(νx).

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SMOOTH WAVELETS ON THE SPHERE

This is a consequence of (2.7). Next we describe a multiresolution analysis on the interval. The starting point for the construction of biorthogonal wavelets on the interval [0, 1] is a multiresolution analysis derived from B-splines; see [1, 5, 11]. Here φj,k := φ(2j x −k) are the dilates and translates of the centralized cardinal B-spline of order d; i.e., φ(x) := Nd (x + bd/2c). According to [11] we define ( 8j := {φj,k : k ∈ 1j },

φj,k :=

X , φj,k

X ∈ {L, R},

φ[j,k] = 2j/2 φ(2j · −k),

k ∈ 1X j , k ∈ 10j .

(2.11)

0 R The index set 1j is divided into sets 1L j , 1j , and 1j , where the former two are associated with the left and right border of the interval and the latter corresponds to the interior functions which do not overlap the border. In our context the explicit form of the edge L and φ R is not needed, so for details we refer to [11] or Appendix A.1. In functions φj,k j,k the following we suppose that j is large enough such that supp φ[j,k] ⊂ [0, 1] for k ∈ 10j . More precisely, we demand

  j ≥ log2 (`˜ + `˜2 − 1) + 1 =: j0 ,

(2.12)

˜ `˜2 depend on the support of the dual function φ; ˜ see Appendix A.1. where `, 3. MULTIRESOLUTION ANALYSIS ON THE SPHERE

We consider the unit sphere  S 2 := x ∈ R3 kxk22 = x12 + x22 + x32 = 1 .

(3.1)

Our aim is to construct a stable wavelet basis on S 2 . For this, we take spherical coordinates (for technical reasons with transformation on the unit square) ρ: [0, 1]2 → S 2 

 cos(2πx) sin(πy)   (x, y) 7→  sin(2πx) sin(πy)  .

(3.2)

cos(πy) In the following we identify S 2 with the unit square [0, 1]2 . Of course not every (differentiable) function on [0, 1]2 corresponds to a (differentiable) function on S 2 . The transformation ρ maps each of the edges {(x, 0) | x ∈ [0, 1]} and {(x, 1) | x ∈ [0, 1]} of the parameter domain to one point on the sphere, namely the north and the south pole. We formulate conditions for functions on [0, 1]2 which guarantee that the corresponding functions on S 2 are differentiable. A function f is continuous on S 2 if the periodicity condition f (0, y) = f (1, y),

0 ≤ y ≤ 1,

(3.3)

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ILONA WEINREICH

holds and if there exist constants SS , SN such that f (x, 0) = SS , f (x, 1) = SN ,

(3.4) 0≤x≤1

(3.5)

are satisfied. Moreover, the function f is differentiable on S 2 if there exist constants AS , BS , AN , BN such that ∂f ∂f (0, y) = (1, y), ∂x ∂x ∂f (x, 0) = AS cos(2πx) + BS sin(2πx), ∂y ∂f (x, 1) = AN cos(2πx) + BN sin(2πx), ∂y

0 ≤ y ≤ 1,

(3.6)

0 ≤ x ≤ 1,

(3.7)

0 ≤ x ≤ 1,

(3.8)

hold. Conditions (3.3)–(3.6) are obvious because of the chosen parameterization. For the derivation of the last two conditions (3.7), (3.8) we refer to [16]. T HEOREM 3.1 [16]. A function f represents a closed surface of the class C 1 (i.e., at each point of f the tangential plane of f varies continuously) if f and the partial derivatives ∂f/∂x and ∂f/∂y are continuous and satisfy the conditions (3.3)–(3.8) with appropriate AS , BS , AN , BN . In the following we construct a wavelet basis of L2 ([0, 1]2) that satisfies the conditions (3.3)–(3.8). As generators we choose tensor products of univariate generators of a multiresolution analysis of L2 ([0, 1]2), where we denote bivariate generators with bold φ and univariate with φ. We consider the tensor product j,o

j

φ k = φk1 ⊗ φj,k2 ,

(3.9) j

where the two component functions are defined as follows. The first component φk1 := P p 2j/2 Cµ/2j (2j · −k1 ) = 2j/2 l∈Z Cµ/2j (2j · −2j l − k1 ) is a periodized E-spline with Cµ (·|X) according to (2.1). We choose X = (1, . . . , 1) and µ1 = 0, µ2 = 2πi, µ3 = −2πi. So the conditions for a multiresolution analysis are satisfied for j > 2; see [9, 10]. The second component is a dilated and translated version of a centered B-spline φj,k2 = 2j/2 Nd (2j · −k2 + b d2 c) of order d with support in the interior of [0, 1]. The indices k2 with the property supp φj,k2 ⊂ [0, 1] belong to the set 10j . j,o

In the following let j ≥ jn with j0 according to (2.12). The functions φ k satisfy j,o

supp φ k ⊂ [0, 1] × (0, 1).

(3.10)

On the edges y = 0 and y = 1 we consider fixed linear combinations of several generators. Therefore we need to know which of the functions φj,k2 , k2 ∈ 1X j , X ∈ {L, R}, defined in [11] do not vanish for y = 0, y = 1.

7

SMOOTH WAVELETS ON THE SPHERE

Remark 3.1. Let φ be a spline of degree d − 1. For the generators φj,k , k ∈ 1X j , X ∈ {L, R}, corresponding to (2.11) one has φj,`−d+r (0) = φj,2j −`−`(d)+d−r (1) =

n

for r = 1, . . . , d − 1, for r = 0.

0 2j/2

(3.11)

For the derivatives—i.e., the one-sided limits for r = 0—of the generators one has (φj,`−d+r )0 (0) = −(φj,2j −`−`(d)+d−r )0 (1) =

n

0 2j ·3/2

for r 6= 1, for r = 1.

(3.12) j

On the border y = 0 of [0, 1]2 we consider linear combinations of all φk1 and the function φj,`−d , which does not vanish at y = 0. We define the combined generator (which we call edge generator in the following)

φ

j,L

:=

j −1 2X

k1 =0

j

λj,L φk1 φj,`−d .

