Continuous wavelet transform on a special

JOURNAL OF MATHEMATICAL PHYSICS

VOLUME 44, NUMBER 9

SEPTEMBER 2003

Continuous wavelet transform on a special homogeneous space M. Fashandi,a) R. A. Kamyabi Gol, A. Niknam, and M. A. Pourabdollah School of Mathematical Sciences, Ferdowsi University, Mashad, 91775, Iran

共Received 22 July 2002; accepted 23 April 2003兲 We consider a semidirect product of two locally compact groups S and T, with S Abelian, denoted by S ␴ T. An action of S ␴ T on S is introduced to make S a homogeneous space of S ␴ T. Then we define a unitary representation from S ␴ T into the unitary group of L 2 (S) which is our main tool for defining the continuous wavelet transform on L 2 (S). Also the main properties of the transform are discussed. We prove the Plancherel and inversion formulas and reproducing kernel’s formula for this transform. This is finally specialized to the case of the continuous wavelet transform on L 2 (R d ). © 2003 American Institute of Physics. 关DOI: 10.1063/1.1591055兴

I. INTRODUCTION AND NOTATIONS

Let G be a locally compact topological group. We denote by d ␮ G a fixed left Haar measure on G and by ⌬ G the modular function. By C c (G) we mean the space of continuous functions of compact support on G. An action of a locally compact group G on a locally compact Hausdorff space S is a continuous map (x,s)哫xs from G⫻S to S such that s哫xs is a homeomorphism of S for each x苸G, and x(ys)⫽(xy)s, for all x,y苸G and s苸S. S is called a transitive G-space if for every s,t苸S there exists x苸G such that xs⫽t. A homogeneous space is a transitive G-space that is isomorphic to a quotient space G/H for some closed subgroup H of G 关for more information see Folland 共1995兲, Chap. 2兴. Let G/H be a homogeneous space. A rho-function for the pair (G,H) is a continuous function ␳ :G哫(0,⬁) such that

␳共 x␰ 兲⫽

⌬ H共 ␰ 兲 ␳共 x 兲, ⌬ G共 ␰ 兲

x苸G, ␰ 苸H.

For any locally compact group G and any closed subgroup H of G, (G,H) admits a rho-function 关see Folland 共1995兲, Proposition 2.54兴. Any homogeneous space G/H has a strongly quasi-invariant measure which arises from a rho-function and for any rho-function for the pair (G,H) there is a strongly quasi-invariant measure 关see Folland 共1995兲, Chap. 2, or Reiter 共1968兲, Chap. 8兴. But here we meet a special case of a homogeneous space which has more familiar measure than strongly quasi-invariant measures. Let G be of the form G⫽G 1 H where G 1 and H are closed subgroups and G 1 艚H⫽ 兵 e 其 , so every x苸G could be written uniquely in the form x⫽gh with g苸G 1 ,h苸H. Suppose further that the map x哫(g,h) of G onto the product G 1 ⫻H is continuous and hence a homeomorphism. Now, the function

␳共 x 兲⫽

⌬ H共 h 兲 , ⌬ G共 h 兲

x⫽gh,g苸G 1 ,h苸H

共1兲

a兲

Electronic mail: [email protected]

0022-2488/2003/44(9)/4260/7/$20.00

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is a rho-function for the pair (G,H). By Reiter 共1968兲, Chap. 8, Sec. 1.4, there is a relatively invariant positive measure d ␳ x˙ on G/H such that for all f 苸C c (G/H):

冕

G/H

f 共 y ⫺1 x˙ 兲 d␳ 共 x˙ 兲 ⫽ ␳ 共 y 兲

冕

G/H

f 共 x˙ 兲 d␳ 共 x˙ 兲 ,

y苸G,

共2兲

But G/H is isomorphic to G 1 as a topological space, so by Reiter 共1968兲, Chap. 8, Sec. 1.5, d␳ x˙ is a left Haar measure on G 1 , which will be shown by d␮ G 1 , and for all f 苸C c (G) we have

冕冕 G1

H

f 共 gh 兲 d␮ H 共 h 兲 d␮ G 1 共 g 兲 ⫽

冕

G

f 共 x 兲 ␳ 共 x 兲 d␮ G 共 x 兲 ,

共3兲

or if we replace f by f / ␳ we obtain

冕冕 G1

H

f 共 gh 兲

⌬ G共 h 兲 d␮ 共 h 兲 d␮ G 1 共 g 兲 ⫽ ⌬ H共 h 兲 H

冕

G

f 共 x 兲 d␮ G 共 x 兲 .

