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Wavelet Transform on Compact Gelfand Pairs and Its Discretization
Journal of Mathematical Analysis and Applications 238, 234᎐258 Ž1999. Article ID jmaa.1999.6530, available online at http:rrwww.idealibrary.com on
Wavelet Transform on Compact Gelfand Pairs and Its Discretization K. Mokni Department of Mathematics, Faculty of Sciences of Monastir, Monastir 5019, Tunisia
and K. Trimeche ` Department of Mathematics, Faculty of Sciences of Tunis, Campus, Tunis 1060, Tunisia Submitted by Joseph D. Ward Received November 17, 1997
Using harmonic analysis on a compact Gelfand pair Ž G, K ., we study the continuous wavelet analysis on the homogeneous space GrK. As example we give the wavelet associated with the K-biinvariant Poisson kernel on G. Next, using the previous results, we define and study three types of wavelet packets on GrK and the corresponding wavelet transforms. We prove for these transforms reconstruction formulas, and we describe the related multiresolution analysis. 䊚 1999 Academic Press
0. INTRODUCTION We consider a compact Gelfand pair Ž G, K .. In this case the Fourier analysis is associated with the spherical functions of the pair Ž G, K ., and the decomposition of the space L2 Ž GrK . Žof square integrable functions on G with respect to the Haar measure of G, which are K-right invariant. into a discrete direct sum of irreducible spaces relative to the regular representation of the group G. Wavelet theory involves breaking up a complicated function Žphenomenon. into many simple pieces at different scales and positions. More explicitly, wavelet analysis allows us to represent a function f in L2 Ž GrK . as a sum of wavelets ⌿ , g s L g D⌿, i.e., left translated and dilated copies of a mother wavelet ⌿, a K-left invariant function in L2 Ž GrK . with 234 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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COMPACT GELFAND PAIRS
vanishing integral. The coefficients of the decomposition are the inner products Ž f, ⌿ , g .L2 ŽG r K . . The objective of this paper is to generalize the important subject of wavelets to the case of the compact Gelfand pair Ž G, K .. We present here a general construction allowing the development of wavelet packets on the homogeneous space GrK from the continuous wavelet analysis on GrK. An analogous study has been made in w3x by Freeden and Windheuser for the compact Gelfand pair Ž SO Ž3., SO Ž2... It is believed that generalized wavelet expansions as discussed here will be a very useful tool in many areas of mathematics. The content of this paper is as follows. We summarize in the first section the main results of the harmonic analysis of the compact Gelfand pairs Ž G, K ., and we give more details about the particular cases Ž SO Ž q ., SO Ž q y 1.., q g ⺞, q G 3 and ŽUŽ q ., UŽ q y 1.., q g ⺞, q G 2. In the second section we study the continuous wavelet analysis on the homogeneous space GrK. As an example we give the wavelets associated with the K-biinvariant Poisson kernel on G. We define and study in the third section three types of wavelet packets on GrK and the corresponding wavelet transforms, and we prove for these transforms reconstruction formulas. The multiresolution analysis by means of wavelet packets on GrK is discussed in the fourth section.
1. HARMONIC ANALYSIS ON COMPACT GELFAND PAIRS Let G be a compact group with a normalized Haar measure dx, and let K be a closed subgroup. We denote by e the identity element of G. The group G acts on the space of complex functions f on G by left translation L g and right translation R g as follows: L g f Ž x . s f Ž gy1 x . and R g f Ž x . s f Ž xg .. Let L p Ž G ., p s 1, 2, be the space of measurable functions f on G such that
HG< f Ž x . <
p
dx - ⬁.
We denote by 䢇
L p Ž GrK . the space of K-right invariant functions f in L p Ž G ., that
is, f Ž gk . s f Ž g .
for all k g K ,
g g G.
MOKNI AND TRIMECHE `
236
The hermitian product of the Hilbert space L2 Ž GrK . is denoted by Ž⭈, ⭈ . 2 , and the corresponding norm by 5 ⭈ 5 2 . L p Ž K _ GrK . the space of K-biinvariant functions f in L p Ž G ., that is, f Ž kgk⬘ . s f Ž g . for all k, k⬘ g K , g g G. 䢇
A K-biinvariant function on G is said to be a zonal function on G. The convolution product is defined, for f, h in L1 Ž G ., by f ) hŽ x . s
y1
HGh Ž y
x . f Ž y . dy,
x g G.
The pair Ž G, K . is said to be a Gelfand pair if L1 Ž K _ GrK ., provided with this convolution product is commutative.
ˆ be the set of equivalence classes of unitary irreducible represenLet G ˆ realized in a Hilbert space Ea; this space is tations of G. Take a in G, automatically finite-dimensional. Let EaK be the subspace of Ea defined by EaK s ¨ g Ea : a Ž k . ¨ s ¨ , for all k g K 4 . We know that Ž G, K . is a Gelfand pair if and only if dimŽ EaK . F 1, for any ˆ Denote by ⌳ the set of a g G, ˆ such that dimŽ EaK . s 1. It is well a g G. known Žsee w4, p. 394x. that the left regular representation of G on L2 Ž GrK . is decomposed in a direct sum of unitary irreducible representations of finite dimension as follows: L2 Ž GrK . s
[ag⌳ Ha .
The representations a are the restrictions to Ha of the regular representation . Denote by a0 the trivial representation; then Ha 0 s ⺓. Let d a be the dimension of Ha , and let Ya, 1 , . . . , Ya, d a4 be an orthonormal basis of Ha with respect to the scalar product of L2 Ž GrK .. For all a g ⌳ there exists a unique normed vector ⑀ a g Ha of class one, that is, Ž k . ⑀ a s ⑀ a , for all k g K. The function a defined by
a Ž g . s Ž Ž g . ⑀ a , ⑀ a . 2 ,
for all g g G,
is a positive-definite spherical function of the Gelfand pair Ž G, K . associated with the representation Ž , Ha .. It satisfies the following properties:
a 0 ' 1 a Ž ⑀ . s 1,
for all a g ⌳
a Ž gy1 . s a Ž g . , 5 a 5 22 s
1 da
5 a 5 ⬁ s 1,
Ž 1.1.
for all a g ⌳ , ,
for all a g ⌳ for all a g ⌳
Ž 1.2. ggG
Ž 1.3. Ž 1.4. Ž 1.5.
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COMPACT GELFAND PAIRS
1
f ) a s
da
b ) a s a Ž xy1 y . s
1 da
for all a g ⌳ ,
f,
1 da
␦ b , a a ,
Ž 1.6.
for all a, b g ⌳
Ž 1.7.
da
for all a g ⌳ ,
Ý Ya, j Ž x . Ya, j Ž y . ,
x, y g G Ž 1.8.
js1
HK Ž xky . dk s Ž x . Ž y . , a
f g Ha
a
for all x, y g G and
a
a g ⌳ . Ž 1.9.
The formula Ž1.9. is called the product formula for the spherical functions. The spherical Fourier transform is defined for all f g L p Ž K _ GrK . by ᭙a g ⌳ ,
fˆŽ a . s
y1
HG f Ž x . Ž x a
. dx.
Ž 1.10.
If f g L2 Ž K _ GrK . we have the inversion formula f Ž ⭈. s
d a fˆŽ a . a Ž ⭈ .
Ý
Ž 1.11.
ag⌳
and the Parseval formula 5 f 5 2L2 ŽG. s
Ý
d a < fˆŽ a . < 2 .
Ž 1.12.
ag⌳
The family Ya, j 4 , a g ⌳, j g 1, . . . , d a4 , is an orthonormal basis of L2 Ž GrK .; so for all f g L2 Ž GrK . we define the Fourier coefficients by F Ž f . Ž a, j . s Ž f , Ya, j . 2 . The map F is an isomorphism isometric from L2 Ž GrK . onto l 2 Ž N ., where N s Ž a, j . : a g ⌳ , j g 1, . . . , d a 4 4 , and l 2 Ž N . is the space of sequences x a, j 4Ž a, j.g N such that da
Ý Ý < x a, j < 2 - ⬁. ag⌳ js1
MOKNI AND TRIMECHE `
238
This result is called the Parse¨ al theorem for F. The inversion formula of the map F is given for all f g L2 Ž GrK . by da
f Ž ⭈. s
Ý Ý F Ž f . Ž a, j . Ya, j Ž ⭈. .
Ž 1.13.
ag⌳ js1
All of these ingredients are well known for any compact Gelfand pair; we can refer to w2x, for example. 1.1. Examples Let ⺖ denote one of the fields ⺢, ⺓, of the real numbers or the complex numbers. For an integer q, let ⺖ q be a q-dimensional vector space over ⺖, with Hermitian product defined for z s Ž z1 , . . . , z q . and w s Ž w 1 , . . . , wq . in ⺖ q by q
Ž z, w . s
Ý
z kw k .
ks0
Let ⑀ 1 , . . . , ⑀ q 4 be the canonical orthonormal basis of ⺖ q. We consider G and K as follows: In the case ⺖ s ⺢, G s SO Ž q . is the group of rotations of ⺢ q. K s SO Ž q y 1. is the subgroup of rotations which leave the vector ⑀ 1 fixed. In the case ⺖ s ⺓, G s UŽ q . is the group of linear transformations of ⺓ q which leave the Hermitian product invariant Žthe unitary group.. K s UŽ q y 1. is the subgroup of elements in UŽ q . leaving fixed the vector ⑀ 1. 䢇
䢇
It is well known that Ž G, K . is a Gelfand pair. G is a connected compact Lie group which acts transitively on the unit sphere ⍀ of ⺖ q, so the homogeneous space GrK can be identified with ⍀. The surface element d on ⍀ of total mass 1 is, up to scalar, the unique G-invariant measure on ⍀. L p Ž GrK . is identified with L p Ž ⍀ . s L p Ž ⍀, d .. We now describe the decomposition of L2 Ž GrK . and the corresponding spherical functions. Case ⺖ s ⺢. We suppose that q G 3. For n integer, we say that a function ⌽ is a spherical harmonic of degree n if it is the restriction to ⍀ of a polynomial F Ž x . s F Ž x 1 , . . . , x q ., which is homogeneous of degree n in x and satisfies the Laplace equation ⌬ F s 0. Denote by Hn the class of all such functions. The restriction of the left regular representation to Hn is irreducible, and we have the decomposition L2 Ž ⍀ . s
⬁
[ ns0 Hn .
Ž 1.14.
COMPACT GELFAND PAIRS
239
The spaces Hn are analogous to the spaces Ha ; they are of finite dimension d n , where dn s
Ž 2 n q q y 2. Ž n q q y 3. ! . n! Ž q y 2 . !
Ž 1.15.
Here the set ⌳ is identified with ⺞. The spherical functions associated with this Gelfand pair are defined for each n g ⺞, by the unique zonal function n in Hn verifying nŽ ⑀ 1 . s 1, and are given, for ¨ g ⍀, by the formula
n Ž ¨ . s PnŽ qy2.r2 Ž ¨ , ⑀ 1 . ,
Ž 1.16.
where PnŽ qy2.r2 is the classical ultraspherical Žor Gegenbauer. polynomial of degree n satisfying PnŽ qy2.r2 Ž1. s 1. When q s 3, the spherical functions are given by Pn1r2 , the Legendre polynomials Žin fact this is the case elaborated in w3x.. Case ⺖ s ⺓. Here we take q G 2. For m, n integer, we say that a function ⌽ belongs to the class HarmŽ m, n., if it is the restriction to ⍀ of a polynomial F Ž z, z . s F Ž z1 , . . . , z q , z1 , . . . , z q . , which is homogeneous of degree m in z and homogeneous of degree n in z, such that
ž
⭸2 ⭸ z1 ⭸ z1
q ⭈⭈⭈ q
⭸2 ⭸ zq ⭸ zq
/
F s 0.
The restriction of the left regular representation to HarmŽ m, n. is irreducible, and we have the decomposition L2 Ž ⍀ . s
⬁
[ m , ns0 Harm Ž m, n . .
Ž 1.17.
The spaces HarmŽ m, n. are analogous to the spaces Ha ; they are of finite dimension dŽ m, n. , where dŽ m , n. s
Ž m q n q q y 1. Ž m q q y 2. ! Ž n q q y 2. ! . m!n! Ž q y 1 . ! Ž q y 2 . !
Ž 1.18.
Here the set ⌳ is identified with ⺞ = ⺞. Consider the map A from G into the closed unit disc D of ⺓ defined by AŽ g . s Ž g ⑀ 1 , ⑀ 1 .. Then f ¬ f ( A is an isometry between the space L p ŽUŽ q y 1. _ UŽ q .rUŽ q y 1.. and the space L p Ž D, dm., of measurable
MOKNI AND TRIMECHE `
240 functions on D such that
HD< f Ž z . <
p
dm Ž z . - ⬁,
with dm Ž x q iy . s
qy1 2
Ž1 y x 2 y y 2 .
qy 2
dx dy.
The spherical functions associated with this Gelfand pair are defined for each Ž m, n. g ⺞ = ⺞, by the unique zonal function ⌽ in HarmŽ m, n. verifying ⌽ Ž ⑀ 1 . s 1, given for re i g D by the formula qy2, < myn<. ⌽ Ž re i . s r < myn< e iŽ myn. R ŽminŽ Ž 2 r 2 y 1. , m , n.
Ž 1.19.
qy2, < myn<. qy2, < myn<. Ž . where R ŽminŽ is the Jacobi polynomial satisfying R ŽminŽ 1 s 1. m, n. m, n. qy 2 This function is denoted R m, n . For more details about the above examples of Gelfand pairs we can see w1, 6, 7x.
