Wavelets and Linear Algebrawala.vru.ac.ir/article_6352_1a0047fe61bbe9c10151dbc7795a6868.pdf · Wavelets and Linear Algebra 1 (2014) 63-73 Vali-e-Asr University ... R. A. Kamyabi Gola,c,

Wavelets and Linear Algebra 1 (2014) 63-73

Vali-e-Asr University

Wavelets and Linear Algebrahttp://wala.vru.ac.ir

Two-wavelet constants for square integrablerepresentations of G/H

R. A. Kamyabi Gola,c, F. Esmaeelzadehb,∗, R. Raisi Tousic

a Department of Mathematics, Center of Excellency in Analysis on AlgebraicStructures(CEAAS), Ferdowsi University Of Mashhad, Mashhad, IslamicRepublic of Iran.bDepartment of Mathematics, Bojnourd Branch, Islamic Azad University,Bojnourd, Islamic Republic of Iran.cDepartment of Mathematics, Ferdowsi University Of Mashhad, Mashhad,Islamic Republic of Iran.

Article InfoArticle history:Received 28 November 2013Accepted 28 April 2014Available online 1 July 2014Communicated by AbdolazizAbdollahi

Keywords:Homogenous spaceIrreduciblerepresentation.

2000 MSC:43A15; 43A25

AbstractIn this paper we introduce two-wavelet constants for square in-tegrable representations of homogeneous spaces. We establishthe orthogonality relations for square integrable representationsof homogeneous spaces which give rise to the existence of aunique self adjoint positive operator on the set of admissiblewavelets. Finally, we show that this operator is a constant mul-tiple of identity operator when G is a semidirect product groupof a unimodular subgroup K and a closed subgroup H.

c© (2014) Wavelets and Linear Algebra

∗Corresponding authorEmail addresses: [email protected] (R. A. Kamyabi Gol),

[email protected] (F. Esmaeelzadeh), [email protected] (R. Raisi Tousi)

c© (2014) Wavelets and Linear Algebra

Kamyabi Gol, Esmaeelzadeh and Raisi Tousi/ Wavelets and Linear Algebra 1 (2014) 63-73 64

1. Introduction

The purpose of this paper is to explore and develop certain results concerning two-waveletconstants in the setting of homogeneous spaces. We emphasize that in [8] the orthogonality re-lations for square integrable representations of locally compact groups has been proved (see also[9, 1, 15]). Our approach leads to extend orthogonality relations for square integrable representa-tions of homogeneous spaces. In several cases the group representations are not square integrablebecause the group is too large and it is necessary to make the group smaller which can be per-formed by factoring out a suitable subgroup, that is one has to work with homogenous spaces. In[2] continuous wavelet transform and square integrable representations have been studied in thesetting of homogeneous spaces. In this note we define two-wavelet constant for a square integrablerepresentation of a homogeneous space. Also we obtain the orthogonality relations for square in-tegrable representations of homogeneous spaces. First we need to introduce the basic definitionsand fix some notations.

Let G be a locally compact group and H be a closed subgroup of G. Consider G/H as ahomogeneous space on which G acts from the left and µ as a Radon measure on it. For g ∈ G andBorel subset E of G/H, the translation µg of µ is defined by µg(E) = µ(gE). A measure µ is saidto be G-invariant if µg = µ, for all g ∈ G. A measure µ is called strongly quasi invariant providedthat a continuous function λ : G ×G/H → (0,∞) exists which satisfies

dµg(kH) = λ(g, kH)dµ(kH),

for all g, k ∈ G. If the functions λ(g, .) reduce to constants, then µ is called relatively invariantunder G (for a detailed account of homogeneous spaces, the reader is referred to [5, 6, 16, 3]).A rho-function for the pair (G,H) is defined to be a continuous function ρ : G → (0,∞) whichsatisfies

ρ(gh) =∆H(h)∆G(h)

ρ(g) (g ∈ G, h ∈ H),

where ∆G, ∆H are the modular functions on G and H, respectively. It is well known that (see [5]),any pair (G,H) admits a rho-function and for each rho-function ρ there is a strongly quasi invariantmeasure µ on G/H such that

dµg

dµ(kH) =

ρ(gk)ρ(k)

(g, k ∈ G).