共4兲

Also we can rewrite 共2兲 as follows, which is valid for all f 苸L 1 (G 1 ),

冕

G1

f 共 y ⫺1 x 兲 d␮ G 1 共 x 兲 ⫽ ␳ 共 y 兲

冕

G1

f 共 x 兲 d␮ G 1 共 x 兲 ,

y苸G.

共5兲

By Reiter 共1968兲, Chap. 8, Sec. 2.3, formula 共3兲 holds for all f 苸L 1 (G). Later we approach a homogeneous space which has the above properties. Let S and T be groups and suppose that there is a homomorphism t哫 ␴ t from T into the group of automorphisms of S. For (s,t), (s´ , ´t )苸S⫻T, define 共 s,t 兲共 s´ , ´t 兲 ⫽ 共 s ␴ t 共 s´ 兲 ,t ´t 兲 .

Then S⫻T is a group, it is called a semidirect product of S and T and is denoted by S ␴ T. Its identity is (e 1 ,e 2 ), where e 1 and e 2 are the identities of S and T, respectively. The inverse of (s,t) is ( ␴ t ⫺1 (s ⫺1 ),t ⫺1 ). ˜ ⫽ 兵 (e 1 ,e 2 ) 其 , and 䉰 and ˜T ⭐G, ˜S 艚T Let ˜S ⫽ 兵 (s,e 2 );s苸S 其 and ˜T ⫽ 兵 (e 1 ,t);t苸T 其 . Then ˜S គG ˜ ˜T . If S and T are locally compact groups and (s,t)哫 ␴ t (s) is continuous then G is a locally G⫽S compact group and ˜S and ˜T are closed in G. ˆ the dual group of a locally compact Abelian group G and by ˆf Fourier We denote by G ˆ , the definition of transform of a function f . For more information about the properties of G Fourier transform and related theorems, we refer the reader to Rudin 共1960兲, Chap. 1 or Folland 共1995兲, Chap. 4. By a 共unitary兲 representation we mean a homomorphism ␲ from a locally compact group G into the group U共H兲 of unitary operators on some nonzero Hilbert space H that is continuous with respect to the strong operator topology. A vector ␾ 苸H is said to be admissible if

冕

G

兩 具 ␾ , ␲ 共 x 兲 ␾ 典 兩 2 d␮ G 共 x 兲 ⬍⫹⬁.

A representation ␲ is called irreducible if 兵0其 and H are the only closed linear subspaces that are invariant under the unitary operators ␲ (x);x苸G. An irreducible representation which has at least one admissible vector is called a square integrable representation. If there exists a vector ␾ 苸H such that the closed linear span of 兵 ␲ (x) ␾ ,x苸G 其 is equal to H, then ␾ is called a cyclic vector and ␲ is called a cyclic representation. In this paper we study the continuous wavelet transform on L 2 (S) using a unitary representation from S ␴ T into the unitary group of L 2 (S). In particular we extend the results of Koorn-

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winder 共1993兲. The idea of working on semidirect product of groups in the framework of wavelet analysis is not new. In Grossman et al. 共1985兲, which is one of the first papers of wavelet analysis, the continuous wavelet transform on L 2 (R) was defined by means of a certain square integrable representation of the semidirect product R ␴ R ⫹ on the unitary group of L 2 (R). The square integrability of this representation guarantees the existence of an inverse wavelet transform and admissibility condition. However, we prove them without square integrability’s assumption, instead we use the fact that S is a homogeneous space of S ␴ T. In recent years, this kind of result to other groups has been extended, especially the construction of wavelet transform and discrete frames from semidirect product of R k and a closed matrix group has been considered 关see, e.g., Aniello et al. 共2001兲, Aniello et al. 共1998兲, Fuehr and Meyer 共2002兲兴. Also some authors have concerned the wavelet transform in the context of square integrable representations of locally compact groups on infinite dimensional separable Hilbert spaces 关see Wong 共2002兲兴. These wavelet transforms are based on coherent states parametrized by elements in the group G. The book by Ali et al. 共2000兲 presents the more general theory of coherent states associated with homogeneous space and contains an extensive list of references on coherent states parametrized by points in a homogeneous space. Torresani 共1995兲 and Antoine and Vandergheynst 共1999兲 have presented some applications of coherent states on manifolds. II. MAIN RESULTS

Let G be semidirect product of two locally compact groups S and T, that S is Abelian, and let •:G⫻S哫S be as follows: 共 a,b 兲 .s⫽a ␴ b 共 s 兲 .