(3.13) j

Further edge generators φ j,Lcos , φ j,Lsin are defined as linear combinations of all φk1 and the function φj,`−d+1 , the derivative of which does not vanish for y = 0. Therefore let φ

j,Lcos

:=

j −1 2X

k1 =0

φ

j,Lsin

:=

j −1 2X

k1 =0

j,Lcos j φk1 φj,`−d+1 ,

(3.14)

j,Lsin j φk1 φj,`−d+1 .

(3.15)

λk1

λk1

Analogously, we define the edge generators for the right border y = 1 of [0, 1]2 φ

j,R

:=

j −1 2X

k1 =0

φ

j,Rcos

:=

j −1 2X

k1 =0

φ j,Rsin :=

j −1 2X

k1 =0

j

λj,R φk1 φj,2j −r ,

(3.16)

j,Rcos j φk1 φj,2j −r−1 ,

(3.17)

j,Rsin j φk1 φj,2j −r−1 ,

(3.18)

λk1

λk1

with φj,k2 , k2 ∈ 1R j according to [11] and τ := ` − d + `(d). Here again d is the order of the employed B-spline and `, `(d) depend on its support; see Appendix A.1. For X ∈ {L, Lcos , Lsin , R, Rcos , Rsin } the coefficients λj,X will be chosen such that the conditions (3.3)–(3.8) for C 1 -continuity on S 2 are satisfied (see Theorem 3.2 below) for φ j,X . Furthermore, the normalization condition kφ j,X kL2 ([0,1]2 ) = 1

(3.19)

is required. The periodicity conditions (3.3) and (3.6) obviously hold because of the j periodicity of φk1 for all φ j,X . Moreover, functions which do not overlap the borders

8

ILONA WEINREICH

y = 0, y = 1 satisfy (3.4) and (3.5). For the functions φ j,X , X ∈ {Lcos , Lsin , Rcos , Rsin } Lemma 3.1 provides φ j,Lcos (x, 0) = φ j,Lsin (x, 0) = φ j,Rcos (x, 1) = φ j,Rsin (x, 1) = 0. Special consideration is necessary for φ j,L and φ j,R . They are supposed to reproduce a constant. Furthermore, they must satisfy the normality conditions (3.19) and therefore φ j,L (x, 0) = 2j/2 ,

φ j,R (x, 1) = 2j/2 .

(3.20)

For interior functions (3.7) and (3.8) are valid with AS = BS = AN = BN = 0. Especially, one has ∂ ∂ j,L φ (x, 0) = φ j,R (x, 1) = 0. ∂y ∂y The remaining functions φ j,Lcos , φ j,Lsin , φ j,Rcos , and φ j,Rsin should satisfy ∂ ∂ j,Lcos φ (x, 0) = φ j,Rcos (x, 1) = 23j/2 cos(2πx), ∂y ∂y ∂ ∂ j,Lsin φ (x, 0) = φ j,Rsin (x, 1) = 23j/2 sin(2πx). ∂y ∂y

(3.21)

T HEOREM 3.2. By coordinate transformation ρ (see (3.2)) the generators φ j,X , X ∈ {L, Lcos , Lsin , R, Rcos , Rsin }, lead to C 1 -functions on S 2 , and kφ j,X kL2 ([0,1]2 ) = 1 holds iff the coefficients are λj,L = λj,R = 2−j/2 Kj , j,Rcos

= 2−j/2 Lk1 ,

(3.23)

j,Rsin

= 2−j/2 Mk1 ,

(3.24)

j,Lcos

= λk1

j,Lsin

= λk1

λk1

λk1

(3.22)

j

j

j

j

for each k1 = 0, . . . , 2j − 1 and j ≥ j0 where Kj , Lk1 , and Mk1 are determined by (2.8), (2.9), and (2.10). Proof. For the norms we have

2j −1

X

j λj,L φk1 ⊗ φj,`−d kφ j,L kL2 ([0,1]2) =

k1 =0

L2 ([0,1]2 )

2j −1

X

= Kj φ j (2j x − k1 ) ⊗ φj,`−d

k1 =0

L2 ([0,1]2 )

= k1 ⊗ φj,`−d kL2 ([0,1]2) = 1,

2j −1

X

j Lk1 φ j (2j x − k1 ) ⊗ φj,`−d+1 kφ j,Lcos kL2 ([0,1]2) =

k1 =0

L2 ([0,1]2 )

9

SMOOTH WAVELETS ON THE SPHERE

= k cos ⊗φj,`−d+1 kL2 ([0,1]2) = 1,

2j −1

X

j Mk1 φ j (2j x − k1 ) ⊗ φj,`−d+1 kφ j,Lsin kL2 ([0,1]2) =

k1 =0

L2 ([0,1]2 )

= k sin ⊗φj,`−d+1 kL2 ([0,1]2 ) = 1. In the same way, we obtain kφ j,X kL2 ([0,1]2 ) = 1 for X ∈ {R, Rcos , Rsin }. To prove the C 1 -continuity conditions (3.3)–(3.8) for all φ j,X , X ∈ Iφ , we still have to consider (3.20) and (3.21). We show (3.20) for φ j,L . First, with φ j,L (x, 0) =

j −1 2X

k1 =0

j

λj,L φk1 (x)φj,`−d (0) = 2j/2

j −1 2X

Kj φ j (2j x − k1 ) = 2j/2

k1 =0

we obtain (3.20). Condition (3.21) arises from 2X −1 ∂ j,Lcos j,L j φ (x, 0) = λk1 cos φk1 (x)(φj,`−d+1 )0 (0) ∂y j

k1 =0

=

j −1 2X

k1 =0 j,Lsin

2−j/2 Lk1 2j/2 φ j (2j x − k1 )23j/2 = 23j/2 cos(2πx). j

j,Rcos

as well as φ and φ j,Rsin the argumentation is analogous. The values of the For φ generators on the interval edges are calculated as follows: 2X −1 ∂ j,Lcos j φ (0, 0) = Lk1 φ j (−k1 )23j/2 = 23j/2 cos(0)23j/2, ∂y j