2. CONTINUOUS WAVELET ANALYSIS 2.1. Continuous Wa¨ elet Transform DEFINITION 2.1. A family ⌿ 4 . gx0, ⬁w, of functions in L2 Ž K _ GrK . is said to be a ‘‘wavelet’’ on GrK, if the following conditions are satisfied:
ˆ Ž a.< 2 d r s 1. Ži. For all a g ⌳, a / a0 , H0⬁ < ⌿ ˆ Ž a0 . s 0. Žii. For all gx0, ⬁w, ⌿ ˆ Ž a.< 2 d r - ⬁. Žiii. For all t gx0, ⬁w, Ý ag ⌳ , a/ a d a Ht⬁ < ⌿ 0 If we call the function ⌿ s ⌿1 the mother wavelet, then we can define the dilation operators D , gx0, ⬁w, by D⌿ s ⌿ . Let ⌿ , g be the function on G defined by ⌿ , g Ž x . s L g D⌿ Ž x . s ⌿ Ž gy1 x .. We consider L2 Žx0, ⬁w=GrK . the Hilbert space of square integrable complex functions on x0, ⬁w=G with respect to the measure d dx , which are K-right invariant relative to the second variable. We have the following lemma.
241
COMPACT GELFAND PAIRS
LEMMA 2.2 ŽParseval identity.. relations
For all F, H in L2 Ž GrK . we ha¨ e the
Ž Ž F , ⌿ . . . . 2 , Ž H , ⌿ . . . . 2 . L Ž x0, ⬁w =GrK . s Ž F , H . 2 2
5 Ž F , ⌿ . . . . 2 5 L2 Žx0, ⬁w=G r K . s 5 F 5 2 . Proof. By the inversion formula Ž1.11. we have ⌿ Ž gy1 x . s
᭙ g , x g G,
ˆ Ž a. a Ž gy1 x . , d a⌿
Ý ag⌳
and by the addition formula Ž1.8. we get ⌿ Ž gy1 x . s
᭙ g , x g G,
da
ˆ Ž a. Ya, k Ž x . Ya, k Ž g . . Ý Ý⌿ ag⌳ ks1
Using the orthogonality of Ya, k 4 we obtain
Ž Ž F , ⌿ . . . . 2 ⭈ Ž H , ⌿ . . . . 2 . L Ž x0, ⬁w =GrK . 2
s s
⬁
HGH0
Ž F , ⌿ , g . 2 Ž H , ⌿ , g . 2
d dg
da
⬁
H0 Ý Ý F Ž F . Ž a, k . F Ž H . Ž a, k . < ⌿ˆ Ž a. <
ag⌳ ks1
2
d
da
s
Ý Ý F Ž F . Ž a, k . F Ž H . Ž a, k . ag⌳ ks1
s Ž F , H .2. DEFINITION 2.3. Let ⌿ 4 , gx0, ⬁w, be a wavelet on GrK. We define the wavelet transform Ž W . ⌿ from L2 Ž GrK . into L2 Žx0, ⬁w=GrK . by
Ž W . ⌿ Ž H . Ž , g . s Ž H , ⌿ , g . 2 for all H in L2 Ž GrK . and Ž , g . in x0, ⬁w=G. DEFINITION 2.4. Let K t 4 , t gx0, ⬁w, be a family of functions in L1 Ž K _ GrK . such that Kˆt Ž a0 . s 1 for all t gx0, ⬁w. We said that K t 4 , t gx0, ⬁w, is an approximate identity in L2 Ž GrK . if for all f g L2 Ž GrK ., lim 5 f y f ) K t 5 2 s 0.
tª0 q
MOKNI AND TRIMECHE `
242
PROPOSITION 2.5. Let K t 4 , t gx0, ⬁w, be a family of functions in L1 Ž K _ GrK ., such that Kˆt Ž a0 . s 1 for all t gx0, ⬁w. We suppose that there is a positi¨ e constant M such that ᭙ t ) 0,
< Kˆt Ž a . < F M ;
a g ⌳,
then K t 4 , t gx0, ⬁w, is an approximate identity if and only if lim Kˆt Ž a . s 1
for all a g ⌳ .
tª0 q
Proof. To prove that lim t ª 0q Kˆt Ž a. s 1 for all a g ⌳, we have just to remark that a ) K t s Kˆt Ž a. ⭈ a and take the limit in L2 Ž G .. Conversely, take a function f in L2 Ž GrK .; then for all Ž a, j . g N , F Ž f ) K t . Ž a, j . s
HGHG f Ž y . K Ž x . Y t
a, j
Ž yx . dy dx
using formula Ž1.6. for Ya, j and the fact that K t is K-left invariant, we get F Ž f ) K t . Ž a, j . sd a
HGHGHGHK f Ž y . K Ž x . Y t
a, j
Ž g . a Ž xy1 kyy1 g . dk dg dy dx.
The product formula Ž1.10. gives F Ž f ) K t . Ž a, j . sd a
HGHGHG f Ž y . K Ž x . Y t
a, j
Ž g . a Ž xy1 . a Ž yy1 g . dg dy dx.
Integrating on x and on g we obtain F Ž f ) K t . Ž a, j . s F Ž f . Ž a, j . Kˆt Ž a . . This relation gives the inequality < F Ž f ) K t . Ž a, j . < 2 F < F Ž f . Ž a, j . Kˆt Ž a . < 2 F M 2 < F Ž f . Ž a, j . < 2 , so the Fourier series of f ) K t tends to the Fourier series of f as t ª 0q, and this is the desired result. We now prove the reconstruction formula for the continuous wavelet transform Ž W . ⌿ . THEOREM 2.6 ŽReconstruction Formula.. Let ⌿ 4 , gx0, ⬁w, be a wa¨ elet on GrK. Then for all H g L2 Ž GrK . with ¨ anishing integral on G,
243
COMPACT GELFAND PAIRS
we ha¨ e the following identity which holds in L2 Ž GrK .: H Ž ⭈. s
⬁
HGH0
d dg
Ž W . ⌿ Ž H . Ž , g . ⌿ , g Ž ⭈ .
.
Proof. Take an arbitrary t ) 0. From Definition 2.3 and the Fubini theorem we obtain ⬁
HGHt
Ž W . ⌿ Ž H . Ž , g . ⌿ , g Ž x . ⬁
s
HGHt HGH Ž y . ⌿
s
HGH Ž y . Ht HG⌿ Ž g
ž
, g
⬁
ž
Ž y . ⌿ , g Ž x . dy y1
d dg
y . ⌿ Ž gy1 x .
/
d dg
d dg
/
dy.
Ž 2.1.
Using the inversion formula for ⌿ and the formula Ž1.7., we can show that the second member of Ž2.1. is equal to ⬁
HGH Ž y . Ht ž Ý d < ⌿ˆ Ž a. < Ž y 2
a
a
y1
x.
ag⌳
/
d dy
.
This can be written in the form H ) K t Ž x . with KtŽ x. s 1 q
Ý
da
ag⌳
⬁
žH t
ˆ Ž a. < 2 <⌿
d
/
a Ž x . .
Ž 2.2.
This series is absolutely and uniformly convergent because for all a g ⌳, 5 a 5 ⬁ F 1, and because of the property Žiii. of the definition of the wavelet ⌿ . By the Parseval formula Ž1.12. the function K t is in L2 Ž K _ GrK .. Moreover, Kˆt Ž a0 . s 1, and, for all a g ⌳, a / a0 , we have Kˆt Ž a . s
⬁
Ht < ⌿ˆ Ž a. <
2
d
,
so the property Ži. of Definition 2.1 gives lim t ª 0q Kˆt Ž a. s 1 for all a g ⌳ and < Kˆt Ž a.< F 1 for all a g ⌳ and t ) 0. So by Proposition 2.5, K t 4 , t gx0, ⬁w, is an approximate identity, and we have lim H ) K t s H.
tª0 q
The limit is in L2 Ž G ..
Ž 2.3.
MOKNI AND TRIMECHE `
244
DEFINITION 2.7. The associated scaling function ⌽t 4 , t gx0, ⬁w, to a given wavelet ⌿ 4 , gx0, ⬁w, on GrK, is defined by ᭙ g g G,
⌽t Ž g . s
Ý
ˆ t Ž a . a Ž g . , d a⌽
ag⌳
ˆ t Ž a0 . s 1 and ⌽ ˆ t Ž a. s Ž Ht⬁ < ⌿ ˆ Ž a.< 2 d .1r2 for all a g ⌳, a / a0 . where ⌽ Remark. We note that ⌽t belongs to L2 Ž K _ GrK ., because ⬁ d ˆ t Ž a. < 2 s 1 q ˆ Ž a. < 2 5 ⌽t 5 22 s Ý d a < ⌽ - ⬁. Ý da < ⌿ t ag⌳ ag⌳ , a/a 0
H
Moreover, using inversion formula Ž1.13. for ⌽t and formula Ž1.7., we obtain ᭙ g g G, ⌽t ) ⌽t Ž g . s K t Ž g . , Ž 2.4. where K t is the function given by the relation Ž2.2.. So ⌽t ) ⌽t 4 , t gx0, ⬁w, is an approximate identity. COROLLARY 2.8. Let ⌿ 4 , gx0, ⬁w, be a wa¨ elet on GrK, and let ⌽t 4 , t gx0, ⬁w, be the associated scaling function. Then for all H g L2 Ž GrK . with ¨ anishing integral on G, the equality holds in L2 Ž GrK ., H s limq tª0
HG Ž H , ⌽
t , g 2 ⌽t , g
.
dg ,
where ⌽t, g , g g G, is the function gi¨ en by ⌽t , g Ž x . s ⌽t Ž gy1 x . .
᭙ x g G,
Proof. Using the Fubini theorem and the property ⌽t Ž gy1 . s ⌽t Ž g . ,
᭙ g g G, we obtain
HG Ž H , ⌽
t , g 2 ⌽t , g
.
Ž x . dg y1
s
HGHGH Ž y . ⌽ Ž g
s
HGH Ž y . HG⌽ Ž y
s
HGH
s
HGH Ž y . ⌽ ) ⌽ Ž y
t
ž Ž . žH y
y . ⌽t Ž gy1 x . dy dg
y1
t
g . ⌽t Ž gy1 x . dg dy
/ /
⌽t Ž g . ⌽t Ž gy1 yy1 x . dg dy
G
t
t
y1
s H ) ⌽t ) ⌽t Ž x . . We deduce the result from relation Ž2.3..
x . dy
245
COMPACT GELFAND PAIRS
The scaling function ⌽t 4 , t gx0, ⬁w, associated with the wavelet ⌿ 4 , gx0, ⬁w, on GrK, is the kernel of an approximate identity Ži.e., ⌽t ) ⌽t is an approximate identity.. Conversely, the following proposition will show that to a given kernel verifying some suitable conditions, one can define an associated wavelet on GrK in the sense of Definition 2.1. PROPOSITION 2.9. Let ⌽t 4 , t gx0, ⬁w, be a kernel of an approximate ˆ t Ž a. is real, identity. Suppose that for all a g ⌳, a / a0 , the function t ¬ ⌽ continuously differentiable, and decreasing. Moreo¨ er, assume that ᭙ t ) 0,
ˆ t Ž a0 . s 1, ⌽
ˆ t Ž a. s 0, lim ⌽
and
for all a g ⌳ , a / a0 .
tª⬁
Then the family ⌿ 4 , gx0, ⬁w, defined by
ˆ Ž a. s y ⌿
᭙a g ⌳ ,
ž
d d
ž
ˆ Ž a. ⌽
2
/
1r2
/
,
Ž 2.5.
is a wa¨ elet on GrK with scaling function ⌽t 4 . Proof. From the relation Ž2.5. we deduce for all t gx0, ⬁w, and a g ⌳, a / a0 , ⬁
Ht < ⌿ˆ Ž a. <
2
d
2
ˆ t Ž a. . . s Ž⌽
So ⌽t 4 is associated with ⌿ 4 . Let us verify the properties Ži., Žii., and Žiii. of Definition 2.1. Ži. For all a g ⌳, a / a0 , we have ⬁
H0 < ⌿ˆ Ž a. <
2
d
2
ˆ t Ž a. . s 1, s limq Ž ⌽ tª0
because ⌽t 4 , t gx0, ⬁w, is the kernel of an approximate identity. Žii. For all gx0, ⬁w, we have
ˆ Ž a0 . s y ⌿
ž
d d
1 s 0.
/
Žiii. For all t gx0, ⬁w, we deduce from the Parseval formula for ⌽t
Ý ag⌳ , a/0
da
⬁
Ht < ⌿ˆ Ž a. <
2
d
s
2
Ý ag⌳ , a/0
ˆ t Ž a. . - ⬁. da Ž ⌽
MOKNI AND TRIMECHE `
246
Moreover, this inequality proves that for almost ) 0, ⌿ belongs to L2 Ž K _ GrK ., and this gives the desired result. 2.2. Multiresolution Analysis We now define the continuous multiresolution analysis of L2 Ž GrK .. Denote by pt , t gx0, ⬁w, the convolution operator of L2 Ž GrK . given by pt Ž H . s H ) ⌽t ) ⌽t ,
for H g L2 Ž GrK . .
Ž 2.6.
The corresponding ‘‘scale space’’ is defined by Vt s pt Ž L2 Ž GrK . . .
Ž 2.7.
LEMMA 2.10. We ha¨ e the following properties for pt and Vt : Ži. g N.