It is easy to check that ‘‘.’’ is an action of G on S. Since for each s 1 , s 2 苸S, (s 2 s ⫺1 1 ,e 2 ).s 1 ˜ 哫S defined by ⫽s 2 , S is a transitive G-space. Also S is a homogeneous space of G, since ⌽:G/T ˜ )⫽s is a homeomorphism. Let ⌽((s,t)T

␳ 共 s,t 兲 ⫽ ␳ 共 e 1 ,t 兲 ⫽

⌬ ˜T 共 e 1 ,t 兲 . ⌬ G 共 e 1 ,t 兲

˜ ). From now on, we use ␳ (t) instead of ␳ (e 1 ,t). ␳ Clearly, ␳ is a rho-function for the pair (G,T satisfies Eq. 共1兲, so there is a left Haar measure d ␮ ˜S (s,e 2 ) on ˜S such that 共4兲 and 共5兲 hold, hence there is a left Haar measure d ␮ S on S, that 共4兲 and 共5兲 could be rewritten as follows:

冕冕 S

冕

S

T

f 共 s,t 兲 d␮ T 共 t 兲 d␮ S 共 s 兲 ⫽ ␳共 t 兲

冕

G

f 共共 a,b 兲 ⫺1 .s 兲 d␮ S 共 s 兲 ⫽ ␳ 共 b 兲

f 共 g 兲 d␮ G 共 g 兲 ,

冕

S

f 苸L 1 共 G 兲 ,

共6兲

f 苸L 1 共 S 兲 .

共7兲

f 共 s 兲 d␮ S 共 s 兲 ,

Now we define ␲ from G into U(L 2 (S)), the unitary group of L 2 (S) such that 关 ␲ 共 a,b 兲 f 兴共 s 兲 ⫽ ␳ 共 b 兲 ⫺1/2 f 共共 a,b 兲 ⫺1 .s 兲 .

共8兲

It can be easily checked that ␲ is a unitary representation. Definition 1: We say ⌿苸L 1 艚L 2 (S) is a wavelet if 储 ⌿ 储 L 2 (S) ⫽1 and there is a constant C ⌿ such that 0⬍C ⌿ ⬍⫹⬁ and for any ␥ 苸Sˆ , C⌿ ª For a wavelet ⌿ we put

冕

T

ˆ 共 ␥ ⴰ ␴ t 兲 兩 2 d␮ T 共 t 兲 . 兩⌿

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⌿ s,t ª ␲ 共 s,t 兲 ⌿,

4263

共 s,t 兲 苸G.

Definition 2: For any f 苸L 2 (S), the continuous wavelet transform of f is defined as follows: W f 共 s,t 兲 ⫽ 具 f ,⌿ s,t 典 L 2 (S) . By Parseval’s formula 关see Rudin 共1960兲, Sec. 1.6.2兴 we also have ˆ 典 2ˆ , W f 共 s,t 兲 ⫽ 具 ˆf ,⌿ s,t L (S )

f 苸L 2 共 S 兲 .

Using the Schwarz inequality it is easy to see that W f is a bounded and continuous function on G. Lemma 3: Let ⌿苸L 2 (S), then ˆ 共 ␥ 兲 ⫽ ␳ 共 t 兲 1/2␥ 共 s 兲 ⌿ ˆ 共 ␥ⴰ␴t兲, ⌿ s,t

␥ 苸Sˆ .

共9兲

Proof: First we assume ⌿苸L 1 艚L 2 (S). Since ⌿ s,t 苸L 1 艚L 2 (S) by definition of Fourier transform 关Rudin 共1960兲, Sec. 1.2.3兴 and using 共7兲 and 共8兲 we have ˆ 共 ␥ 兲⫽ ⌿ s,t