k1

∂ j,Lsin 1 φ ( 4 , 0) = ∂y

j −1 2X

j

Mk1 φ j (2j 41 − k1 )23j/2 = 23j/2 sin(2π 14 ) = 23j/2 ,

k1

∂ ∂ j,Rcos φ (0, 1) = φ j,Rsin ( 14 , 1) = 23j/2 . ∂y ∂y So (3.3)–(3.8) are valid for all φ j,X , X ∈ {L, Lcos , Lsin , R, Rcos , Rsin }, which means that the corresponding functions give rise to smooth functions on the sphere. In the following we choose the generators φ j,X , X ∈ {L, Lcos , Lsin , R, Rcos , Rsin }, according to Theorem 3.2 such that they represent C 1 -functions on S 2 . The inner functions j,o φ k from (3.9) satisfy the conditions for C 1 -continuity automatically, because their value j,o on the border of the interval [0, 1] is zero. The same is true for those φ k , k = (k1 , k2 ), 0 j j / 1j and k2 6= ` − d, ` − d + 1, 2 − τ − 1, 2 − τ . Let the following index set Ijo with k2 ∈ be associated with these functions, which do not overlap the borders of the interval Ijo := {k ∈ Z2 | k1 = 0, . . . , 2j − 1, k2 ∈ 1j \{` − d, ` − d + 1, 2j − τ − 1, 2j − τ }}. j,o

By Sjdiff we denote the space spanned by all edge generators φ j,X , X ∈ Iφ , and by all φ k , diff k ∈ Ijo . By 8diff j we denote the following basis of Sj j,o

j,L 8diff , φ j,Lcos , φ j,Lsin , φ j,R , φ j,Rcos , φ j,Lsin , φ k | k ∈ Ijo }. j := {φ

(3.25)

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ILONA WEINREICH

The first question is whether the so-constructed spaces are nested and therefore may give rise to a multiresolution analysis. We have to check whether · · · ⊂ Sjdiff ⊂ Sjdiff +1 ⊂ · · ·

(3.26)

is valid. Therefore, we show that φ j,L , φ j,Lcos , φ j,Lsin , φ j,R , φ j,Rcos , and φ j,Rsin as well as j,o φ k , k ∈ Ijo , are refinable. j,o

Remark 3.2. The inner functions φ k , k ∈ Ijo , are obviously refinable because they are a tensor product where both component functions are refinable. The first component function, the E-spline, is nonstationary refinable; see (2.4). We show the refinability of φ j,L and φ j,Lcos . The arguments for the remaining edge generator are analogous. T HEOREM 3.3. Let φ j,L , φ j,Lcos , φ j,Lsin ∈ Sjdiff , j ≥ j0 , be defined as in (3.13)–(3.15) j,Lcos

with coefficients λj,L , λk1 refinement equations

j,Lsin

, and λk1

as in Theorem 3.2. The functions satisfy the

φ j,L = 2−1/2 φ j +1,L +

X

j +1,o

L γj,s φs

,

(3.27)

s∈Ij+1

φ j,Lcos = 2−3/2 φ j +1,Lcos +

X

j +1,o

Lcos γj,s φs

,

(3.28)

,

(3.29)

s∈Ij+1

φ j,Lsin = 2−3/2 φ j +1,Lsin +

X

L

j +1,o

γj,ssin φ s

s∈Ij+1

with L = 2−1/2λj +1,L σ˜ s2 ,0 , γj,s 1 ,s2

Lcos γj,s = 2−1/2 λj +1,Lcos σ˜ s2 ,1 , 1 ,s2

γj,ssin1 ,s2 = 2−1/2λj +1,Lsin σ˜ s2 ,1 . L

(3.30)

Here s ∈ Ij +1 := {s ∈ Z2 | s1 = 0, . . . , 2j +1 − 1, s2 = `, . . . , 2` + `2 − 2} and ( σ˜ s2 ,r =

α˜ s2 ,r

for s2 = `, . . . , 2` + `1 − 1,

2r+1/2β˜s2 ,r

for s2 = 2` + `1 , . . . , 2` + `2 − 2

(3.31)

with α˜ s2 ,r , as in (A.45) and β˜s2 ,r as in (A.51). Proof. First we show (3.27). To do so we have to represent φ j,L as a linear combination of φ

j +1,L

=

2j+1 X−1

j +1

λj +1,L φs1 φj +1,`−d

s1 =0 j +1,o

and inner functions φ k

. Using the refinement equation j φk1

=

2j+1 X−1 s1 =0

j +1

2−1/2 as1 −2k1 φs1 j

(3.32)

11

SMOOTH WAVELETS ON THE SPHERE

j

for the periodic E-splines φk1 as well as Eq. (A.51) for φj,`−d we obtain φ

j,L

=

j −1 2X

k1 =0

=

j −1 2X

j

λj,L φk1 φj,`−d

λj,L

k1 =0

=

2j+1 X−1 s1 =0

2j+1 X−1

j −1 2X

s1 =0

k1 =0

2−1/2 φj +1,`−d +

2`+` 2 −2 X

X

−1 2 −2 2X X

s1 =0

s2 =`

! σ˜ s2 ,0 φj +1,s2

s2 =`

!

j λj,L 2−1 as1 −2k1

2j+1 −1 2`+`

+

! j j +1 2−1/2 as1 −2k1 φs1

j +1

φs1 φj +1,`−d !

j

j λj,L as1 −2k1 2−1 σ˜ s2 ,0

k1 =0

j +1

φs1 φj +1,s2 .