ˆ t Ž a.. 2 , for all t gx0, ⬁w, and Ž a, j . F Ž pt Ž H ..Ž a, j . s F Ž H .Ž a, j .Ž ⌽
Žii. Vt ; Vt ⬘ , for t G t⬘. Žiii. Vt , t gx0, ⬁w, is globally in¨ ariant under the action of the group G. Proof. Ži. We have ᭙ t g x 0, ⬁ w , F Ž pt Ž H . . Ž a, j . s
Ž a, j . g N , y1
HGHG⌽ ) ⌽ Ž g t
t
x . H Ž g . Ya, j Ž x . dg dx
Using the facts that ᭙g,
x g G,
⌽t ) ⌽t Ž gy1 x . s
d a⬘
Ý Ý
$
⌽t ) ⌽t Ž a⬘ . Ya⬘, j⬘ Ž x . Ya⬘, j⬘ Ž g .
a⬘g⌳ j⬘s1
and $
2
ˆ t Ž a⬘ . . , ⌽t ) ⌽t Ž a⬘ . s Ž ⌽
᭙a⬘ g ⌳ ,
and the orthogonality of the family Ya, j 4 we can easily check that ᭙ t g x 0, ⬁ w ,
ˆ t Ž a. . Ž a, j . g N , F Ž pt Ž H . . Ž a, j . s Ž ⌽
2
HGH Ž g . Y
a, j
Ž g . dg ,
and this gives the desired result. Žii. Take t, t⬘ gx0, ⬁w, such that t G t⬘, and F s pt Ž H . g Vt . From the inversion formula Ž1.13. for pt Ž H . we obtain da
Fs
Ý Ý F Ž H . Ž a, j . Ž ⌽ˆ t Ž a. . ag⌳ js1
2
Ya, j .
247
COMPACT GELFAND PAIRS
Thus ᭙ Ž a, j . g N ,
2
ˆ t Ž a. . . F Ž F . Ž a, j . s F Ž H . Ž a, j . Ž ⌽
Now define the sequence A a, j 4 , Ž a, j . g N , by A a, j s
ˆ t Ž a. . F Ž H . Ž a, j . Ž ⌽
Ž ⌽ˆ t ⬘ Ž a. .
2
2
.
Ž 2.8.
ˆ t Ž a. is decreasing, and F Ž H .Ž a, j .4 is in l 2 Ž N .; then The function t ¬ ⌽ the sequence A a, j 4 belongs to l 2 Ž N .. Thus from the Parseval theorem for F there exists a function f g L2 Ž GrK ., such that ᭙ Ž a, j . g N ,
F Ž f . Ž a, j . s A a, j .
Ž 2.9.
The relation da
Fs
Ý Ý A a, j Ž ⌽ˆ t ⬘ Ž a. .
2
Ya, j
Ž 2.10.
ag⌳ js1
implies that F s pt ⬘Ž f . belongs to Vt ⬘. Žiii. Let F s pt Ž H . for some H g L2 Ž GrK .. The invariance of the measure dg under the left translation of G gives L g Ž pt Ž H . . s pt Ž L g Ž H . . ,
for g g G.
Ž 2.11.
Hence we obtain the desired result. PROPOSITION 2.11. The scale spaces Vt , t gx0, ⬁w, satisfy the following properties: 䢇 䢇
Ha 0 ; Vt ; Vt ⬘ ; L2 Ž GrK . for 0 - t⬘ F t - ⬁. F Vt s Ha0 . t)0
䢇
D Vt
is dense in L2 Ž GrK ..
t)0
Proof. We deduce the results from Lemma 2.10 and the relation da
lim
Ý Ý F Ž H . Ž a, j . Ž ⌽ˆ t Ž a. .
tª⬁ ag⌳ js1
2
s
HG H Ž g . dg.
MOKNI AND TRIMECHE `
248
Remark. A family of scale spaces satisfying those properties is called a continuous multiresolution analysis of L2 Ž GrK .. 2.3. The Poisson Kernel and the Associated Wa¨ elet Suppose that G is a compact connected Lie group, and K is a closed subgroup. Let DŽ G . be the set of all left invariant differential operators on G, and let DK Ž G . be the subspace of those which are also K-right invariant. The spherical functions of the Gelfand pair Ž G, K . are eigenfunctions of any D in DK Ž G . Žsee w4x.. Take a Laplacian ⌬ on G in the sense of w8, Chap. Ix; then one has ⌬ a s a a .
᭙a g ⌳ ,
The eigenvalues a are nonnegative, and a 0 s 0. Let t gx0, ⬁w. The K-biinvariant Poisson kernel on G, denoted by Pt , is defined by the formula ᭙ g g G,
Pt Ž g . s
Ý
ey a t d a a Ž g . .
Ž 2.12.
ag⌳
According to w8x this series converges absolutely and uniformly for all g g G and t G b ) 0; so it defines a continuous function relative to the variables g and t. We have the following properties for the family Pt 4 :
HGP Ž g . dg s 1, for all t gx0, ⬁w.
䢇
Pˆt Ž a0 . s
䢇
Pt ) Pt ⬘ s Ptqt ⬘ , for all t, t⬘ gx0, ⬁w.
t
Moreover, the kernel Pt is a solution of the heat equation ⭸⭸ut s ⌬ u; more precisely, if f belongs to L1 Ž K _ GrK ., then uŽ g, t . s Pt ) f Ž g . is a solution verifying the boundary condition in L1 Ž G .: lim u Ž ⭈, t . s f .
tª0 q
So Pt 4 , t gx0, ⬁w, is an approximate identity. From the relation Ž2.12. we deduce that Pˆt Ž a . s ey a t ,
for all t g x 0, ⬁ w , a g ⌳ .
So, for all a g ⌳, a / a0 , the function t ¬ Pˆt Ž a. is real, differentiable, and decreasing, and lim Pˆt Ž a . s 1,
tª0 q
lim Pˆt Ž a . s 0.
tª⬁
249
COMPACT GELFAND PAIRS
By Proposition 2.9, the family Pt 4 , t gx0, ⬁w, is the scale function of a wavelet ⌿ 4 , gx0, ⬁w, on GrK, defined by ᭙a g ⌳ ,
ˆ Ž a. s y ⌿
ž
d d
ž
Pˆ Ž a .
2
/
1r2
/
s 2 a ey a , Ž 2.13.
'
and then ᭙ g g G,
⌿ Ž g . s
Ý
'2
a
d a ⑀y a a Ž g . .
Ž 2.14.
ag⌳
In the particular case of the group G s SO Ž q . the Poisson kernel Pt 4 , t gx0, ⬁w, is given on wy1, 1x by the formula ᭙ r g w y1, 1 x ,
Pt Ž r . s
eyn Ž nqqy2.t d n PnŽ qy2.r2 Ž r . . Ž 2.15.
Ý ng⺞
And in the case G s UŽ q . the Poisson kernel Pt 4 , t gx0, ⬁w, is given on the unit disc D by the formula Žsee w5x. ᭙ z g D,
Pt Ž z . s
qy 2 eyt Ž m k n. dŽ m , n. R m , n Ž z. ,
Ý
Ž 2.16.
m, ng⺞
where m k n s maxŽ m, n..
3. DISCRETIZATION BY WAVELET PACKETS We now define three types of wavelet packets on GrK: P-wavelet, M-wavelet, and S-wavelet. 3.1. The P-Wa¨ elet Packet on GrK DEFINITION 3.1. Let ⌿ 4 , gx0, ⬁w, be a wavelet on GrK. The P-wavelet packet on GrK, denoted ⌿lP 4 , l g ⺪, is defined by ᭙a g ⌳ ,
ˆ lP Ž a . s ⌿
žH
l
lq1
ˆ Ž a. < 2 <⌿
d
1r2
/
,
where l 4 , l g ⺪, is a strictly decreasing sequence such that lim l s 0 and
lª⬁
lim l s ⬁.
lªy⬁
The function ⌿lP is called the P-wa¨ elet packet member of step l.
MOKNI AND TRIMECHE `
250 Remarks.
Ži. Note that, for all a g ⌳, a / a0 , we have ⬁
⬁
ˆ lP Ž a . < 2 s H < ⌿ ˆ Ž a. < 2 Ý <⌿ 0
lsy⬁
d
s 1.
Žii. For all g g G, we define the function ⌿l,P g by ⌿lP, g Ž x . s ⌿lP Ž gy1 x . .
᭙ x g G,
Žiii. For all F, H in L2 Ž GrK . we obtain from Lemma 2.2 ⬁
Ý
H Ž F, ⌿ . Ž H, ⌿ . P l, g 2
lsy⬁ G
⬁
Ý lsy⬁
P l, g 2
HG Ž F , ⌿ .
P l, g 2
2
dg s Ž F , H . 2
Ž 3.1.
dg s 5 F 5 22 .
Ž 3.2.
Notation. We denote by L2 Ž⺪ = GrK . the space of measurable functions F on ⺪ = G, K-right invariant relative to the second variable, and satisfying ⬁
HG< F Ž l, g . <
Ý lsy⬁
2
dg - ⬁.
DEFINITION 3.2. Let ⌿lP 4 , l g ⺪, be a P-wavelet packet on GrK. We define the P-wavelet packet transform Ž W . P⌿ from L2 Ž GrK . into L2 Ž⺪ = GrK . by P
Ž W . ⌿ Ž H . Ž l, g . s Ž H , ⌿lP, g . 2 for all H in L2 Ž GrK . and Ž l, g . g ⺪ = G. THEOREM 3.3 ŽReconstruction Formula.. Let ⌿lP 4 , l g ⺪, be a P-wa¨ elet packet on GrK. Then for all H in L2 Ž GrK . with ¨ anishing integral on G, we ha¨ e the following identity which holds in L2 Ž GrK .: ⬁
H Ž ⭈. s
Ý lsy⬁
HGŽ W .
P ⌿
Ž H . Ž l, g . ⌿lP, g Ž ⭈ . dg.
Proof. By using Theorem 2.6, we have just to remark that H Ž x . s lim
⬁
HH Lª⬁ G
L
Ž W . ⌿ Ž H . Ž , g . ⌿ , g Ž x .
d dg
251
COMPACT GELFAND PAIRS
and that ⬁
HGH
Ž W . ⌿ Ž H . Ž , g . ⌿ , g Ž x .
L
Ly1
s
Ý lsy⬁
HGŽ W .
P ⌿
d dg
Ž H . Ž l, g . ⌿lP, g Ž x . dg.
We shall now define the scale discrete scaling function associated with a given P-wavelet packet on GrK. DEFINITION 3.4. The scale discrete scaling function ⌽LP 4 , L g ⺪, associated with a P-wavelet packet ⌿lP 4 , l g ⺪, on GrK, is defined by ᭙ g g G,
⌽LP Ž g . s
ˆ LP Ž a. a Ž g . , d a⌽
Ý ag⌳
ˆ LP Ž a0 . s 1 and where ⌽ Ly1
᭙a g ⌳ ,
ˆ LP Ž a. s ⌽
a / a0 ,
žÝ
1r2
ˆ lP Ž a. < 2 <⌿
lsy⬁
/
ˆ LŽ a. . s⌽
THEOREM 3.5. Let ⌽LP 4 , L g ⺪, be a scale discrete scaling function. Then for all H g L2 Ž GrK . with ¨ anishing integral on G, we ha¨ e the following equality which holds in L2 Ž GrK .: H Ž ⭈ . s lim
H Ž H, ⌽ .
P P L , g 2 ⌽L , g
Lª⬁ ⍀
Ž ⭈ . dg ,
Ž 3.3.
P where ⌽L, g , g g G, is the function gi¨ en by
⌽LP , g Ž x . s ⌽LP Ž gy1 x . .
᭙ x g G,
Proof. We deduce the result from the fact that ᭙a g ⌳ ,
ˆ LP Ž a. s ⌽ ˆ LŽ a. ⌽
and Corollary 2.8. PROPOSITION 3.6. Ži. We ha¨ e ᭙a g ⌳ ,
Ž ⌽ˆ LP Ž a. .
2
q
⬁
ˆ lP Ž a. < 2 s 1. Ý <⌿ lsL
Ž 3.4.
MOKNI AND TRIMECHE `
252 Žii. We ha¨ e ᭙a g ⌳ ,
⬁
P Ž a. . Ý ž Ž ⌽ˆ lq1
a / a0 ,
2
ˆ lP Ž a. . y Ž⌽
lsy⬁
2
/ s 1.
Ž 3.5.
Proof. Ži. For all a g ⌳, a / a0 , we have ⬁
Ly1
⬁
lsy⬁
lsy⬁
lsL
ˆ lP Ž a. < 2 s Ý < ⌿ ˆ lP Ž a. < 2 q Ý < ⌿ ˆ lP Ž a. < 2 ; Ý <⌿ thus
Ž ⌽ˆ LP Ž a. .
2
⬁
ˆ lP Ž a. < 2 s 1. Ý <⌿
q
lsL
This equality holds also for a0 . Žii. For all a g ⌳, a / a0 , and all l g ⺪, we have 2
2
P ˆ lP Ž a. < 2 s Ž ⌽ ˆ lq1 ˆ lP Ž a. . . <⌿ Ž a. . y Ž ⌽
We deduce the result from this relation. PROPOSITION 3.7. H Ž ⭈. s
For all H g L2 Ž GrK . we ha¨ e
HG Ž H , ⌽ .