冕

S

⌿ s,t 共 x 兲 ␥ 共 x 兲 d␮ S 共 x 兲 ⫽ ␳ 共 t 兲 1/2

冕

S

ˆ 共 ␥ⴰ␴t兲. ⌿ 共 x 兲 ␥ 共共 s,t 兲 .x 兲 d␮ S 共 x 兲 ⫽ ␳ 共 t 兲 1/2␥ 共 s 兲 ⌿

Now, if ⌿苸L 2 (S), by definition of Fourier transform of functions in L 2 (S) 关see the proof of Theorem 1.6.1 in Rudin 共1960兲兴 formula 共9兲 obtains. 䊐 Theorem 4: Let ⌿ be a wavelet and f ,g苸L 2 (S). Then we have 2 2 共i兲 (Plancherel formula) 储 W f 储 L 2 (G) ⫽C ⌿ 储 f 储 L 2 (S) , 共ii兲 (Parseval formula) 具 W f ,Wg 典 L 2 (G) ⫽C ⌿ 具 f ,g 典 L 2 (S) . Proof: Put ¯ˆ 共 ␥ ⴰ ␴ 兲 , F t 共 ␥ 兲 ⫽ ˆf 共 ␥ 兲 ⌿ t

t苸T, ␥ 苸Sˆ .

By Holder’s inequality F t 苸L 1 (Sˆ ). Also we can easily check that F t 共 ␥ 兲 ⫽ ␳ 共 t ⫺1 兲 关 f 쐓 共 ⌿ⴰ ␴ t ⫺1 兲 쐓 兴 ˆ 共 ␥ 兲 , where ⌿ 쐓 (s)⫽⌿(s ⫺1 ). Since f 苸L 2 (S) and (⌿ⴰ ␴ t ⫺1 ) 쐓 苸L 1 艚L 2 (S) by Folland 共1995兲, Proposition 2.39, f 쐓(⌿ⴰ ␴ t ⫺1 ) 쐓 苸L 2 (S), thus its Fourier transform is defined, so F t 苸L 2 (Sˆ ) and by Pontriagin Duality Theorem 关Rudin 共1960兲, Theorem 1.7.2兴, Fˆ t 苸L 2 (S). Also we put G t ( ␥ ) ¯ˆ ( ␥ ⴰ ␴ ), so G satisfies the above results about F . Therefore by 共9兲 ⫽gˆ ( ␥ ) ⌿ t t t ˆ 典 2 ˆ ⫽ ␳ 共 t 兲 1/2 W f 共 s,t 兲 ⫽ 具 ˆf ,⌿ s,t L (S )

冕

Sˆ

ˆ t 共 s ⫺1 兲 . F t 共 ␥ 兲 ␥ 共 s 兲 d␮ Sˆ 共 ␥ 兲 ⫽ ␳ 共 t 兲 1/2 F

ˆ (s ⫺1 ). So we have Similarly, Wg(s,t)⫽ ␳ (t) 1/2 G t

冕冕 T

S

W f 共 s,t 兲 Wg 共 s,t 兲 ␳ 共 t 兲 ⫺1 d␮ S 共 s 兲 d␮ T 共 t 兲 ⫽ ⫽

冕具 冕冕 T

T

ˆ t 典 L 2 (S) d␮ T 共 t 兲 Fˆ t ,G

Sˆ

F t 共 ␥ 兲 G t 共 ␥ 兲 d␮ Sˆ 共 ␥ 兲 d ␮ T 共 t 兲

⫽C ⌿ 具 ˆf ,gˆ 典 L 2 (Sˆ ) ⫽C ⌿ 具 f ,g 典 L 2 (S) . So by 共6兲 we obtain

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冕

G

W f 共 s,t 兲 Wg 共 s,t 兲 d␮ G 共 s,t 兲 ⫽C ⌿ 具 f ,g 典 L 2 (S) .

共10兲

If f ⫽g, then 2

2

储 W f 储 L 2 (G) ⫽C ⌿ 储 f 储 L 2 (S) .

That means W f 苸L 2 (G), so we have the Parseval formula by 共10兲. Remark 5: For a moment, let us define the continuous wavelet transform as follows: W f 共 s,t 兲 ⫽

1

冑C ⌿

具 f ,⌿ s,t 典 ,

f 苸L 2 共 S 兲 .

䊐

共11兲

By Theorem 4 we can easily conclude that W is an isometry of L 2 (S) into its range. Theorem 7 will specify the range of continuous wavelet transform which is a closed subspace of L 2 (G). The following lemma will be used for proving Theorem 6, we refer the reader to Koornwinder 共1993兲, Lemma 4.1, for its proof. Lemma 6: Let H1 and H2 be Hilbert spaces and let ⌽:H1 →H2 be an isometry into H2 . Then ⌽⌽ 쐓 :H2 →H2 is the orthogonal projection of H2 onto ⌽(H2 ). Theorem 7: (Reproducing kernel formula) Let h苸L 2 (G). Then h⫽W f , for some f 2 苸L (S) if and only if h 共 s´ , ´t 兲 ⫽1/C ⌿

冕

G

h 共 s,t 兲 具 ⌿ s,t ,⌿ s´ , ´t 典 L 2 (S) d␮ G 共 s,t 兲 .