We denote L1(G/H, µ) simply by L1(G/H), where µ is a strongly quasi invariant measure on G/H.The mapping T : L1(G)→ L1(G/H) defined by

T f (gH) =

∫H

f (gh)ρ(gh)

dh, (1.1)

is a surjective bounded linear operator with ‖T‖ ≤ 1. More precisely, for all φ ∈ L1(G/H) thereexists some f ∈ L1(G) such that φ = T f .

Let K and H be two locally compact groups with identity elements eK and eH, respectively andh 7→ τh be a homomorphism of H into the group of automorphisms of K. Then the set K × Hendowed with the operation

(k1, h1)(k2, h2) = (k1τh1(k2), h1h2)


is a group in which the identity element is (eK , eH) and the inverse of an element (k, h) is

(k, h)−1 = (τh−1(k−1), h−1).

This group, which is denoted by K ×τ H, is called the semidirect product of K and H with respectto τ. Equip K × H with the product topology. If the mapping (k, h) 7→ τh(k) from K × H onto K iscontinuous, then the semidirect product K ×τ H of K and H with respect to τ is a locally compactgroup [10]. It has been shown in [13] that if G is the semidirect product of two locally compactgroups K, H respectively, then any x ∈ G has a unique decomposition x = kh, for k ∈ K, h ∈ H andG/H has a relatively invariant Radon measure, which arises from the rho function ρ : G → (0,∞)defined by ρ(x) =

∆H(h)∆G(h) , where x ∈ G, h ∈ H, and x = kh for some k ∈ K. Moreover, there exists a

left Haar measure dk on K for which∫G

f (x)dx =

∫K

∫H

f (kh)δ(h)dhdk, f ∈ L1(G), (1.2)

where δ : H → (0,∞) is a homomorphism such that δ(h) =∆G(h)∆H(h) [4, 13, 11, 16].

In this paper we study two-wavelet constants and the orthogonality relations for a square inte-grable representation $ of a homogeneous space G/H which results to the existence of a uniqueself adjoint positive operator on the set of admissible wavelets for $. This paper is organized asfollows.

In Section 2, two-wavelet constant is introduced and it is shown that there exists a uniqueself adjoint positive operator on the set of admissible wavelets which it grantees orthogonalityrelations. In Section 3, we show that if G is a semidirect product of two locally compact groupsH, K and K is unimodular, then the unique self adjoint positive operator on the set of admissiblewavelets is a constant multiple of identity operator. Finally we give some examples which supportour discussion.Throughout this paper we assume that G is a locally compact group and H is a compact subgroupof G.

2. Two-Wavelet Constant

An admissible wavelet in the setting of homogeneous space G/H is studied by the authors in[2]. For the readers’ convenience we need to recall some basic concepts from [2] concerning thetheory of unitary representations and continuous wavelet transforms on homogeneous spaces.

A continuous unitary representation of a homogeneous space G/H is a map $ from G/H intothe group U(H), all unitary operators on some nonzero Hilbert space H , for which the functiongH 7→< $(gH)x, y > is continuous , for each x, y ∈ H and

$(gkH) = $(gH)$(kH), $(g−1H) = $(gH)∗,

for each g, k ∈ G. Moreover, a closed subspace M of H is said to be invariant with respect to $if $(gH)M ⊆ M, for all g ∈ G. A continuous unitary representation is called irreducible if theonly invariant subspaces of H are {0} and H (in the sequel we always mean by a representation,


a continuous unitary representation). An irreducible representation $ of G/H on H is said to besquare integrable if there exists a nonzero element ζ ∈ H such that∫