(3.33)

Expression (3.33) corresponds to the refinement equation (3.27) if and only if j −1 2X

k1 =0

λj,L 2−1 as1 −2k1 = 2−1/2 λj +1,L j

(3.34)

L is of the form (3.30). According to Theorem 3.2, φ j,L corresponds to a is valid and γj,s C 1 -function on S 2 if the equation λj,L = 2−j/2 Kj holds, where the Kj are the coefficients from the representation of the unity (2.8). With the refinement equation (3.32) for E-splines in (2.8) we obtain j −1 2X

k1 =0

=

Kj

2j+1 X−1

as1 φ j +1 (2j +1 x − 2k1 − s1 ) j

s1 =0 2j+1 −1

X

j −1 2X

s1 =0

k1 =0

! j Kj as1 −2k1

φ j +1 (2j +1 x − s1 ) = 1.

This gives Kj +1 =

j −1 2X

k1 =0

j

Kj as1 −2k1

(3.35)

and the relation j −1 2X

k1 =0

j λj,L as1 −2k1 2−1

=

j −1 2X

k1 =0

2−(j/2+1) Kj as1 −2k1 j

= 2−(j/2+1)Kj +1 = 2−1/2 λj +1,L L for the coefficients λj,L . Therefore (3.34) and the refinement equation (3.27) with γj,s according to (3.30) are valid. An application of the refinement equations (A.51) and (3.32) together with the representation formula for the cosine with E-splines yields the proof

12

ILONA WEINREICH

of (3.29). The refinement equation (3.29) is valid iff j −1 2X

k1 =0

j,Lcos −2 j 2 as1 −2k1

λk1

j +1,Lcos

= 2−3/2λs1

.

(3.36) j,L

The coefficients of φ j,Lcos are chosen according to Theorem 3.2 and we obtain λk1 cos = j j 2−j/2 Lk1 , where Lk1 are the coefficients of the cosine representation (2.9) with E-splines. Similar to the calculation of Kj , we receive j +1 Ls1

=

j −1 2X

k1 =0 j,Lcos

For the coefficients λk1

j −1 2X

k1 =0

j

j

Lk1 as1 −2k1 .

(3.37)

we obtain the relation

j,L j λk1 cos ar1 −2k1 2−2

=

j −1 2X

k1 =0

2−2 2−j/2 Lk1 as1 −2k1 j

j

j +1

= 2−(j +1)/22−3/2 Ls1 j +1,Lcos

= 2−3/2 λs1

.

We have shown that (3.36) is valid. Thus the statement follows. The proof of the generator φ j,Lsin is analogous to that of φ j,Lcos by employing the corresponding sine representation formula with E-splines. For the dual multiresolution analysis we consider two-dimensional functions based on a tensor product of dual E-splines and dual B-splines. Dual functions corresponding to E-splines are constructed as generalizations of the stationary case [3] in [26, Chap. 3]. The biorthogonal B-splines on the interval are built in [11]. In this case a biorthogonalization procedure is performed for obtaining the desired biorthogonal system. A similar procedure is necessary in the present case. Under the assumption that the ˜ j of 8diff . corresponding biorthogonalization matrix is nonsingular we obtain a dual basis 8 j For details we refer to [26]. There a biorthogonal basis is constructed such that

 ˜ 8diff j , 8j = I

(3.38)

is valid. In the present paper we do not need that basis explicitly, but assume its existence. 4. WAVELETS ON S 2 BY STABLE COMPLETIONS

For the construction of the wavelet spaces we employ the concept of stable completions which has been introduced by Carnicer et al. in [4]. The principal idea is to start with a refinement matrix Mj,0 and to complete it with a matrix Mj,1 such that the matrix Mj = (Mj,0 , Mj,1 ) is uniformly bounded and invertible. More precisely, we consider a dense subset S of closed nested subspaces of Sj of a Hilbert space H with a stable basis 8j of Sj . Let

SMOOTH WAVELETS ON THE SPHERE

13

9j ⊂ Sj +1 be a set of functions spanning a complement of Sj in Sj +i . The set {9j ∪ 8j } is uniformly stable iff 9jT = 8Tj+1 Mj,1

(4.1)

for Mj,1 ∈ [`2 , (∇j ), `2 (1j +1 )] such that Mj = (Mj,0 , Mj,1 ) ∈ [`2 (1j ∪ ∇j ), `2 (1j +1 )]

(4.2)

is uniformly bounded and invertible. This means that there exists a matrix Gj ∈ [`2 (1j +1 ), `2 (1j ∪ ∇j )] which can be represented as Gj,0 Gj = Gj,1

!

and has the following properties Gj Mj = Mj Gj = I

(4.3)

kMj k, kGj k = O(1),

j ∈ N0 .

(4.4)

Condition (4.3) is equivalent to     I 0 Gj,0 Mj,0 Gj,0 Mj,1 = , Gj,1 Mj,0 Gj,1 Mj,1 0 I

Mj,0 Gj,0 + Mj,1 Gj,1 = I.

(4.5)

In particular, the reconstruction property 8Tj+1 = 8Tj Gj,0 + 9jT Gj,1 follows. Each matrix Mj,1 with the properties mentioned above is called the stable completion of Mj,0 . In [13], a method for how to obtain a stable completion for B-splines with a factorization method of the refinement matrix is developed. Once given a so-called initial stable completion we can obtain each desired stable completion of Mj,0 by the method described ˜ j = {φ˜ j,k | k ∈ ∇˜ j } is a biorthogonal basis of 8j , i.e., in [4]. For example, if 8 hφj,k , φ˜ j,k 0 i = δkk 0 ,

˜ j, k ∈ 1j , k 0 ∈ 1

(4.6)

which is also refinable and stable, which means ˜ j,0 ˜ Tj+1 M ˜ Tj = 8 8 and X k∈1j

(4.7)

X

2

˜ |ck | ∼ ck φj,k

, 2

k∈1j

(4.8)

H

then the induced projectors Qn satisfy a commutator property (see [8]) which is essential ˜ j }. The following proposition describes for the global stability of the basis {8j } and {8 how to achieve another stable completion Mj,1 which corresponds to the complements ˇ j,1 of Mj,0 . (Qj +1 − Qj )Sj +1 from an initial stable completion M

14

ILONA WEINREICH

˜ j,0 be associated as above and ˜ j }, Mj,0 , and M P ROPOSITION 4.1 [4]. Let {8j }, {8 −1 ˇ ˇ ˇ Mj,1 be a stable completion of Mj,0 and let Gj = Mj . Then ˇ j,1 ˜ T )M Mj,1 := (I − Mj,0 M j,0

(4.9)

is a stable completion as well and Gj = M−1 j is of the form ˜T M j,0 Gj = ˇ j,1 G

! .