Proof: Let us consider W as in 共11兲, so by Lemma 6, WW 쐓 is an orthogonal projection of L (G) onto W(L 2 (S)). Now, if h苸L 2 (G) we have 2

共 WW 쐓 h 兲共 s´ , ´t 兲 ⫽

⫽

1

冑C ⌿ 1

冑C ⌿

⫽1/C ⌿

具 W 쐓 h,⌿ s´ , ´t 典 L 2 (S) 具 h,W⌿ s´ , ´t 典 L 2 (G)

冕

G

h 共 s,t 兲 具 ⌿ s,t ,⌿ s´ , ´t 典 L 2 (S) d␮ G 共 s,t 兲 . 䊐

Lemma 8: Let f and ⌿苸L 1 艚L 2 (S). Put h t (s)⫽W f (s,t), for a fixed t苸T, then ¯ˆ 共 ␥ ⴰ ␴ 兲 , hˆ t 共 ␥ 兲 ⫽ ␳ 共 t 兲 1/2 ˆf 共 ␥ 兲 ⌿ t

␥ 苸Sˆ .

Proof: By 共9兲 we have h t 共 s 兲 ⫽W f 共 s,t 兲 ⫽ ␳ 共 t 兲 1/2

冕 ␥ 冕

⫽ ␳ 共 t 兲 ⫺1/2

Sˆ

¯ˆ 共 ␥ ⴰ ␴ 兲 ␥ 共 s 兲 d␮ ˆ 共 ␥ 兲 ˆf 共 兲 ⌿ t S

Sˆ

关 f 쐓 共 ⌿ⴰ ␴ t ⫺1 兲 쐓 兴 ˆ 共 ␥ 兲 ␥ 共 s 兲 d␮ Sˆ 共 ␥ 兲

⫽ ␳ 共 t 兲 ⫺1/2 关 f 쐓 共 ⌿ⴰ ␴ t ⫺1 兲 쐓 兴共 s 兲 .

共12兲

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In the last equality we applied Fourier Inversion formula 关see Folland 共1995兲, Theorem 2.40兴 which is valid here because 关 f 쐓 ( ⌿ⴰ ␴ t ⫺1 ) 쐓 兴 苸L 1 艚L 2 (S)艚C 0 (S), also 关 f 쐓(⌿ⴰ ␴ t ⫺1 ) 쐓 兴 ˆ 苸L 1 (Sˆ ). Therefore h t 苸L 1 艚L 2 (S) and 共12兲 obtains. 䊐 1 2 1 ˆ ˆ Theorem 9: (Inversion formula) Let f 苸L 艚L (S) and f 苸L (S ), then for each u苸S, f 共 u 兲 ⫽1/C ⌿

冕

G

W f 共 s,t 兲 ⌿ s,t 共 u 兲 d␮ G 共 s,t 兲 .

Proof: By Lemma 8 we have

冕

S

¯ˆ 共 ␥ ⴰ ␴ 兲 . W f 共 s,t 兲 ␥ 共 s ⫺1 兲 d␮ S 共 s 兲 ⫽ ␳ 共 t 兲 1/2 ˆf 共 ␥ 兲 ⌿ t

ˆ ( ␥ ⴰ ␴ t ) and integrating over T we get Multiplying both sides of the above equality by ␳ (t) ⫺1/2⌿ ˆf 共 ␥ 兲 ⫽1/C ⌿