P P L , g 2 ⌽L , g
Ž ⭈ . dg q
⬁
Ý H Ž W . P Ž H . Ž l, g . ⌿lP, g Ž ⭈. dg.
lsL G
Ž 3.6. Proof. The relation Ž3.4. gives the result. The relation Ž3.5. leads us to an other type of wavelet packet on GrK: the modified wavelet packet, also called the M-wavelet packet on GrK. 3.2. The M-Wa¨ elet Packet on GrK DEFINITION 3.8. Let ⌽LP 4 , L g ⺪, be a scale-discrete scaling function, associated with a P-wavelet packet ⌿lP 4 , l g ⺪, on GrK. Ži. The corresponding M-wavelet packet ⌿lM 4 , l g ⺪, on GrK, and ˜ lM 4 are defined by its dual ⌿ ᭙a g ⌳ ,
P ˆ lM Ž a. s ⌽ ˆ lq1 ˆ lP Ž a. ⌿ Ž a. y ⌽
᭙a g ⌳ ,
P ˆ ˜ lM Ž a. s ⌽ ˆ lq1 ˆ lP Ž a. . ⌿ Ž a. q ⌽
253
COMPACT GELFAND PAIRS
2Ž Žii. The M-wavelet packet transform Ž W . M . ⌿ is defined from L GrK 2Ž into L ⺪ = GrK . by M
Ž W . ⌿ Ž H . Ž l, g . s Ž H , ⌿lM, g . 2 for all H g L2 Ž GrK . and Ž l, g . g ⺪ = G. Remark. By construction of the M-wavelet packet on GrK, we have the relation 1r2
Ly1
ˆ LP Ž a. s ⌽
ž
ˆ˜ M Ž a. ˆ lM Ž a. ⌿ Ý ⌿ l
lsy⬁
/
,
Ž 3.7.
which implies that Ly1
⌽LP ) ⌽LP s
˜ lM ⌿lM ) ⌿
Ý
Ž 3.8.
lsy⬁
the relation Ž3.8. gives immediately the following
˜ lM 4 be, respecti¨ ely, an M-wa¨ elet packet THEOREM 3.9. Let ⌿lM 4 and ⌿ on GrK and its dual. Then for all H in L2 Ž GrK . with ¨ anishing integral on G, we ha¨ e the following identity which holds in L2 Ž GrK .: H Ž ⭈. s
⬁
Ý lsy⬁
HGŽ W .
M ⌿
˜ lM, g Ž ⭈ . dg. Ž H . Ž l, g . ⌿
Remark. In the same way as for the P-wavelet packet on GrK, we get, for all H g L2 Ž GrK ., H Ž ⭈. s
HG Ž H , ⌽ .
P P L , g 2 ⌽L , g
Ž ⭈ . dg q
⬁
˜ lM, g Ž ⭈ . dg. Ý H Ž W . M Ž H . Ž l, g . ⌿
lsL G
3.3. The S-Wa¨ elet Packet on GrK Remark that for all a g ⌳, a / a 0 , we have ⬁
ˆ ˆ lM Ž a. ⌿ ˜ lM Ž a. s 1. Ý ⌿
Ž 3.9.
lsy⬁
Note that this equality is the key to the reconstruction formula for the M-wavelet transform. Suppose, now, that for a given sequence ⌿lS 4 , l g ⺪, of functions ˜ lS 4, l g ⺪, in in L2 Ž K _ GrK ., one can define another sequence ⌿ 2Ž L K _ GrK . such that for all a g H, a / a0 : ⬁
ˆ ˆ lS Ž a. ⌿ ˜ lS Ž a. s 1. Ý ⌿ lsy⬁
MOKNI AND TRIMECHE `
254
Then we can obtain the same results as above. In fact this is possible under some conditions verified by ⌿lS 4 . DEFINITION 3.10. A family ⌿lS 4 , l g ⺪, in L2 Ž K _ GrK . is said to be ˆ lS is real, ⌿ ˆ lS Ž a0 . s 0, and satisfies the an S-wa¨ elet packet on GrK if ⌿ following conditions: Ži. There are positive real constants, 0 - A F B - ⬁, such that ᭙a g ⌳ ,
AF
⬁
2 ˆ lS Ž a. . F B. Ý Ž⌿
lsy⬁
These inequalities are called ‘‘stability conditions.’’ Žii. For all L g ⺪, da
Ý ag⌳ , a/a 0
Ly1 Ý lsy⬁ Ž ⌿ˆ lS Ž a. .
2
ˆ lS Ž a. . Ý⬁lsy⬁Ž ⌿
2
-⬁
Žiii. There exists M ) 0 such that for all L g ⺪ and g g G,
HG Ý d ag⌳
a
Ly1 Ý lsy⬁ Ž ⌿ˆ lS Ž a. .
2
ˆ lS Ž a. . Ý⬁lsy⬁Ž ⌿
2
a Ž g . dg F M.
The S-wavelet transform Ž W . S⌿ , from L2 Ž GrK . into L2 Ž⺪ = GrK ., is defined by S
Ž W . ⌿ Ž H . Ž l, g . s Ž H , ⌿lS, g . 2 for all H g L2 Ž GrK . and Ž l, g . g ⺪ = G. S PROPOSITION 3.11. The linear operator Ž W . ⌿ from L2 Ž GrK . into L ⺪ = GrK . ¨ erifies 2Ž
A 5 H 5 22 F 5 Ž W . ⌿ Ž H . 5 2L2 Ž⺪=G r K . F B 5 H 5 22 S
for all H g L2 Ž GrK . with ¨ anishing integral on G. In particular, it is a continuous operator. Proof. By applying the inversion formula Ž1.11. to ⌿lS and the Parseval identity relative to the system Ya, j 4 , Ž a, j . g N , to H, we show that the property A 5 H 5 22 F 5 Ž W . ⌿ Ž H . 5 2L2 Ž⺪=G r K . F B 5 H 5 22 S
255
COMPACT GELFAND PAIRS
is equivalent to da
A
Ý Ý < F Ž H . Ž a, j . < 2 ag⌳ js1 ⬁
da
ˆ lS Ž a. . Ý Ý < F Ž H . Ž a, j . < 2 Ý Ž ⌿
F
ž
ag⌳ js1
2
lsy⬁
/
da
FB
Ý Ý < F Ž H . Ž a, j . < 2 . ag⌳ js1
Then the stability conditions give the result. The constant A is positive, so the operator Ž W . S⌿ is injective and then possesses a continuous linear inverse operator, which we shall describe now.
˜ lS 4, l g ⺪, on GrK, DEFINITION 3.12. Ži. The dual S-wavelet packet ⌿ S4 corresponding to a given S-wavelet packet ⌿l , l g ⺪, on GrK, is defined ˆ ˜ lS Ž a0 . s 0 and by ⌿ ᭙a g ⌳ ,
ˆ ˜ lS Ž a. s ⌿
a / a0 ,
ˆ lS Ž a. ⌿ ˆ lS Ž a. . Ý⬁lsy⬁Ž ⌿
.
2
Žii. The S-scaling function ⌽LS 4 , L g ⺪, associated with ⌿lS 4 , l g ⺪, is defined by
ˆ
⌽LS
¡ ¢1
Ý Ž a. s~ ž
ˆ ˆ Ž a. ⌿ ˜ lS Ž a. /
Ly 1 S lsy⬁⌿l
1r2
for all a g ⌳ , a / a0 for a s a0 .
˜ lS 4, l g ⺪, Remark. From the definition of the dual S-wavelet packet ⌿ on GrK, Ži. We have ⬁
2 ˆ S Ž a. s ˜ Ý ž⌿ / l
᭙a g ⌳ ,
lsy⬁
1
ˆ lS Ž a. . Ý⬁lsy⬁Ž ⌿
2
.
Ž 3.10.
Then 1 B
F
⬁
1
lsy⬁
A
2 ˆ ˜ lS Ž a. / F Ý ž⌿
,
where A, B are the constants given in Definition 3.10.
Ž 3.11.
MOKNI AND TRIMECHE `
256 Žii. We have
Ly1 ˆ ˜ lS Ž a. Ý lsy⬁ ⌿
᭙a g ⌳ ,
Ý⬁lsy⬁
ž / ž ⌿˜ˆ Ž a. / S l
2
2
s
Ly 1 Ý lsy⬁ Ž ⌿ˆ lS Ž a. .
2
ˆ lS Ž a. . Ý⬁lsy⬁Ž ⌿
2
.
Ž 3.12.
LEMMA 3.13. The S-scaling function ⌽LS 4 , L g ⺪, associated with an S-wa¨ elet packet ⌿lS 4 , l g ⺪, on GrK, is the kernel of an approximate identity in the sense that, for all H g L2 Ž GrK ., lim H ) ⌽LS ) ⌽LS s H.
Lª⬁
The limit is in L2 Ž GrK .. Proof. From Definition 3.12Žii. we have Ly1
2
Ž ⌽ˆ LS Ž a. . s
᭙a g ⌳ ,
ˆ˜ S Ž a. . ˆ lS Ž a. ⌿ Ý ⌿ l
lsy⬁
Then ᭙a g ⌳ ,
2
ˆ LS Ž a. . s 1. lim Ž ⌽
Lª⬁
So, to prove the lemma, it suffices to show that for all L g ⺪, the function ⌽LS is uniformly bounded in L1 Ž G . and continuous. We deduce these properties by applying the inversion formula Ž1.11. to ⌽LS , the stability conditions Ži. and Žiii. of Definition 3.10. The reconstruction formula for the S-wavelet transform arises immediately from the above, exactly as for the M-wavelet transform, and we obtain the following:
˜ lS 4, l g ⺪, be, respecti¨ ely, an S-wa¨ elet THEOREM 3.14. Let ⌿lS 4 and ⌿ packet on GrK and its dual. Then for all H in L2 Ž GrK . with ¨ anishing integral o¨ er G, we ha¨ e the following identity, which holds in L2 Ž GrK .: H Ž ⭈. s
⬁
Ý lsy⬁
HGŽ W .
S ⌿
˜ lS, g Ž ⭈ . dg. Ž H . Ž l, g . ⌿
4. MULTIRESOLUTION ANALYSIS BY WAVELET PACKETS The discrete multiresolution analysis will be defined by means of convolution operators, as for the continuous multiresolution analysis.
COMPACT GELFAND PAIRS
257
Let ⌿l 4 , l g ⺪, be a P-, M-, or S-wavelet packet on GrK, and let ⌽L 4 , L g ⺪, be the corresponding scaling function. For all of those wavelet packets on GrK we have, according to the relation Ž3.8., the equality Ly1
⌽L ) ⌽L s
Ý
˜L , ⌿L ) ⌿
Ž 4.1.
lsy⬁
˜ l 4 is the dual wavelet packet in case M and S, and the P-wavelet where ⌿ packet in case P. From this relation we deduce ˜L . ⌽Lq 1 ) ⌽Lq1 s ⌽L ) ⌽L q ⌿L ) ⌿
Ž 4.2.
Now define the convolution operators pL and q L , L g ⺪, by pLŽ H . s ⌽L ) ⌽L ) H
˜L ) H , q LŽ H . s ⌿L ) ⌿ and the spaces VL s pL Ž L2 Ž GrK . . WL s q L Ž L2 Ž GrK . . . The discrete family of scale spaces VL 4 , L g ⺪, verifies, as in the continuous case, the following properties: 䢇 䢇
Ha 0 ; VL ; VL⬘ ; L2 Ž GrK . for y⬁ - L F L⬘ - ⬁. F VL s Ha0 . Lg⺪
䢇
D VL
is dense in L2 Ž GrK ..
Lg⺪
Moreover, by the relation Ž2.12., every VL is globally invariant under the left translation of the group G. A function H in L2 Ž GrK . with vanishing integral over G can be approximated by its ‘‘projections’’ on the scale spaces VL ; in fact, lim pLŽ H . s H.
Lª⬁
We say that pLŽ H . is the approximation of H at the step L. Note that, by the relation Ž3.14., we have VLq 1 s VL q WL . So to go from an approximation at step L to an approximation at step L q 1, one needs the space WL .
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Suppose now that the scale function ⌽L 4 , L g ⺪, verifies the relation ⌽L ) ⌽L s ⌽Ly1 . So pL ( pL s pLy1 and pLŽ VL . s VLy1 , and then one can pass directly from a resolution at step L to a resolution at step L y 1. EXAMPLE. We know that the Poisson kernel Pt 4 , t gx0, ⬁w, verifies the relation Pt ) Pt ⬘ s Ptqt ⬘ ,
for all t , t⬘ g x 0, ⬁ w .
Take the dyadic sequence L 4 defined by L s 2yL for L g ⺪, and let ⌽L s P L , the associated scale-discrete scaling function. Then we have ⌽L ) ⌽L s P2yL ) P2yL s P2yL q2y L s P2yŽ Ly 1. s ⌽Ly1 , where ⌽L 4 , L g ⺪, is said to be a reproducing scaling function. REFERENCES 1. H. Annabi and K. Trimeche, Convolution generalisee ` ´ ´ ´ sur le disque unite, ´ C. R. Acad. Sci. Paris 278 Ž1974., 21᎐24. 2. J. Faraut, ‘‘Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques,’’ Les cours du C.I.M.P.A., Nancy, 1980. 3. W. Freeden and U. Windheuser, Spherical wavelet transform and its discretization, Ad¨ . Comput Math. 5 Ž1996., 51᎐94. 4. S. Helgason, ‘‘Groups and Geometric Analysis, Integral Geometry, Invariant Differential Operators, and Spherical Functions,’’ Academic Press, New York, 1984. 5. Y. Kanjin, A convolution measure algebra on the unit disc, Tohoku Math. J. 28 Ž1976., ˆ 105᎐115. 6. T. H. Koornwinder, ‘‘The Addition Formula for Jacobi Polynomials. II. The Laplace Type Integral Representation and the Product Formula,’’ Mathematisch Centrum, Amsterdam, TW 133r72, April 1972. 7. T. H. Koornwinder, The addition formula for Jacobi polynomials. I. Summary of results, Indag. Math. 34 Ž1972., 188᎐191. 8. E. L. Stein, Topics in harmonic analysis related to the Littelwood᎐Paley theory, in ‘‘Annals of Mathematical Studies,’’ Vol. 63, Princeton Univ. Press, Princeton, NJrUniv. of Tokyo Press, Tokyo, 1970.