(4.10)

The sets ˇ j,1 , 8 ˜ j := G ˜ j +1 9

9j := MTj,1 8j +1 ,

(4.11)

form biorthogonal systems ˜ j i = I, h9j , 9

˜ j i = h8j 9 ˜ ji = 0 h9j , 8

(4.12)

such that (Q∗j +1 − Q∗j )S˜j +1 = W˜ j .

(Qj +1 − Qj )Sj +1 = Wj ,

(4.13)

With these results we are prepared to construct bases for the complement spaces diff Wjdiff = Sjdiff +1 Sj .

(4.14)

As a basis of Sjdiff we again consider 8diff j as defined in (3.25). We want to characterize the wavelet basis of the complement spaces Wjdiff by stable completions; see [4]. For this, we first construct an initial stable completion. Our proceeding is based on a matrix factorization method of Dahmen and Micchelli [13]. The generators of {Sjdiff }j ≥j0 are tensor products of two one-dimensional generators except for the edge generators φ j,L , φ j,Lcos , φ j,Lcos , φ j,Rcos , φ j,Rsin , φ j,R . Hence we need to modify the procedure only concerning these edge functions. In [26] we showed that under certain conditions (which are satisfied in our context) E be the E-spline the matrix factorization method is applicable also for E-splines. Let Mj,0 E E E refinement matrix, Mj,1 a stable completion, and Gj,0 , Gj,1 the reconstruction matrices, with j

E (2 ) , GE j,0 Mj,0 = I

E GE j,0 Mj,1 = 0,

E GE j,1 Mj,0 = 0,

E (2 ) GE , j,1 Mj,1 = I

E E E 2 GE Mj,0 j,0 + Mj,1 Gj,1 = I

j

j+1

,

kMjE k, kGE j k = O(1).

B be the refinement matrix of the interval B-spline. In [11] an initial stable Let Mj,0 B is constructed. According to Proposition 4.6 [11] there exist matrices completion Mj,1 B B B Mj,1 , Gj,0 , Gj,1 with

15

SMOOTH WAVELETS ON THE SPHERE

B #1j , GB j,0 Mj,0 = I

B GB j,0 Mj,1 = 0,

B GB j,1 Mj,0 = 0,

B (2 ) GB , j,1 Mj,1 = I

B B B #1j +1 GB , Mj,0 j,0 + Mj,1 Gj,1 = I

kMjB k, kGB j k = O(1).

j

we first consider the reduced For the construction of matrices corresponding to 8diff j B,red refinement matrix Mj,0 , which is the internal (#1j +1 − 4) × (#1j − 4) submatrix B . For this submatrix a stable completion can easily be constructed, because it of Mj,0 corresponds to the inner functions of the multiresolution analysis of the interval [0, 1]. B,red B,red , GB,red Finally we obtain matrices Mj,1 j,0 , Gj,1 which constitute a stable completion of B Mj,0 ; i.e., B,red #1j , GB,red j,0 Mj,0 = I

B,red GB,red j,0 Mj,1 = 0,

B,red GB,red j,1 Mj,0 = 0,

B,red (2 ) GB,red . j,1 Mj,1 = I

j

(4.15)

For a more detailed presentation we refer to Appendix A.2. To construct an initial stable completion for the sphere let the refinement matrix Mj,0 for the generators of Sjdiff be given by 

φ j,L φ j,Lcos φ j,Lsin j,o φ 0,`−d+2 .. .





φ j +1,L φ j +1,Lcos φ j +1,Lsin j +1,o φ 0,`−d+2 .. .



                                 =: MT  ,  j,0       j +1,o   j,o  φ 2j+1 −1,2j+1 −τ −2   φ 2j −1,2j −τ −2          φ j,Rcos φ j +1,Rsin         j,Rsin j +1,Rcos     φ φ j,R j +1,R φ φ

(4.16)

where we order the interior generators j,o

j

φ k = φk1 ⊗ φj,k2

(4.17)

lexicographically with k = (k1 , k2 ) ∈ Ijo . Remark 4.1. Note that the size of the refinement matrix Mj,0 is (2j +1 (2j +1 − σ − 4) + 6) × (2j (2j − σ − 4) + 6) with σ := 2` + `(d) − 2d − 1. Proof. The generators in the interior consist of tensor products (4.17), where all 2j j translates of the E-spline component φk1 and #1j − 4 = 2j + 2d + 2` − `(d) + 1 − 4 = 2j − σ − 4φj,k2 (all, except the four with value or derivative value zero 0 or 1) are considered. All together one has 2j (2j − σ − 4) + 6 basis elements including the six edge generators φ j,L , φ j,Lcos , φ j,Lsin , φ j,Rcos , φ j,Rsin , φ j,R at the j th level.

16

ILONA WEINREICH

With α = 2−1/2 and β = 2−3/2 the matrix Mj,0 has the representation

(4.18)

with submatrices Nj,L , Nj,R , M− j,0 , where Nj,L results from the refinement equations (3.27), (3.29), and Nj,R from corresponding equations for the right edge generators 0 φ j,R , φ j,R . Finally, M− j,0 is the Kronecker product of the involved refinement matrices for the one-dimensional generators; i.e., B,red E M− j,0 = Mj,0 ⊗ Mj,0 ,

(4.19)

E is the refinement matrix of the periodic E-spline and M B,red is defined as where Mj,0 j,0 in (A.56). In view of the construction of an initial stable completion of Mj,0 , we first look for a matrix Gj,0 such that

Gj,0 Mj,0 = I (2

j (2j −σ −4)+6)

.