G/H

ρ(e)ρ(g)

| < ζ,$(gH)ζ > |2dµ(gH) < ∞, (2.1)

where µ is a relatively invariant measure on G/H which arises from a rho function ρ : G → (0,∞).If ζ satisfies (2.1), it is called an admissible vector for $ in Hilbert spaceH . An admissible vectorζ ∈ H is called admissible wavelet if ‖ζ‖ = 1. In this case, we define the wavelet constant cζ as

cζ :=∫

G/H

ρ(e)ρ(g)

| < ζ,$(gH)ζ > |2dµ(gH). (2.2)

We call cζ the wavelet constant associated to the admissible wavelet ζ. It is worthwhile to notethat there is a close relation between the representations of homogeneous spaces G/H, where H isa compact subgroup of G, and the representations of G. More precisely if $ is a representation ofG/H, then it defines a representation π of G in which the subgroup H is considered to be containedin kernel of π. Conversely, any representation π of G which is trivial on H induces a representation$ of G/H, by letting $(gH) = π(g).Let $ be a representation of G/H on a Hilbert space H and ζ be an admissible wavelet for $.We define the continuous wavelet transform associated to the admissible wavelet ζ as the linearoperator Wζ : H → C(G/H) defined by

(Wζx)(gH) =1√cζ

(ρ(e)ρ(g)

)1/2 < x, $(gH)ζ >,

for all x ∈ H , g ∈ G where cζ is the wavelet constant associated to ζ as in (2.2). Note that if $ is asquare integrable representation of G/H on H and ζ is an admissible wavelet for $, then Wζ is abounded linear operator fromH into L2(G/H).

Our aim in this section is to obtain the orthogonality relations for $ and to show the exis-tence of a self adjoint positive operator on the set of the admissible wavelets which satisfies theorthogonality relations.

Theorem 2.1. (orthogonality relations) Let $ be a square integrable representation of G/H onHilbert spaceH and ζ, ξ be two admissible wavelets for $. Then the following hold.

(i) For all x ,y inH ,∫G/H

ρ(e)ρ(g)

< x, $(gH)ζ >< $(gH)ξ, y > dµ(gH) = cζ,ξ < x, y > (2.3)

wherecζ,ξ =

∫G/H

ρ(e)ρ(g)

< ζ,$(gH)ζ >< $(gH)ξ, ζ > dµ(gH). (2.4)

We call cζ,ξ two-wavelet constant.(ii) There exists a unique self adjoint positive operator C onH such that:


(a) The set of all admissible wavelets coincides with the domain of C.(b) For all x ,y inH∫

G/H

ρ(e)ρ(g)

< x, $(gH)ζ >< $(gH)ξ, y > dµ(gH) =< Cξ,Cζ >< x, y > .

Proof. To prove (i), let ζ be an admissible wavelet. Define the linear operator

Aζ : M → L2(G/H), Aζ(x)(gH) = (ρ(e)ρ(g)

)1/2 < x, $(gH)ζ >, (2.5)

whereM = {x ∈ H ,

∫G/H

ρ(e)ρ(g)

| < x, $(gH)ζ > |2dµ(gH) < ∞}.

As has been shown in [2, Lemma 2.5], M is a closed invariant subspace ofH under $ and by theirreducibility of $ we have M = H . Since Aζ is an intertwining operator for $ and `, where ` isthe left regular representation of G on L2(G/H) such that `(g)ϕ = ( ρ(e)

ρ(g) )1/2lgϕ, lgϕ(gH) = ϕ(g−1gH)

(see more details in [2]), we get

< A∗ξAζ$(gH)x, y > = < A∗ξ`(g)Aζx, y >

= < `(g)Aζx, Aξy >

= < Aζx, `(g−1)Aξy >

= < Aζx, Aξ$(g−1H)y >

= < $(gH)A∗ξAζx, y >

So, A∗ξAζ$(gH) = $(gH)A∗ξAζ for every g ∈ G. By Schur’s Lemma for homogeneous spaces [2]there exists cζ,ξ such that A∗ξAζ = cζ,ξI. This means that for every x, y inH ,∫