Wavelet Transform on Compact Gelfand Pairs and Its Discretization K. Mokni Department of Mathematics, Faculty of Sciences of Monastir, Monastir 5019, Tunisia
and K. Trimeche ` Department of Mathematics, Faculty of Sciences of Tunis, Campus, Tunis 1060, Tunisia Submitted by Joseph D. Ward Received November 17, 1997
Using harmonic analysis on a compact Gelfand pair Ž G, K ., we study the continuous wavelet analysis on the homogeneous space GrK. As example we give the wavelet associated with the K-biinvariant Poisson kernel on G. Next, using the previous results, we define and study three types of wavelet packets on GrK and the corresponding wavelet transforms. We prove for these transforms reconstruction formulas, and we describe the related multiresolution analysis. 䊚 1999 Academic Press
0. INTRODUCTION We consider a compact Gelfand pair Ž G, K .. In this case the Fourier analysis is associated with the spherical functions of the pair Ž G, K ., and the decomposition of the space L2 Ž GrK . Žof square integrable functions on G with respect to the Haar measure of G, which are K-right invariant. into a discrete direct sum of irreducible spaces relative to the regular representation of the group G. Wavelet theory involves breaking up a complicated function Žphenomenon. into many simple pieces at different scales and positions. More explicitly, wavelet analysis allows us to represent a function f in L2 Ž GrK . as a sum of wavelets ⌿ , g s L g D⌿, i.e., left translated and dilated copies of a mother wavelet ⌿, a K-left invariant function in L2 Ž GrK . with 234 0022-247Xr99 $30.00 Copyright 䊚 1999 by Academic Press All rights of reproduction in any form reserved.
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COMPACT GELFAND PAIRS
vanishing integral. The coefficients of the decomposition are the inner products Ž f, ⌿ , g .L2 ŽG r K . . The objective of this paper is to generalize the important subject of wavelets to the case of the compact Gelfand pair Ž G, K .. We present here a general construction allowing the development of wavelet packets on the homogeneous space GrK from the continuous wavelet analysis on GrK. An analogous study has been made in w3x by Freeden and Windheuser for the compact Gelfand pair Ž SO Ž3., SO Ž2... It is believed that generalized wavelet expansions as discussed here will be a very useful tool in many areas of mathematics. The content of this paper is as follows. We summarize in the first section the main results of the harmonic analysis of the compact Gelfand pairs Ž G, K ., and we give more details about the particular cases Ž SO Ž q ., SO Ž q y 1.., q g ⺞, q G 3 and ŽUŽ q ., UŽ q y 1.., q g ⺞, q G 2. In the second section we study the continuous wavelet analysis on the homogeneous space GrK. As an example we give the wavelets associated with the K-biinvariant Poisson kernel on G. We define and study in the third section three types of wavelet packets on GrK and the corresponding wavelet transforms, and we prove for these transforms reconstruction formulas. The multiresolution analysis by means of wavelet packets on GrK is discussed in the fourth section.
1. HARMONIC ANALYSIS ON COMPACT GELFAND PAIRS Let G be a compact group with a normalized Haar measure dx, and let K be a closed subgroup. We denote by e the identity element of G. The group G acts on the space of complex functions f on G by left translation L g and right translation R g as follows: L g f Ž x . s f Ž gy1 x . and R g f Ž x . s f Ž xg .. Let L p Ž G ., p s 1, 2, be the space of measurable functions f on G such that
HG< f Ž x . <
p
dx - ⬁.
We denote by 䢇
L p Ž GrK . the space of K-right invariant functions f in L p Ž G ., that
is, f Ž gk . s f Ž g .
for all k g K ,
g g G.
MOKNI AND TRIMECHE `
236
The hermitian product of the Hilbert space L2 Ž GrK . is denoted by Ž⭈, ⭈ . 2 , and the corresponding norm by 5 ⭈ 5 2 . L p Ž K _ GrK . the space of K-biinvariant functions f in L p Ž G ., that is, f Ž kgk⬘ . s f Ž g . for all k, k⬘ g K , g g G. 䢇
A K-biinvariant function on G is said to be a zonal function on G. The convolution product is defined, for f, h in L1 Ž G ., by f ) hŽ x . s
y1
HGh Ž y
x . f Ž y . dy,
x g G.
The pair Ž G, K . is said to be a Gelfand pair if L1 Ž K _ GrK ., provided with this convolution product is commutative.
ˆ be the set of equivalence classes of unitary irreducible represenLet G ˆ realized in a Hilbert space Ea; this space is tations of G. Take a in G, automatically finite-dimensional. Let EaK be the subspace of Ea defined by EaK s ¨ g Ea : a Ž k . ¨ s ¨ , for all k g K 4 . We know that Ž G, K . is a Gelfand pair if and only if dimŽ EaK . F 1, for any ˆ Denote by ⌳ the set of a g G, ˆ such that dimŽ EaK . s 1. It is well a g G. known Žsee w4, p. 394x. that the left regular representation of G on L2 Ž GrK . is decomposed in a direct sum of unitary irreducible representations of finite dimension as follows: L2 Ž GrK . s
[ag⌳ Ha .
The representations a are the restrictions to Ha of the regular representation . Denote by a0 the trivial representation; then Ha 0 s ⺓. Let d a be the dimension of Ha , and let Ya, 1 , . . . , Ya, d a4 be an orthonormal basis of Ha with respect to the scalar product of L2 Ž GrK .. For all a g ⌳ there exists a unique normed vector ⑀ a g Ha of class one, that is, Ž k . ⑀ a s ⑀ a , for all k g K. The function a defined by
a Ž g . s Ž Ž g . ⑀ a , ⑀ a . 2 ,
for all g g G,
is a positive-definite spherical function of the Gelfand pair Ž G, K . associated with the representation Ž , Ha .. It satisfies the following properties:
a 0 ' 1 a Ž ⑀ . s 1,
for all a g ⌳
a Ž gy1 . s a Ž g . , 5 a 5 22 s
1 da
5 a 5 ⬁ s 1,
Ž 1.1.
for all a g ⌳ , ,
for all a g ⌳ for all a g ⌳
Ž 1.2. ggG
Ž 1.3. Ž 1.4. Ž 1.5.
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COMPACT GELFAND PAIRS
1
f ) a s
da
b ) a s a Ž xy1 y . s
1 da
for all a g ⌳ ,
f,
1 da
␦ b , a a ,
Ž 1.6.
for all a, b g ⌳
Ž 1.7.
da
for all a g ⌳ ,
Ý Ya, j Ž x . Ya, j Ž y . ,
x, y g G Ž 1.8.
js1
HK Ž xky . dk s Ž x . Ž y . , a
f g Ha
a
for all x, y g G and
a
a g ⌳ . Ž 1.9.
The formula Ž1.9. is called the product formula for the spherical functions. The spherical Fourier transform is defined for all f g L p Ž K _ GrK . by ᭙a g ⌳ ,
fˆŽ a . s
y1
HG f Ž x . Ž x a
. dx.
Ž 1.10.
If f g L2 Ž K _ GrK . we have the inversion formula f Ž ⭈. s
d a fˆŽ a . a Ž ⭈ .
Ý
Ž 1.11.
ag⌳
and the Parseval formula 5 f 5 2L2 ŽG. s
Ý
d a < fˆŽ a . < 2 .
Ž 1.12.
ag⌳
The family Ya, j 4 , a g ⌳, j g 1, . . . , d a4 , is an orthonormal basis of L2 Ž GrK .; so for all f g L2 Ž GrK . we define the Fourier coefficients by F Ž f . Ž a, j . s Ž f , Ya, j . 2 . The map F is an isomorphism isometric from L2 Ž GrK . onto l 2 Ž N ., where N s Ž a, j . : a g ⌳ , j g 1, . . . , d a 4 4 , and l 2 Ž N . is the space of sequences x a, j 4Ž a, j.g N such that da
Ý Ý < x a, j < 2 - ⬁. ag⌳ js1
MOKNI AND TRIMECHE `
238
This result is called the Parse¨ al theorem for F. The inversion formula of the map F is given for all f g L2 Ž GrK . by da
f Ž ⭈. s
Ý Ý F Ž f . Ž a, j . Ya, j Ž ⭈. .
Ž 1.13.
ag⌳ js1
All of these ingredients are well known for any compact Gelfand pair; we can refer to w2x, for example. 1.1. Examples Let ⺖ denote one of the fields ⺢, ⺓, of the real numbers or the complex numbers. For an integer q, let ⺖ q be a q-dimensional vector space over ⺖, with Hermitian product defined for z s Ž z1 , . . . , z q . and w s Ž w 1 , . . . , wq . in ⺖ q by q
Ž z, w . s
Ý
z kw k .
ks0
Let ⑀ 1 , . . . , ⑀ q 4 be the canonical orthonormal basis of ⺖ q. We consider G and K as follows: In the case ⺖ s ⺢, G s SO Ž q . is the group of rotations of ⺢ q. K s SO Ž q y 1. is the subgroup of rotations which leave the vector ⑀ 1 fixed. In the case ⺖ s ⺓, G s UŽ q . is the group of linear transformations of ⺓ q which leave the Hermitian product invariant Žthe unitary group.. K s UŽ q y 1. is the subgroup of elements in UŽ q . leaving fixed the vector ⑀ 1. 䢇
䢇
It is well known that Ž G, K . is a Gelfand pair. G is a connected compact Lie group which acts transitively on the unit sphere ⍀ of ⺖ q, so the homogeneous space GrK can be identified with ⍀. The surface element d on ⍀ of total mass 1 is, up to scalar, the unique G-invariant measure on ⍀. L p Ž GrK . is identified with L p Ž ⍀ . s L p Ž ⍀, d .. We now describe the decomposition of L2 Ž GrK . and the corresponding spherical functions. Case ⺖ s ⺢. We suppose that q G 3. For n integer, we say that a function ⌽ is a spherical harmonic of degree n if it is the restriction to ⍀ of a polynomial F Ž x . s F Ž x 1 , . . . , x q ., which is homogeneous of degree n in x and satisfies the Laplace equation ⌬ F s 0. Denote by Hn the class of all such functions. The restriction of the left regular representation to Hn is irreducible, and we have the decomposition L2 Ž ⍀ . s
⬁
[ ns0 Hn .
Ž 1.14.
COMPACT GELFAND PAIRS
239
The spaces Hn are analogous to the spaces Ha ; they are of finite dimension d n , where dn s
Ž 2 n q q y 2. Ž n q q y 3. ! . n! Ž q y 2 . !
Ž 1.15.
Here the set ⌳ is identified with ⺞. The spherical functions associated with this Gelfand pair are defined for each n g ⺞, by the unique zonal function n in Hn verifying nŽ ⑀ 1 . s 1, and are given, for ¨ g ⍀, by the formula
n Ž ¨ . s PnŽ qy2.r2 Ž ¨ , ⑀ 1 . ,
Ž 1.16.
where PnŽ qy2.r2 is the classical ultraspherical Žor Gegenbauer. polynomial of degree n satisfying PnŽ qy2.r2 Ž1. s 1. When q s 3, the spherical functions are given by Pn1r2 , the Legendre polynomials Žin fact this is the case elaborated in w3x.. Case ⺖ s ⺓. Here we take q G 2. For m, n integer, we say that a function ⌽ belongs to the class HarmŽ m, n., if it is the restriction to ⍀ of a polynomial F Ž z, z . s F Ž z1 , . . . , z q , z1 , . . . , z q . , which is homogeneous of degree m in z and homogeneous of degree n in z, such that
ž
⭸2 ⭸ z1 ⭸ z1
q ⭈⭈⭈ q
⭸2 ⭸ zq ⭸ zq
/
F s 0.
The restriction of the left regular representation to HarmŽ m, n. is irreducible, and we have the decomposition L2 Ž ⍀ . s
⬁
[ m , ns0 Harm Ž m, n . .
Ž 1.17.
The spaces HarmŽ m, n. are analogous to the spaces Ha ; they are of finite dimension dŽ m, n. , where dŽ m , n. s
Ž m q n q q y 1. Ž m q q y 2. ! Ž n q q y 2. ! . m!n! Ž q y 1 . ! Ž q y 2 . !
Ž 1.18.
Here the set ⌳ is identified with ⺞ = ⺞. Consider the map A from G into the closed unit disc D of ⺓ defined by AŽ g . s Ž g ⑀ 1 , ⑀ 1 .. Then f ¬ f ( A is an isometry between the space L p ŽUŽ q y 1. _ UŽ q .rUŽ q y 1.. and the space L p Ž D, dm., of measurable
MOKNI AND TRIMECHE `
240 functions on D such that
HD< f Ž z . <
p
dm Ž z . - ⬁,
with dm Ž x q iy . s
qy1 2
Ž1 y x 2 y y 2 .
qy 2
dx dy.
The spherical functions associated with this Gelfand pair are defined for each Ž m, n. g ⺞ = ⺞, by the unique zonal function ⌽ in HarmŽ m, n. verifying ⌽ Ž ⑀ 1 . s 1, given for re i g D by the formula qy2, < myn<. ⌽ Ž re i . s r < myn< e iŽ myn. R ŽminŽ Ž 2 r 2 y 1. , m , n.
Ž 1.19.
qy2, < myn<. qy2, < myn<. Ž . where R ŽminŽ is the Jacobi polynomial satisfying R ŽminŽ 1 s 1. m, n. m, n. qy 2 This function is denoted R m, n . For more details about the above examples of Gelfand pairs we can see w1, 6, 7x.