(4.20)

Regarding

(4.21)

17

SMOOTH WAVELETS ON THE SPHERE

where



1 α

˜ j,L = Nj,L  N 0 0

0

0



1 β

0 ,

0

1 β



1 β

˜ j,R = Nj,R  N 0 0

0

0



1 β

 0 ,

0

1 α

we obtain

(4.22)

ˇ j,0 , M ˇ j,1 , For the construction of a stable completion of Mj,0 we define the matrices G ˇ j,1 as follows: G B,red E G− j,0 := Gj,0 ⊗ Gj,0 ,

(4.23)

Gdj,0 :=

(4.24)

ˇ j,0 = Gd Hj,M , G j,0

(4.25)

E red E red E red M− j,1 := (Mj,0 ⊗ Mj,1 , Mj,1 ⊗ Mj,0 , Mj,1 ⊗ Mj,1 ),

(4.26)

(4.27)

18

ILONA WEINREICH

(4.28)

where we denote the interior (3 · 22j − 2j (σ + 4)) × (2j +1 (2j +1 − σ − 4) + 6)-block with G− j,1 . Note that ˇ ˇ j,1 = H−1 M M j,M j,1

ˇ j,1 = G ˇ j,1 Hj,M . G

and

(4.29)

Altogether, the preceding constructions lead to a stable completion of Mj,0 . This is stated in the following theorem. The proof can be done straightforward by employing the onedimensional components of the involved matrices, from which the desired properties are known. For details we refer to Appendix A.3. ˇ j,1 (4.27) is a stable completion of Mj,0 (4.18); i.e., T HEOREM 4.1. The matrix M (i) ˇ j,0 Mj,0 = I2j (2j −σ −4)+6 G

(4.30)

ˇ j,1 = 0 ˇ j,0 M G

(4.31)

ˇ j,1 Mj,0 = 0 G ˇ j,1 = I3·2 ˇ j,1 M G

(4.32) 2j

−2j (σ +4)

(4.33)

(ii) ˇ j,1 G ˇ j,0 + M ˇ j,1 = I(2 Mj,0 G

j+1 (2j+1 −σ −4)+6)

(4.34)

(iii) kMj k, kGj k = O(1).

(4.35)

˜ j a dual basis. For the construction of a be defined according to (3.25) and 8 Let 8diff j biorthogonal wavelet basis we define the projectors X X j,o j,o j,X hf, φ˜ k iφ k + hf, φ˜ iφ j,X (4.36) Qj f = X∈{L,L0 ,R 0 ,R}

k∈Ijo

Q∗j f =

X

k∈Ijo

hf, φ k iφ˜ k + j,o

j,o

X

hf, φ j,X iφ˜

j,X

.

(4.37)

X∈{L,L0 ,R 0 ,R}

˜ Remark 4.2. If both of the sets 8diff j , 8j are l2 -stable, then the sequences of linear projectors defined by (4.36) and (4.37) are uniformly bounded operators. The

SMOOTH WAVELETS ON THE SPHERE

19

biorthogonality (3.38) implies Ql Qj = Ql ,

Q∗l Q∗j = Q∗l

(4.38)

for l ≤ j . ˜ j,0 we denote the refinement matrix corresponding to 8 ˜ j and write the refinement By M equation as ˜ j,0 . ˜ j +1 )T M ˜ j )T = (8 (8

(4.39)

The construction of the biorthogonal wavelet bases is a direct consequence of Proposition 4.1. ˜ C OROLLARY 4.1. Let 8diff j be defined by (3.25) and 8j a dual basis. Let Mj,0 be the ˇ j,1 be an ˜ j,0 a refinement matrix according to (4.39). Let M refinement matrix (4.18) and M initial stable completion corresponding to Theorem 4.1. Then ˜ j,0 )T )M ˇ j,1 Mj,1 := (I − Mj,0 (M

(4.40)

is a stable completion of Mj,0 as well. The sets 9jdiff := MTj,1 8diff j +1 ,

ˇT 8 ˜ ˜ j := G 9 j,1 j +1

(4.41)

build biorthogonal systems ˜ ji = 1 h9jdiff , 9 ˜ j i = h8diff ˜ h9jdiff , 8 j , 9j i = 0. Complement spaces are given by diff (Qj +1 − Qj )Sjdiff +1 = Wj

(Q∗j +1 − Q∗j )S˜j +1 = W˜ j . Under the assumption that the biorthogonalization matrix for the construction of the ˜ j is invertible and has a uniform bounded condition number one can biorthogonal system 8 show the Riesz basis property for L2 (S 2 ) by verifying the Jackson and Bernstein estimates following arguments developed by Dahmen [7]. For details see [26]. In the present paper we sketched some of the results of [26]. First, we have focused on the construction of the multiresolution analysis leading to C 1 -functions on the sphere. We showed refinability of the constructed combined tensor product generators. Second, we employed a matrix factorization method for the wavelet construction. With this method we achieved an initial stable completion. A desired stable completion can be gained by the method of Carnicer et al. [4]; see Proposition 4.1 in this paper. The result is a biorthogonal wavelet basis leading to C 1 -functions on the sphere. APPENDIX

A.1. Multiresolution Analysis on an Interval Here we describe a multiresolution analysis on the interval. The starting point is a multiresolution analysis derived from B-splines. To obtain moment conditions for the

20

ILONA WEINREICH

wavelets one also needs a reproduction of polynomials for the dual generator. We consider cardinal B-splines Nd of order d with supp Nd = [0, d]. Let the generator φ be a centered B-spline    d φ(x) := Nd x + 2

(A.42)

with support [−b d2 c, d d2 e] =: [`1 , `2 ]. We have the symmetry property φ(x + `(d)) = φ(−x), where `(d) = d mod 2. It is well known that B-splines generate a multiresolution analysis. Dual functions to φ are φ˜ = d,d˜ , φ˜ with d˜ ≥ d and d + d˜ even, as constructed in [3] such that ˜ − k)i = δ0k . hφ, φ(·

(A.43)