冕冕␳ T

S

ˆ 共 ␥ ⴰ ␴ t 兲 ␥ 共 s ⫺1 兲 d␮ S 共 s 兲 d␮ T 共 t 兲 . 共 t 兲 ⫺1/2 W f 共 s,t 兲 ⌿

Now, by the inverse Fourier transform, Lemma 3 and 共6兲 we have f 共 u 兲⫽

冕

Sˆ

ˆf 共 ␥ 兲 ␥ 共 u 兲 d ␮ Sˆ ( ␥ )⫽

1 C⌿

⫽

1 C⌿

⫽

1 C⌿

冕␳ 冕 冕 冕␳ 冕 T

T

G

共 t 兲 ⫺1

S

S

W f 共 s,t 兲

冕

Sˆ

ˆ 共 ␥ 兲 ␥ 共 u 兲 d ␮ ˆ 共 ␥ 兲 d␮ 共 s 兲 d␮ 共 t 兲 ⌿ s,t S S T

共 t 兲 ⫺1 W f 共 s,t 兲共 u 兲 ⌿ s,t 共 u 兲 d␮ S 共 s 兲 d␮ T 共 t 兲

W f 共 s,t 兲 ⌿ s,t 共 u 兲 d␮ G 共 s,t 兲 . 䊐

III. EXAMPLE AND REMARKS

As an example of the continuous wavelet transform discussed in Sec. II we consider semidirect product of the additive group R d and the multiplicative group R ⫹ ª(0,⬁) under homomorphism ␴ :R ⫹ →Aut(R d ) defined by a苸R ⫹ ,b苸R d .

␴ a 共 b 兲 ⫽ab,

In order to compare our result with Koornwinder 共1993兲, we reverse the order of R d and R ⫹ and we put G⫽R ⫹ ␴ R d with the following multiplication and inversion; 共 a,b 兲共 a´ ,b´ 兲 ⫽ 共 aa´ ,b⫹ab´ 兲 ,

共 a,b 兲 ⫺1 ⫽ 共 a ⫺1 ,⫺ba ⫺1 兲 .

Since (a,b)→ab is continuous so G is a locally compact group with modular function ⌬ G (a,b)⫽a ⫺d and with Haar measure a ⫺d⫺1 da db, where da is the Lebesgue measure on R and db is the Lebesgue measure on R d 关for more information about computing the Haar measure for semidirect product of two locally compact groups see Hewitt and Ross 共1985兲, Chap. 5, Sec. 15.29兴. G acts on R d with the following action: G⫻R d →R d ,

共 a,b 兲 .x⫽ax⫹b,

that are the affine transformations. With this action R d is a homogeneous space of G. Now formulas 共6兲 and 共7兲 are valid with the rho-function ␳ (a)⫽a d ,a苸R ⫹ and d␮ R d appearing in them is equal to the Lebesgue measure on R d . The representation ␲ defined by 共8兲 is as follows: 关 ␲ 共 a,b 兲 f 兴共 x 兲 ⫽a ⫺d/2 f

冉 冊

x⫺b , a

共 a,b 兲 苸G,x苸R d , f 苸L 2 共 R d 兲 .

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Let ⌿ be a wavelet as in Definition 1, so C ⌿⫽

冕

Rd

ˆ 共 a ␰ 兲兩 2 兩⌿

da , a

᭙ ␰ 苸R ⫹ ,

and ⌿ a,b 共 x 兲 ⫽a ⫺d/2⌿

冉 冊

x⫺b , a

共 a,b 兲 苸G,x苸R d .

We refer the reader to Theorems 2.2, 2.3, 2.5 and Proposition 4.4 in Koornwinder 共1993兲 to compare them with properties of this transform that we obtained in Sec. II. Remark 10: Theorem 4 implies that any wavelet ⌿ is an admissible and cyclic vector for the representation ␲ defined by 共8兲. Remark 11: If G⫽R ␴ R 쐓 where R 쐓 ⫽R⫺ 兵 0 其 , then the representation ␲ defined by 共8兲 is irreducible 关see Koornwinder 共1993兲, Sec. 5兴; however, if G⫽R ␴ R ⫹ then

兵 f 苸L 2 共 R 兲 ;support ˆf 債 关 0,⬁ 兲 其 is a proper closed invariant subspace for ␲, so in this case ␲ is not irreducible. Therefore, it is not possible to say anything about irreducibility of ␲ defined by 共8兲. Remark 12: Let G be a locally compact group, H a Hilbert space, and ␲ a square integrable representation of G on U共H兲 and g be an admissible vector for it. In Grossman et al. 共1985兲, the continuous wavelet transform of any f 苸H is defined by Wg f 共 x 兲⫽具 f ,␲共 x 兲g典, which satisfies in plancherel formula, inversion formula, and reproducing kernel formula because of irreducibility of ␲ 关the latest results in this context could be found in Wong 共2002兲兴. We defined the continuous wavelet transform in the same way, but without irreducibility’s assumption. ACKNOWLEDGMENT

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