2. CONTINUOUS WAVELET ANALYSIS 2.1. Continuous Wa¨ elet Transform DEFINITION 2.1. A family ⌿ 4 . gx0, ⬁w, of functions in L2 Ž K _ GrK . is said to be a ‘‘wavelet’’ on GrK, if the following conditions are satisfied:
ˆ Ž a.< 2 d r s 1. Ži. For all a g ⌳, a / a0 , H0⬁ < ⌿ ˆ Ž a0 . s 0. Žii. For all gx0, ⬁w, ⌿ ˆ Ž a.< 2 d r - ⬁. Žiii. For all t gx0, ⬁w, Ý ag ⌳ , a/ a d a Ht⬁ < ⌿ 0 If we call the function ⌿ s ⌿1 the mother wavelet, then we can define the dilation operators D , gx0, ⬁w, by D⌿ s ⌿ . Let ⌿ , g be the function on G defined by ⌿ , g Ž x . s L g D⌿ Ž x . s ⌿ Ž gy1 x .. We consider L2 Žx0, ⬁w=GrK . the Hilbert space of square integrable complex functions on x0, ⬁w=G with respect to the measure d dx , which are K-right invariant relative to the second variable. We have the following lemma.
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COMPACT GELFAND PAIRS
LEMMA 2.2 ŽParseval identity.. relations
For all F, H in L2 Ž GrK . we ha¨ e the
Ž Ž F , ⌿ . . . . 2 , Ž H , ⌿ . . . . 2 . L Ž x0, ⬁w =GrK . s Ž F , H . 2 2
5 Ž F , ⌿ . . . . 2 5 L2 Žx0, ⬁w=G r K . s 5 F 5 2 . Proof. By the inversion formula Ž1.11. we have ⌿ Ž gy1 x . s
᭙ g , x g G,
ˆ Ž a. a Ž gy1 x . , d a⌿
Ý ag⌳
and by the addition formula Ž1.8. we get ⌿ Ž gy1 x . s
᭙ g , x g G,
da
ˆ Ž a. Ya, k Ž x . Ya, k Ž g . . Ý Ý⌿ ag⌳ ks1
Using the orthogonality of Ya, k 4 we obtain
Ž Ž F , ⌿ . . . . 2 ⭈ Ž H , ⌿ . . . . 2 . L Ž x0, ⬁w =GrK . 2
s s
⬁
HGH0
Ž F , ⌿ , g . 2 Ž H , ⌿ , g . 2
d dg
da
⬁
H0 Ý Ý F Ž F . Ž a, k . F Ž H . Ž a, k . < ⌿ˆ Ž a. <
ag⌳ ks1
2
d
da
s
Ý Ý F Ž F . Ž a, k . F Ž H . Ž a, k . ag⌳ ks1
s Ž F , H .2. DEFINITION 2.3. Let ⌿ 4 , gx0, ⬁w, be a wavelet on GrK. We define the wavelet transform Ž W . ⌿ from L2 Ž GrK . into L2 Žx0, ⬁w=GrK . by
Ž W . ⌿ Ž H . Ž , g . s Ž H , ⌿ , g . 2 for all H in L2 Ž GrK . and Ž , g . in x0, ⬁w=G. DEFINITION 2.4. Let K t 4 , t gx0, ⬁w, be a family of functions in L1 Ž K _ GrK . such that Kˆt Ž a0 . s 1 for all t gx0, ⬁w. We said that K t 4 , t gx0, ⬁w, is an approximate identity in L2 Ž GrK . if for all f g L2 Ž GrK ., lim 5 f y f ) K t 5 2 s 0.
tª0 q
MOKNI AND TRIMECHE `
242
PROPOSITION 2.5. Let K t 4 , t gx0, ⬁w, be a family of functions in L1 Ž K _ GrK ., such that Kˆt Ž a0 . s 1 for all t gx0, ⬁w. We suppose that there is a positi¨ e constant M such that ᭙ t ) 0,
< Kˆt Ž a . < F M ;
a g ⌳,
then K t 4 , t gx0, ⬁w, is an approximate identity if and only if lim Kˆt Ž a . s 1
for all a g ⌳ .
tª0 q
Proof. To prove that lim t ª 0q Kˆt Ž a. s 1 for all a g ⌳, we have just to remark that a ) K t s Kˆt Ž a. ⭈ a and take the limit in L2 Ž G .. Conversely, take a function f in L2 Ž GrK .; then for all Ž a, j . g N , F Ž f ) K t . Ž a, j . s
HGHG f Ž y . K Ž x . Y t
a, j
Ž yx . dy dx
using formula Ž1.6. for Ya, j and the fact that K t is K-left invariant, we get F Ž f ) K t . Ž a, j . sd a
HGHGHGHK f Ž y . K Ž x . Y t
a, j
Ž g . a Ž xy1 kyy1 g . dk dg dy dx.
The product formula Ž1.10. gives F Ž f ) K t . Ž a, j . sd a
HGHGHG f Ž y . K Ž x . Y t
a, j
Ž g . a Ž xy1 . a Ž yy1 g . dg dy dx.
Integrating on x and on g we obtain F Ž f ) K t . Ž a, j . s F Ž f . Ž a, j . Kˆt Ž a . . This relation gives the inequality < F Ž f ) K t . Ž a, j . < 2 F < F Ž f . Ž a, j . Kˆt Ž a . < 2 F M 2 < F Ž f . Ž a, j . < 2 , so the Fourier series of f ) K t tends to the Fourier series of f as t ª 0q, and this is the desired result. We now prove the reconstruction formula for the continuous wavelet transform Ž W . ⌿ . THEOREM 2.6 ŽReconstruction Formula.. Let ⌿ 4 , gx0, ⬁w, be a wa¨ elet on GrK. Then for all H g L2 Ž GrK . with ¨ anishing integral on G,
243
COMPACT GELFAND PAIRS
we ha¨ e the following identity which holds in L2 Ž GrK .: H Ž ⭈. s
⬁
HGH0
d dg
Ž W . ⌿ Ž H . Ž , g . ⌿ , g Ž ⭈ .
.
Proof. Take an arbitrary t ) 0. From Definition 2.3 and the Fubini theorem we obtain ⬁
HGHt
Ž W . ⌿ Ž H . Ž , g . ⌿ , g Ž x . ⬁
s
HGHt HGH Ž y . ⌿
s
HGH Ž y . Ht HG⌿ Ž g
ž
, g
⬁
ž
Ž y . ⌿ , g Ž x . dy y1
d dg
y . ⌿ Ž gy1 x .
/
d dg
d dg
/
dy.
Ž 2.1.
Using the inversion formula for ⌿ and the formula Ž1.7., we can show that the second member of Ž2.1. is equal to ⬁
HGH Ž y . Ht ž Ý d < ⌿ˆ Ž a. < Ž y 2
a
a
y1
x.
ag⌳
/
d dy
.
This can be written in the form H ) K t Ž x . with KtŽ x. s 1 q
Ý
da
ag⌳
⬁
žH t
ˆ Ž a. < 2 <⌿
d
/
a Ž x . .
Ž 2.2.
This series is absolutely and uniformly convergent because for all a g ⌳, 5 a 5 ⬁ F 1, and because of the property Žiii. of the definition of the wavelet ⌿ . By the Parseval formula Ž1.12. the function K t is in L2 Ž K _ GrK .. Moreover, Kˆt Ž a0 . s 1, and, for all a g ⌳, a / a0 , we have Kˆt Ž a . s
⬁
Ht < ⌿ˆ Ž a. <
2
d
,
so the property Ži. of Definition 2.1 gives lim t ª 0q Kˆt Ž a. s 1 for all a g ⌳ and < Kˆt Ž a.< F 1 for all a g ⌳ and t ) 0. So by Proposition 2.5, K t 4 , t gx0, ⬁w, is an approximate identity, and we have lim H ) K t s H.
tª0 q
The limit is in L2 Ž G ..
Ž 2.3.
MOKNI AND TRIMECHE `
244
DEFINITION 2.7. The associated scaling function ⌽t 4 , t gx0, ⬁w, to a given wavelet ⌿ 4 , gx0, ⬁w, on GrK, is defined by ᭙ g g G,
⌽t Ž g . s
Ý
ˆ t Ž a . a Ž g . , d a⌽
ag⌳
ˆ t Ž a0 . s 1 and ⌽ ˆ t Ž a. s Ž Ht⬁ < ⌿ ˆ Ž a.< 2 d .1r2 for all a g ⌳, a / a0 . where ⌽ Remark. We note that ⌽t belongs to L2 Ž K _ GrK ., because ⬁ d ˆ t Ž a. < 2 s 1 q ˆ Ž a. < 2 5 ⌽t 5 22 s Ý d a < ⌽ - ⬁. Ý da < ⌿ t ag⌳ ag⌳ , a/a 0
H
Moreover, using inversion formula Ž1.13. for ⌽t and formula Ž1.7., we obtain ᭙ g g G, ⌽t ) ⌽t Ž g . s K t Ž g . , Ž 2.4. where K t is the function given by the relation Ž2.2.. So ⌽t ) ⌽t 4 , t gx0, ⬁w, is an approximate identity. COROLLARY 2.8. Let ⌿ 4 , gx0, ⬁w, be a wa¨ elet on GrK, and let ⌽t 4 , t gx0, ⬁w, be the associated scaling function. Then for all H g L2 Ž GrK . with ¨ anishing integral on G, the equality holds in L2 Ž GrK ., H s limq tª0
HG Ž H , ⌽
t , g 2 ⌽t , g
.
dg ,
where ⌽t, g , g g G, is the function gi¨ en by ⌽t , g Ž x . s ⌽t Ž gy1 x . .
᭙ x g G,
Proof. Using the Fubini theorem and the property ⌽t Ž gy1 . s ⌽t Ž g . ,
᭙ g g G, we obtain
HG Ž H , ⌽
t , g 2 ⌽t , g
.
Ž x . dg y1
s
HGHGH Ž y . ⌽ Ž g
s
HGH Ž y . HG⌽ Ž y
s
HGH
s
HGH Ž y . ⌽ ) ⌽ Ž y
t
ž Ž . žH y
y . ⌽t Ž gy1 x . dy dg
y1
t
g . ⌽t Ž gy1 x . dg dy
/ /
⌽t Ž g . ⌽t Ž gy1 yy1 x . dg dy
G
t
t
y1
s H ) ⌽t ) ⌽t Ž x . . We deduce the result from relation Ž2.3..
x . dy
245
COMPACT GELFAND PAIRS
The scaling function ⌽t 4 , t gx0, ⬁w, associated with the wavelet ⌿ 4 , gx0, ⬁w, on GrK, is the kernel of an approximate identity Ži.e., ⌽t ) ⌽t is an approximate identity.. Conversely, the following proposition will show that to a given kernel verifying some suitable conditions, one can define an associated wavelet on GrK in the sense of Definition 2.1. PROPOSITION 2.9. Let ⌽t 4 , t gx0, ⬁w, be a kernel of an approximate ˆ t Ž a. is real, identity. Suppose that for all a g ⌳, a / a0 , the function t ¬ ⌽ continuously differentiable, and decreasing. Moreo¨ er, assume that ᭙ t ) 0,
ˆ t Ž a0 . s 1, ⌽
ˆ t Ž a. s 0, lim ⌽
and
for all a g ⌳ , a / a0 .
tª⬁
Then the family ⌿ 4 , gx0, ⬁w, defined by
ˆ Ž a. s y ⌿
᭙a g ⌳ ,
ž
d d
ž
ˆ Ž a. ⌽
2
/
1r2
/
,
Ž 2.5.
is a wa¨ elet on GrK with scaling function ⌽t 4 . Proof. From the relation Ž2.5. we deduce for all t gx0, ⬁w, and a g ⌳, a / a0 , ⬁
Ht < ⌿ˆ Ž a. <
2
d
2
ˆ t Ž a. . . s Ž⌽
So ⌽t 4 is associated with ⌿ 4 . Let us verify the properties Ži., Žii., and Žiii. of Definition 2.1. Ži. For all a g ⌳, a / a0 , we have ⬁
H0 < ⌿ˆ Ž a. <
2
d
2
ˆ t Ž a. . s 1, s limq Ž ⌽ tª0
because ⌽t 4 , t gx0, ⬁w, is the kernel of an approximate identity. Žii. For all gx0, ⬁w, we have
ˆ Ž a0 . s y ⌿
ž
d d
1 s 0.
/
Žiii. For all t gx0, ⬁w, we deduce from the Parseval formula for ⌽t
Ý ag⌳ , a/0
da
⬁
Ht < ⌿ˆ Ž a. <
2
d
s
2
Ý ag⌳ , a/0
ˆ t Ž a. . - ⬁. da Ž ⌽
MOKNI AND TRIMECHE `
246
Moreover, this inequality proves that for almost ) 0, ⌿ belongs to L2 Ž K _ GrK ., and this gives the desired result. 2.2. Multiresolution Analysis We now define the continuous multiresolution analysis of L2 Ž GrK .. Denote by pt , t gx0, ⬁w, the convolution operator of L2 Ž GrK . given by pt Ž H . s H ) ⌽t ) ⌽t ,
for H g L2 Ž GrK . .
Ž 2.6.
The corresponding ‘‘scale space’’ is defined by Vt s pt Ž L2 Ž GrK . . .
Ž 2.7.
LEMMA 2.10. We ha¨ e the following properties for pt and Vt : Ži. g N.