An important property of B-splines is their approximation quality which depends on the reproduction of polynomials. B-splines of order d are exact of degree d − 1; i.e., xr =

X ˜ − k)iφ(x − k) h(·)r , φ(·

(A.44)

k∈Z

for all r = 0, . . . , d − 1. The duals φ˜ are exact of degree d˜ − 1. Following [11], the idea for constructing a multiresolution analysis on the interval [0, 1] is to keep all functions φ[j,k] = 2j/2 φ(2j · −k), with support in [0, 1]. For retaining the optimal approximation it is necessary to take functions which overlap the borders of [0, 1] into consideration also. By defining L ˜ − m)i := α˜ m,r := h(·)r , φ(· α˜ j,m,r

R α˜ j,m,r := α˜ 2j −m−`(d),r ,

and

(A.45)

for r = 0, . . . , d − 1, the relations X

L α˜ j,m,r φ[j,m] (x) = 2j (r+1/2)x r ,

(A.46)

m∈Z

X

R α˜ j,m,r φ[j,m] (x) = 2j (r+1/2)(1 − x)r ,

r = 0, . . . , d − 1

(A.47)

m∈Z

are valid. As in [1, 5] these relations can be employed to construct basis functions on the interval [0, 1]. Again following [11] we have supp φ˜ = [`1 − d˜ + 1, `2 + d˜ − 1] =: [`˜1 , `˜2 ]. 0 Let `˜ > |`˜1 |, `˜2 . With ` := `˜ − (d˜ − d) the index sets 1L j := {` − d, . . . , ` − 1}, 1j := j j {`, . . . , 2j − ` − `(d)}, 1R j := {2 − ` + 1 − `(d), . . ., 2 − ` + d − `(d)} are defined. The generators on the right and left border of [0, 1] are defined by L φj,`−d+r

:=

`−1 X m=−`2 +1

R φj,2 j −`+d−`(d)−r :=

α˜ m,r φ[j,m]

2j −` 1 −1 X m=2j −`−`(d)+1

,

(A.48)

[0,1]

R α˜ j,m,r φ[j,m]

(A.49) [0,1]

21

SMOOTH WAVELETS ON THE SPHERE

for r = 0, . . . , d − 1, and for the functions in the interior of [0, 1], i.e., for k ∈ 10j , we write φj,k := φ[j,k] . On the basis of this background we define ( 8j := {φj,k : k ∈ 1j },

where φj,k :=

X , φj,k φ[j,k] ,

k ∈ 1X j , X ∈ {L, R} k ∈ 10j .

(A.50)

L R and φj,2 In [11] refinement equations for the generators φj,`−d+r j −`+d−`(d)−r are proved. For the left border, for instance, we have

L φj,`−d+r

=2

−(r+1/2)

φjL+1,`−d+r

2`+` 1 −1 X

+

! α˜ m,r φ[j +1,m]

+

m=`

2`+` 2 −2 X

β˜m,r φ[j +1,m]

m=2`+`1

(A.51)

with L = 2−1/2 β˜m,r

`−1 X

α˜ q,r am−2q

for r = 0, . . . , d − 1.

q=d(m−`2 )/2e

A.2. Construction of Stable Completion we need reduced refinement For the construction of matrices corresponding to 8diff j B,red B,red B,red B,red matrices Mj,0 , Mj,1 , Gj,0 , Gj,1 which are defined below. B . Let m`−d , We eliminate all nondiagonal elements from the first two columns of Mj,0

B and let e be m`−d+1 be the first two and m2 −r−1 , m2 −r the last two columns of Mj,0 k the kth unit vector of length #1j +1 . We define j

j

m ˜ r := diag(−21/2, −23/2, 1, . . . , 1, −23/2, −21/2)mr for r = ` − d, ` − d + 1, 2j − τ − 1, 2j − τ . Moreover, we define the elimination matrix B in the first two and last two columns apart from HM , which eliminates all entries of Mj,0 the diagonal: ˜ `−d , −23/2 m ˜ `−d+1, e3 , . . . , e#1j+1 −2 , −23/2m ˜2 HM := (−21/2m

j −τ −1

, −21/2 m ˜2

j −τ

),

B,∧ B by M B,red we denote, the inner (#1 := HM Mj,0 HM ∈ R(#1j+1 )×(#1j+1 ) . For Mj,0 j +1 − j,0 B,∧ 4) × (#1j − 4)-block of Mj,0 . Thus one gets

(A.52)

22

ILONA WEINREICH

Similarly we proceed with GB j,0 and define −1 B GB,∧ j,0 := Gj,0 HM .

B #1j it follows that From GB j,0 Mj,0 = I

B,∧ #1j . GB,∧ j,0 Mj,0 = I

(A.53)

B,∧ Dividing GB,∧ j,0 similarly to Mj,0 and denoting the inner (#1j − 4) × (#1j +1 − 4)-block by GB,red j,0 , we obtain

(A.54)

Finally, (A.53) implies B,red #1j −4 . GB,red j,0 Mj,0 = I

(A.55)

B as well as For our construction we need the fact that the first and the last d rows of Mj,1 B the first and the last d columns of Gj,1 are identically zero; see [11, Chap. 4.1]. For the spherical construction we always choose d > 2 because we want to represent differentiable B,red , GB,red functions. Next, reduced matrices Mj,1 j,1 are considered. B,red B The matrix Mj,1 arises from Mj,1 by neglecting the first two and the last two rows

(A.56)

23

SMOOTH WAVELETS ON THE SPHERE

B and GB,red j,1 is obtained from Gj,1 by neglecting the first and last two columns

(A.57)

For the reduced matrices we obviously have B,red #1j , GB,red j,0 Mj,0 = I

B,red GB,red j,0 Mj,1 = 0,

B,red GB,red j,1 Mj,0 = 0,

B,red (2 ) GB,red . j,1 Mj,1 = I

j

A.3. Proof of Theorem 4.1 ˇ j,0 Mj,0 = Gd Md . Consider the interior blocks of Gd Proof. (i) We have G j,0 j,0 j,0 d and Mj,0 : B,red B,red − E E G− j,0 Mj,0 = (Gj,0 ⊗ Gj,0 )(Mj,0 ⊗ Mj,0 ) B,red B,red E = (GE j,0 Mj,0 ) ⊗ (Gj,0 Mj,0 ) j

= I (2 ) ⊗ I (2 = I (2

j −σ −4)

j (2j −σ −4))

.

E is a stable completion of M E which Relation (4.30) follows directly. The matrix Mj,1 j,0 B,red B,red E E yields Gj,0 Mj,0 = 0. Gj,0 Mj,1 = 0 follows from formula (4.15) for the reduced refinement matrices of the B-splines. Taking into account the rules for computation with the Kronecker product we obtain B,red B,red E (GE j,0 ⊗ Gj,0 )(Mj,0 ⊗ Mj,0 ) = 0, B,red B,red E (GE j,0 ⊗ Gj,0 )(Mj,1 ⊗ Mj,0 ) = 0, B,red B,red E (GE j,0 ⊗ Gj,0 )(Mj,1 ⊗ Mj,1 ) = 0, − and therefore G− j,0 Mj,1 = 0. Equations (4.29) and (4.25) yield the assertion (4.31). With B,red B,red E E Gj,1 Mj,0 = 0 and Gj,1 Mj,0 = 0 we receive B,red B,red E (GE j,0 ⊗ Gj,1 )(Mj,0 ⊗ Mj,0 ) = 0, B,red B,red E (GE j,1 ⊗ Gj,0 )(Mj,0 ⊗ Mj,0 ) = 0, B,red B,red E (GE j,1 ⊗ Gj,1 )(Mj,0 ⊗ Mj,0 ) = 0,

24

ILONA WEINREICH

− and therefore G− j,1 Mj,0 = 0. Equation (4.22) and again (4.29) yield (4.32). The remaining assertion (4.33) follows analogously to (4.30)–(4.32). (ii) We transform the left-hand side of (4.34) as follows:

ˇ j,1 G ˇ j,0 + M ˇ j,1 = H−1 Md Gd Hj,M + H−1 M ˇ ˇ Mj,0 G j,M j,0 j,0 j,M j,1 Gj,1 Hj,M d d ˇ ˇ = H−1 j,M (Mj,0 Gj,0 + Mj,1 Gj,1 )Hj,M . − Again we consider the interior blocks M− j,0 , Gj,0 . We have B,red B,red − E E M− j,0 Gj,0 = Mj,0 Gj,0 ⊗ Mj,0 Gj,0 .

(A.58)

− For the interior blocks M− j,1 , Gj,1 the following is true B,red B,red − E E M− j,1 Gj,1 = Mj,0 Gj,0 ⊗ Mj,1 Gj,1 B,red B,red E + Mj,1 GE j,1 ⊗ Mj,0 Gj,0 B,red B,red E + Mj,1 GE j,1 ⊗ Mj,1 Gj,1 .

Therefore, we have B,red B,red B,red B,red − − − E E M− j,0 Gj,0 + Mj,1 Gj,1 = Mj,0 Gj,0 ⊗ (Mj,0 Gj,0 + Mj,1 Gj,1 ) B,red B,red B,red B,red E GE + Mj,1 j,1 ⊗ (Mj,0 Gj,0 + Mj,1 Gj,1 ) j+1 −σ −4

= I2

j+1

= I2

j+1 (2j+1 −σ −4)

⊗ I2

.

The claim is proved for the interior blocks. Clearly, the identity (4.34) follows. (iii) For the interior blocks the statement is clear because of the tensor product structure where for each of the components the statement is already proved. We extend − the matrix M− j,1 with six zero rows and Gj,1 with six zero columns, respectively, and ˇ j,1 (see (4.28)). For these matrices the uniform boundedness ˇ j,1 (see (4.27)) and G obtain M follows directly from the uniform boundedness of the component functions. The matrices − M− j,0 , Gj,0 are extended by six rows (six columns, respectively) with unit vectors (or an appropriate multiple) to Mdj,0 (see (4.22)), Gdj,0 (see (4.24)). This procedure does not change anything with respect to the uniform boundedness. Finally, we obtain Mj,0 and Gj,0 from Mdj,0 , Gdj,0 by multiplication by Hj,M , respectively H−1 j,M . It remains to show −1 kHj,M k, kHj,M k = O(1). We consider the k · k∞ -norm. With (4.21) we have ˜ j,L k∞ + kN ˜ j,R k∞ + 1. 1 ≤ kHj,M k∞ ≤ kN

(A.59)

˜ j,R from above. W.l.o.g. consider ˜ j,L , N It suffices to give an estimate of the entries of N L ˜ Nj,L ; the first column consists of the coefficients γj,s of the refinement equation (3.27). Lcos of the refinement equation (3.29) In the second column there are the coefficients γj,0 Lcos L = 2−1/2 λj +1,L σ˜ s2 ,0 , γj,s = and in the third column γj,0sin of (3.29). We have γj,s 1 ,s2 1 ,s2 L

2−1/2 λj +1,Lcos σ˜ s2 ,1 , and

L γj,ssin1 ,s2

= 2−1/2 λj +1,Lsin σ˜ s2 ,1 .

Applying Theorem 3.2 given λj,L = 2−j/2 Kj , λj,Lcos = 2−j/2 Lk1 , and λj,Lsin = j j j −j/2 Mk1 for the coefficients, where the quantities Kj , Lk1 , Mk1 are bounded independent 2 j

SMOOTH WAVELETS ON THE SPHERE

25

from j ; see [26]. The quantities σ˜ s2 ,0 , σ˜ s2 ,1 come from the refinement equations for the interval B-splines. According to [11] they are bounded (independent of j ). kHj,M k2 = O(1) follows and assertion (4.35) with it. ACKNOWLEDGMENTS This work partially represents results of the author’s PhD thesis which has been performed under the advice of W. Dahmen. I thank him for his support.

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