ˆ t Ž a.. 2 , for all t gx0, ⬁w, and Ž a, j . F Ž pt Ž H ..Ž a, j . s F Ž H .Ž a, j .Ž ⌽
Žii. Vt ; Vt ⬘ , for t G t⬘. Žiii. Vt , t gx0, ⬁w, is globally in¨ ariant under the action of the group G. Proof. Ži. We have ᭙ t g x 0, ⬁ w , F Ž pt Ž H . . Ž a, j . s
Ž a, j . g N , y1
HGHG⌽ ) ⌽ Ž g t
t
x . H Ž g . Ya, j Ž x . dg dx
Using the facts that ᭙g,
x g G,
⌽t ) ⌽t Ž gy1 x . s
d a⬘
Ý Ý
$
⌽t ) ⌽t Ž a⬘ . Ya⬘, j⬘ Ž x . Ya⬘, j⬘ Ž g .
a⬘g⌳ j⬘s1
and $
2
ˆ t Ž a⬘ . . , ⌽t ) ⌽t Ž a⬘ . s Ž ⌽
᭙a⬘ g ⌳ ,
and the orthogonality of the family Ya, j 4 we can easily check that ᭙ t g x 0, ⬁ w ,
ˆ t Ž a. . Ž a, j . g N , F Ž pt Ž H . . Ž a, j . s Ž ⌽
2
HGH Ž g . Y
a, j
Ž g . dg ,
and this gives the desired result. Žii. Take t, t⬘ gx0, ⬁w, such that t G t⬘, and F s pt Ž H . g Vt . From the inversion formula Ž1.13. for pt Ž H . we obtain da
Fs
Ý Ý F Ž H . Ž a, j . Ž ⌽ˆ t Ž a. . ag⌳ js1
2
Ya, j .
247
COMPACT GELFAND PAIRS
Thus ᭙ Ž a, j . g N ,
2
ˆ t Ž a. . . F Ž F . Ž a, j . s F Ž H . Ž a, j . Ž ⌽
Now define the sequence A a, j 4 , Ž a, j . g N , by A a, j s
ˆ t Ž a. . F Ž H . Ž a, j . Ž ⌽
Ž ⌽ˆ t ⬘ Ž a. .
2
2
.
Ž 2.8.
ˆ t Ž a. is decreasing, and F Ž H .Ž a, j .4 is in l 2 Ž N .; then The function t ¬ ⌽ the sequence A a, j 4 belongs to l 2 Ž N .. Thus from the Parseval theorem for F there exists a function f g L2 Ž GrK ., such that ᭙ Ž a, j . g N ,
F Ž f . Ž a, j . s A a, j .
Ž 2.9.
The relation da
Fs
Ý Ý A a, j Ž ⌽ˆ t ⬘ Ž a. .
2
Ya, j
Ž 2.10.
ag⌳ js1
implies that F s pt ⬘Ž f . belongs to Vt ⬘. Žiii. Let F s pt Ž H . for some H g L2 Ž GrK .. The invariance of the measure dg under the left translation of G gives L g Ž pt Ž H . . s pt Ž L g Ž H . . ,
for g g G.
Ž 2.11.
Hence we obtain the desired result. PROPOSITION 2.11. The scale spaces Vt , t gx0, ⬁w, satisfy the following properties: 䢇 䢇
Ha 0 ; Vt ; Vt ⬘ ; L2 Ž GrK . for 0 - t⬘ F t - ⬁. F Vt s Ha0 . t)0
䢇
D Vt
is dense in L2 Ž GrK ..
t)0
Proof. We deduce the results from Lemma 2.10 and the relation da
lim
Ý Ý F Ž H . Ž a, j . Ž ⌽ˆ t Ž a. .
tª⬁ ag⌳ js1
2
s
HG H Ž g . dg.
MOKNI AND TRIMECHE `
248
Remark. A family of scale spaces satisfying those properties is called a continuous multiresolution analysis of L2 Ž GrK .. 2.3. The Poisson Kernel and the Associated Wa¨ elet Suppose that G is a compact connected Lie group, and K is a closed subgroup. Let DŽ G . be the set of all left invariant differential operators on G, and let DK Ž G . be the subspace of those which are also K-right invariant. The spherical functions of the Gelfand pair Ž G, K . are eigenfunctions of any D in DK Ž G . Žsee w4x.. Take a Laplacian ⌬ on G in the sense of w8, Chap. Ix; then one has ⌬ a s a a .
᭙a g ⌳ ,
The eigenvalues a are nonnegative, and a 0 s 0. Let t gx0, ⬁w. The K-biinvariant Poisson kernel on G, denoted by Pt , is defined by the formula ᭙ g g G,
Pt Ž g . s
Ý
ey a t d a a Ž g . .
Ž 2.12.
ag⌳
According to w8x this series converges absolutely and uniformly for all g g G and t G b ) 0; so it defines a continuous function relative to the variables g and t. We have the following properties for the family Pt 4 :
HGP Ž g . dg s 1, for all t gx0, ⬁w.
䢇
Pˆt Ž a0 . s
䢇
Pt ) Pt ⬘ s Ptqt ⬘ , for all t, t⬘ gx0, ⬁w.
t
Moreover, the kernel Pt is a solution of the heat equation ⭸⭸ut s ⌬ u; more precisely, if f belongs to L1 Ž K _ GrK ., then uŽ g, t . s Pt ) f Ž g . is a solution verifying the boundary condition in L1 Ž G .: lim u Ž ⭈, t . s f .
tª0 q
So Pt 4 , t gx0, ⬁w, is an approximate identity. From the relation Ž2.12. we deduce that Pˆt Ž a . s ey a t ,
for all t g x 0, ⬁ w , a g ⌳ .
So, for all a g ⌳, a / a0 , the function t ¬ Pˆt Ž a. is real, differentiable, and decreasing, and lim Pˆt Ž a . s 1,
tª0 q
lim Pˆt Ž a . s 0.
tª⬁
249
COMPACT GELFAND PAIRS
By Proposition 2.9, the family Pt 4 , t gx0, ⬁w, is the scale function of a wavelet ⌿ 4 , gx0, ⬁w, on GrK, defined by ᭙a g ⌳ ,
ˆ Ž a. s y ⌿
ž
d d
ž
Pˆ Ž a .
2
/
1r2
/
s 2 a ey a , Ž 2.13.
'
and then ᭙ g g G,
⌿ Ž g . s
Ý
'2
a
d a ⑀y a a Ž g . .
Ž 2.14.
ag⌳
In the particular case of the group G s SO Ž q . the Poisson kernel Pt 4 , t gx0, ⬁w, is given on wy1, 1x by the formula ᭙ r g w y1, 1 x ,
Pt Ž r . s
eyn Ž nqqy2.t d n PnŽ qy2.r2 Ž r . . Ž 2.15.
Ý ng⺞
And in the case G s UŽ q . the Poisson kernel Pt 4 , t gx0, ⬁w, is given on the unit disc D by the formula Žsee w5x. ᭙ z g D,
Pt Ž z . s
qy 2 eyt Ž m k n. dŽ m , n. R m , n Ž z. ,
Ý
Ž 2.16.
m, ng⺞
where m k n s maxŽ m, n..
3. DISCRETIZATION BY WAVELET PACKETS We now define three types of wavelet packets on GrK: P-wavelet, M-wavelet, and S-wavelet. 3.1. The P-Wa¨ elet Packet on GrK DEFINITION 3.1. Let ⌿ 4 , gx0, ⬁w, be a wavelet on GrK. The P-wavelet packet on GrK, denoted ⌿lP 4 , l g ⺪, is defined by ᭙a g ⌳ ,
ˆ lP Ž a . s ⌿
žH
l
lq1
ˆ Ž a. < 2 <⌿
d
1r2
/
,
where l 4 , l g ⺪, is a strictly decreasing sequence such that lim l s 0 and
lª⬁
lim l s ⬁.
lªy⬁
The function ⌿lP is called the P-wa¨ elet packet member of step l.
MOKNI AND TRIMECHE `
250 Remarks.
Ži. Note that, for all a g ⌳, a / a0 , we have ⬁
⬁
ˆ lP Ž a . < 2 s H < ⌿ ˆ Ž a. < 2 Ý <⌿ 0
lsy⬁
d
s 1.
Žii. For all g g G, we define the function ⌿l,P g by ⌿lP, g Ž x . s ⌿lP Ž gy1 x . .
᭙ x g G,
Žiii. For all F, H in L2 Ž GrK . we obtain from Lemma 2.2 ⬁
Ý
H Ž F, ⌿ . Ž H, ⌿ . P l, g 2
lsy⬁ G
⬁
Ý lsy⬁
P l, g 2
HG Ž F , ⌿ .
P l, g 2
2
dg s Ž F , H . 2
Ž 3.1.
dg s 5 F 5 22 .
Ž 3.2.
Notation. We denote by L2 Ž⺪ = GrK . the space of measurable functions F on ⺪ = G, K-right invariant relative to the second variable, and satisfying ⬁
HG< F Ž l, g . <
Ý lsy⬁
2
dg - ⬁.
DEFINITION 3.2. Let ⌿lP 4 , l g ⺪, be a P-wavelet packet on GrK. We define the P-wavelet packet transform Ž W . P⌿ from L2 Ž GrK . into L2 Ž⺪ = GrK . by P
Ž W . ⌿ Ž H . Ž l, g . s Ž H , ⌿lP, g . 2 for all H in L2 Ž GrK . and Ž l, g . g ⺪ = G. THEOREM 3.3 ŽReconstruction Formula.. Let ⌿lP 4 , l g ⺪, be a P-wa¨ elet packet on GrK. Then for all H in L2 Ž GrK . with ¨ anishing integral on G, we ha¨ e the following identity which holds in L2 Ž GrK .: ⬁
H Ž ⭈. s
Ý lsy⬁
HGŽ W .
P ⌿
Ž H . Ž l, g . ⌿lP, g Ž ⭈ . dg.
Proof. By using Theorem 2.6, we have just to remark that H Ž x . s lim
⬁
HH Lª⬁ G
L
Ž W . ⌿ Ž H . Ž , g . ⌿ , g Ž x .
d dg
251
COMPACT GELFAND PAIRS
and that ⬁
HGH
Ž W . ⌿ Ž H . Ž , g . ⌿ , g Ž x .
L
Ly1
s
Ý lsy⬁
HGŽ W .
P ⌿
d dg
Ž H . Ž l, g . ⌿lP, g Ž x . dg.
We shall now define the scale discrete scaling function associated with a given P-wavelet packet on GrK. DEFINITION 3.4. The scale discrete scaling function ⌽LP 4 , L g ⺪, associated with a P-wavelet packet ⌿lP 4 , l g ⺪, on GrK, is defined by ᭙ g g G,
⌽LP Ž g . s
ˆ LP Ž a. a Ž g . , d a⌽
Ý ag⌳
ˆ LP Ž a0 . s 1 and where ⌽ Ly1
᭙a g ⌳ ,
ˆ LP Ž a. s ⌽
a / a0 ,
žÝ
1r2
ˆ lP Ž a. < 2 <⌿
lsy⬁
/
ˆ LŽ a. . s⌽
THEOREM 3.5. Let ⌽LP 4 , L g ⺪, be a scale discrete scaling function. Then for all H g L2 Ž GrK . with ¨ anishing integral on G, we ha¨ e the following equality which holds in L2 Ž GrK .: H Ž ⭈ . s lim
H Ž H, ⌽ .
P P L , g 2 ⌽L , g
Lª⬁ ⍀
Ž ⭈ . dg ,
Ž 3.3.
P where ⌽L, g , g g G, is the function gi¨ en by
⌽LP , g Ž x . s ⌽LP Ž gy1 x . .
᭙ x g G,
Proof. We deduce the result from the fact that ᭙a g ⌳ ,
ˆ LP Ž a. s ⌽ ˆ LŽ a. ⌽
and Corollary 2.8. PROPOSITION 3.6. Ži. We ha¨ e ᭙a g ⌳ ,
Ž ⌽ˆ LP Ž a. .
2
q
⬁
ˆ lP Ž a. < 2 s 1. Ý <⌿ lsL
Ž 3.4.
MOKNI AND TRIMECHE `
252 Žii. We ha¨ e ᭙a g ⌳ ,
⬁
P Ž a. . Ý ž Ž ⌽ˆ lq1
a / a0 ,
2
ˆ lP Ž a. . y Ž⌽
lsy⬁
2
/ s 1.
Ž 3.5.
Proof. Ži. For all a g ⌳, a / a0 , we have ⬁
Ly1
⬁
lsy⬁
lsy⬁
lsL
ˆ lP Ž a. < 2 s Ý < ⌿ ˆ lP Ž a. < 2 q Ý < ⌿ ˆ lP Ž a. < 2 ; Ý <⌿ thus
Ž ⌽ˆ LP Ž a. .
2
⬁
ˆ lP Ž a. < 2 s 1. Ý <⌿
q
lsL
This equality holds also for a0 . Žii. For all a g ⌳, a / a0 , and all l g ⺪, we have 2
2
P ˆ lP Ž a. < 2 s Ž ⌽ ˆ lq1 ˆ lP Ž a. . . <⌿ Ž a. . y Ž ⌽
We deduce the result from this relation. PROPOSITION 3.7. H Ž ⭈. s
For all H g L2 Ž GrK . we ha¨ e
HG Ž H , ⌽ .
P P L , g 2 ⌽L , g
Ž ⭈ . dg q
⬁
Ý H Ž W . P Ž H . Ž l, g . ⌿lP, g Ž ⭈. dg.
lsL G
Ž 3.6. Proof. The relation Ž3.4. gives the result. The relation Ž3.5. leads us to an other type of wavelet packet on GrK: the modified wavelet packet, also called the M-wavelet packet on GrK. 3.2. The M-Wa¨ elet Packet on GrK DEFINITION 3.8. Let ⌽LP 4 , L g ⺪, be a scale-discrete scaling function, associated with a P-wavelet packet ⌿lP 4 , l g ⺪, on GrK. Ži. The corresponding M-wavelet packet ⌿lM 4 , l g ⺪, on GrK, and ˜ lM 4 are defined by its dual ⌿ ᭙a g ⌳ ,
P ˆ lM Ž a. s ⌽ ˆ lq1 ˆ lP Ž a. ⌿ Ž a. y ⌽
᭙a g ⌳ ,
P ˆ ˜ lM Ž a. s ⌽ ˆ lq1 ˆ lP Ž a. . ⌿ Ž a. q ⌽
253
COMPACT GELFAND PAIRS
2Ž Žii. The M-wavelet packet transform Ž W . M . ⌿ is defined from L GrK 2Ž into L ⺪ = GrK . by M
Ž W . ⌿ Ž H . Ž l, g . s Ž H , ⌿lM, g . 2 for all H g L2 Ž GrK . and Ž l, g . g ⺪ = G. Remark. By construction of the M-wavelet packet on GrK, we have the relation 1r2
Ly1
ˆ LP Ž a. s ⌽
ž
ˆ˜ M Ž a. ˆ lM Ž a. ⌿ Ý ⌿ l
lsy⬁
/
,
Ž 3.7.
which implies that Ly1
⌽LP ) ⌽LP s
˜ lM ⌿lM ) ⌿
Ý
Ž 3.8.
lsy⬁
the relation Ž3.8. gives immediately the following
˜ lM 4 be, respecti¨ ely, an M-wa¨ elet packet THEOREM 3.9. Let ⌿lM 4 and ⌿ on GrK and its dual. Then for all H in L2 Ž GrK . with ¨ anishing integral on G, we ha¨ e the following identity which holds in L2 Ž GrK .: H Ž ⭈. s
⬁
Ý lsy⬁
HGŽ W .
M ⌿
˜ lM, g Ž ⭈ . dg. Ž H . Ž l, g . ⌿
Remark. In the same way as for the P-wavelet packet on GrK, we get, for all H g L2 Ž GrK ., H Ž ⭈. s
HG Ž H , ⌽ .
P P L , g 2 ⌽L , g
Ž ⭈ . dg q
⬁
˜ lM, g Ž ⭈ . dg. Ý H Ž W . M Ž H . Ž l, g . ⌿
lsL G
3.3. The S-Wa¨ elet Packet on GrK Remark that for all a g ⌳, a / a 0 , we have ⬁
ˆ ˆ lM Ž a. ⌿ ˜ lM Ž a. s 1. Ý ⌿
Ž 3.9.
lsy⬁
Note that this equality is the key to the reconstruction formula for the M-wavelet transform. Suppose, now, that for a given sequence ⌿lS 4 , l g ⺪, of functions ˜ lS 4, l g ⺪, in in L2 Ž K _ GrK ., one can define another sequence ⌿ 2Ž L K _ GrK . such that for all a g H, a / a0 : ⬁
ˆ ˆ lS Ž a. ⌿ ˜ lS Ž a. s 1. Ý ⌿ lsy⬁
MOKNI AND TRIMECHE `
254
Then we can obtain the same results as above. In fact this is possible under some conditions verified by ⌿lS 4 . DEFINITION 3.10. A family ⌿lS 4 , l g ⺪, in L2 Ž K _ GrK . is said to be ˆ lS is real, ⌿ ˆ lS Ž a0 . s 0, and satisfies the an S-wa¨ elet packet on GrK if ⌿ following conditions: Ži. There are positive real constants, 0 - A F B - ⬁, such that ᭙a g ⌳ ,
AF
⬁
2 ˆ lS Ž a. . F B. Ý Ž⌿
lsy⬁
These inequalities are called ‘‘stability conditions.’’ Žii. For all L g ⺪, da
Ý ag⌳ , a/a 0
Ly1 Ý lsy⬁ Ž ⌿ˆ lS Ž a. .
2
ˆ lS Ž a. . Ý⬁lsy⬁Ž ⌿
2
-⬁
Žiii. There exists M ) 0 such that for all L g ⺪ and g g G,
HG Ý d ag⌳
a
Ly1 Ý lsy⬁ Ž ⌿ˆ lS Ž a. .
2
ˆ lS Ž a. . Ý⬁lsy⬁Ž ⌿
2
a Ž g . dg F M.
The S-wavelet transform Ž W . S⌿ , from L2 Ž GrK . into L2 Ž⺪ = GrK ., is defined by S
Ž W . ⌿ Ž H . Ž l, g . s Ž H , ⌿lS, g . 2 for all H g L2 Ž GrK . and Ž l, g . g ⺪ = G. S PROPOSITION 3.11. The linear operator Ž W . ⌿ from L2 Ž GrK . into L ⺪ = GrK . ¨ erifies 2Ž
A 5 H 5 22 F 5 Ž W . ⌿ Ž H . 5 2L2 Ž⺪=G r K . F B 5 H 5 22 S
for all H g L2 Ž GrK . with ¨ anishing integral on G. In particular, it is a continuous operator. Proof. By applying the inversion formula Ž1.11. to ⌿lS and the Parseval identity relative to the system Ya, j 4 , Ž a, j . g N , to H, we show that the property A 5 H 5 22 F 5 Ž W . ⌿ Ž H . 5 2L2 Ž⺪=G r K . F B 5 H 5 22 S
255
COMPACT GELFAND PAIRS
is equivalent to da
A
Ý Ý < F Ž H . Ž a, j . < 2 ag⌳ js1 ⬁
da
ˆ lS Ž a. . Ý Ý < F Ž H . Ž a, j . < 2 Ý Ž ⌿
F
ž
ag⌳ js1
2
lsy⬁
/
da
FB
Ý Ý < F Ž H . Ž a, j . < 2 . ag⌳ js1
Then the stability conditions give the result. The constant A is positive, so the operator Ž W . S⌿ is injective and then possesses a continuous linear inverse operator, which we shall describe now.
˜ lS 4, l g ⺪, on GrK, DEFINITION 3.12. Ži. The dual S-wavelet packet ⌿ S4 corresponding to a given S-wavelet packet ⌿l , l g ⺪, on GrK, is defined ˆ ˜ lS Ž a0 . s 0 and by ⌿ ᭙a g ⌳ ,
ˆ ˜ lS Ž a. s ⌿
a / a0 ,
ˆ lS Ž a. ⌿ ˆ lS Ž a. . Ý⬁lsy⬁Ž ⌿
.
2
Žii. The S-scaling function ⌽LS 4 , L g ⺪, associated with ⌿lS 4 , l g ⺪, is defined by
ˆ
⌽LS
¡ ¢1
Ý Ž a. s~ ž
ˆ ˆ Ž a. ⌿ ˜ lS Ž a. /
Ly 1 S lsy⬁⌿l
1r2
for all a g ⌳ , a / a0 for a s a0 .
˜ lS 4, l g ⺪, Remark. From the definition of the dual S-wavelet packet ⌿ on GrK, Ži. We have ⬁
2 ˆ S Ž a. s ˜ Ý ž⌿ / l
᭙a g ⌳ ,
lsy⬁
1
ˆ lS Ž a. . Ý⬁lsy⬁Ž ⌿
2
.
Ž 3.10.
Then 1 B
F
⬁
1
lsy⬁
A
2 ˆ ˜ lS Ž a. / F Ý ž⌿
,
where A, B are the constants given in Definition 3.10.
Ž 3.11.
MOKNI AND TRIMECHE `
256 Žii. We have
Ly1 ˆ ˜ lS Ž a. Ý lsy⬁ ⌿
᭙a g ⌳ ,
Ý⬁lsy⬁
ž / ž ⌿˜ˆ Ž a. / S l
2
2
s
Ly 1 Ý lsy⬁ Ž ⌿ˆ lS Ž a. .
2
ˆ lS Ž a. . Ý⬁lsy⬁Ž ⌿
2
.
Ž 3.12.
LEMMA 3.13. The S-scaling function ⌽LS 4 , L g ⺪, associated with an S-wa¨ elet packet ⌿lS 4 , l g ⺪, on GrK, is the kernel of an approximate identity in the sense that, for all H g L2 Ž GrK ., lim H ) ⌽LS ) ⌽LS s H.
Lª⬁
The limit is in L2 Ž GrK .. Proof. From Definition 3.12Žii. we have Ly1
2
Ž ⌽ˆ LS Ž a. . s
᭙a g ⌳ ,
ˆ˜ S Ž a. . ˆ lS Ž a. ⌿ Ý ⌿ l
lsy⬁
Then ᭙a g ⌳ ,
2
ˆ LS Ž a. . s 1. lim Ž ⌽
Lª⬁
So, to prove the lemma, it suffices to show that for all L g ⺪, the function ⌽LS is uniformly bounded in L1 Ž G . and continuous. We deduce these properties by applying the inversion formula Ž1.11. to ⌽LS , the stability conditions Ži. and Žiii. of Definition 3.10. The reconstruction formula for the S-wavelet transform arises immediately from the above, exactly as for the M-wavelet transform, and we obtain the following:
˜ lS 4, l g ⺪, be, respecti¨ ely, an S-wa¨ elet THEOREM 3.14. Let ⌿lS 4 and ⌿ packet on GrK and its dual. Then for all H in L2 Ž GrK . with ¨ anishing integral o¨ er G, we ha¨ e the following identity, which holds in L2 Ž GrK .: H Ž ⭈. s
⬁
Ý lsy⬁
HGŽ W .
S ⌿
˜ lS, g Ž ⭈ . dg. Ž H . Ž l, g . ⌿
4. MULTIRESOLUTION ANALYSIS BY WAVELET PACKETS The discrete multiresolution analysis will be defined by means of convolution operators, as for the continuous multiresolution analysis.
COMPACT GELFAND PAIRS
257
Let ⌿l 4 , l g ⺪, be a P-, M-, or S-wavelet packet on GrK, and let ⌽L 4 , L g ⺪, be the corresponding scaling function. For all of those wavelet packets on GrK we have, according to the relation Ž3.8., the equality Ly1
⌽L ) ⌽L s
Ý
˜L , ⌿L ) ⌿
Ž 4.1.
lsy⬁
˜ l 4 is the dual wavelet packet in case M and S, and the P-wavelet where ⌿ packet in case P. From this relation we deduce ˜L . ⌽Lq 1 ) ⌽Lq1 s ⌽L ) ⌽L q ⌿L ) ⌿
Ž 4.2.
Now define the convolution operators pL and q L , L g ⺪, by pLŽ H . s ⌽L ) ⌽L ) H
˜L ) H , q LŽ H . s ⌿L ) ⌿ and the spaces VL s pL Ž L2 Ž GrK . . WL s q L Ž L2 Ž GrK . . . The discrete family of scale spaces VL 4 , L g ⺪, verifies, as in the continuous case, the following properties: 䢇 䢇
Ha 0 ; VL ; VL⬘ ; L2 Ž GrK . for y⬁ - L F L⬘ - ⬁. F VL s Ha0 . Lg⺪
䢇
D VL
is dense in L2 Ž GrK ..
Lg⺪
Moreover, by the relation Ž2.12., every VL is globally invariant under the left translation of the group G. A function H in L2 Ž GrK . with vanishing integral over G can be approximated by its ‘‘projections’’ on the scale spaces VL ; in fact, lim pLŽ H . s H.
Lª⬁
We say that pLŽ H . is the approximation of H at the step L. Note that, by the relation Ž3.14., we have VLq 1 s VL q WL . So to go from an approximation at step L to an approximation at step L q 1, one needs the space WL .
258
MOKNI AND TRIMECHE `
Suppose now that the scale function ⌽L 4 , L g ⺪, verifies the relation ⌽L ) ⌽L s ⌽Ly1 . So pL ( pL s pLy1 and pLŽ VL . s VLy1 , and then one can pass directly from a resolution at step L to a resolution at step L y 1. EXAMPLE. We know that the Poisson kernel Pt 4 , t gx0, ⬁w, verifies the relation Pt ) Pt ⬘ s Ptqt ⬘ ,
for all t , t⬘ g x 0, ⬁ w .
Take the dyadic sequence L 4 defined by L s 2yL for L g ⺪, and let ⌽L s P L , the associated scale-discrete scaling function. Then we have ⌽L ) ⌽L s P2yL ) P2yL s P2yL q2y L s P2yŽ Ly 1. s ⌽Ly1 , where ⌽L 4 , L g ⺪, is said to be a reproducing scaling function. REFERENCES 1. H. Annabi and K. Trimeche, Convolution generalisee ` ´ ´ ´ sur le disque unite, ´ C. R. Acad. Sci. Paris 278 Ž1974., 21᎐24. 2. J. Faraut, ‘‘Analyse harmonique sur les paires de Guelfand et les espaces hyperboliques,’’ Les cours du C.I.M.P.A., Nancy, 1980. 3. W. Freeden and U. Windheuser, Spherical wavelet transform and its discretization, Ad¨ . Comput Math. 5 Ž1996., 51᎐94. 4. S. Helgason, ‘‘Groups and Geometric Analysis, Integral Geometry, Invariant Differential Operators, and Spherical Functions,’’ Academic Press, New York, 1984. 5. Y. Kanjin, A convolution measure algebra on the unit disc, Tohoku Math. J. 28 Ž1976., ˆ 105᎐115. 6. T. H. Koornwinder, ‘‘The Addition Formula for Jacobi Polynomials. II. The Laplace Type Integral Representation and the Product Formula,’’ Mathematisch Centrum, Amsterdam, TW 133r72, April 1972. 7. T. H. Koornwinder, The addition formula for Jacobi polynomials. I. Summary of results, Indag. Math. 34 Ž1972., 188᎐191. 8. E. L. Stein, Topics in harmonic analysis related to the Littelwood᎐Paley theory, in ‘‘Annals of Mathematical Studies,’’ Vol. 63, Princeton Univ. Press, Princeton, NJrUniv. of Tokyo Press, Tokyo, 1970.