G/H

ρ(e)ρ(g)

< x, $(gH)ζ >< $(gH)ξ, y > dµ(gH) = cζ,ξ < x, y >,

wherecζ,ξ =

1‖x‖2

∫G/H

ρ(e)ρ(g)

< x, $(gH)ζ >< $(gH)ξ, x > dµ(gH). (2.6)

For (ii), we denote the set of admissible wavelets by A. Define

q : A × A→ C, q(ζ, ξ) = cζ,ξ, (2.7)

for ζ, ξ ∈ A. It is clear that q(ζ, ξ) is a positive, symmetric quadratic form with domain A. To showthat q is closed, consider the norm ‖.‖q on A defined by,

‖ζ‖2q = ‖ζ‖2 + q(ζ, ζ). (2.8)


Let {ζk} be a Cauchy sequence in A with respect to the norm ‖.‖q . Then {ζk} converges to ζ inH ,as k → ∞. For any x ∈ H and k ∈ N, we define the function ψx

k on G/H by

ψxk(gH) =

1‖x‖

.(ρ(e)ρ(g)

)1/2 < x, $(gH)ζk > . (2.9)

Then by (2.8), (2.9) we get

‖ψxj − ψ

xk‖

2L2(G/H) = 1

‖x‖2

∫G/H

ρ(e)ρ(g) | < x, $(gH)(ζ j − ζk) > |2dµ(gH)

= q(ζ j − ζk, ζ j − ζk)

≤ ‖ζ j − ζk‖2q → 0,

as j, k → ∞. Thus {ψxk}∞k=1 is a Cauchy sequence in L2(G/H) for all x ∈ H which implies that

ψxk → ψ for some ψ in L2(G/H) as k → ∞. Also by the Schwartz inequality

ψxk(gH) = 1

‖x‖ .(ρ(e)ρ(g) )

1/2 < x, $(gH)ζk >

→ 1‖x‖ .(

ρ(e)ρ(g) )

1/2 < x, $(gH)ζ >,

as k → ∞. Then1‖x‖

(ρ(e)ρ(g)

)1/2 < x, $(gH)ζ >= ψ(gH),

and hence ∫G/H

ρ(e)ρ(g)

| < x, $(gH)ζ > |2dµ(gH) < ∞,

for all x ∈ H . Therefore ∫G/H

ρ(e)ρ(g)

| < ζ,$(gH)ζ > |2dµ(gH) < ∞,

i.e ζ ∈ A. Furthermore,

‖ζk − ζ‖2q = ‖ζk − ζ‖

2 + q(ζk − ζ, ζk − ζ)

= ‖ζk − ζ‖2 + 1

‖x‖2

∫G/H

ρ(e)ρ(g) | < x, $(gH)(ζk − ζ) > |2dµ(gH)→ 0

as k → ∞. It implies that A is complete and then q is closed. By the second representation theorem[14], there exists a unique self adjoint positive operator C with domain A such that q(ζ, ξ) = cζ,ξ =<Cζ,Cξ > and the proof is complete.

In the following proposition, we introduce orthogonal subspaces of L2(G/H).

Proposition 2.2. Let $1 and $2 be two square integrable representations of G/H on Hilbertspaces H1,H2 with admissible wavelets ζ and ξ, respectively. If $1 and $2 are not equivalent,then the images of Wζ and Wξ are orthogonal subspaces of L2(G/H). That is

< Wζx1,Wξx2 >= 0,

for all x1 ∈ H1, x2 ∈ H2.


Proof. Define the linear mapping β : H1 ×H2 → C as follows,

β(x1, x2) :=< Wζx1,Wξx2 >L2(G/H)=

∫G/H

(Wζx1)(gH)(Wξx2)(gH)dµ(gH).

Since the wavelet transform is continuous, it follows that

|β(x1, x2)| = | < Wζx1,Wξx2 > |

≤ ‖Wζx1‖L2(G/H)‖Wξx2‖L2(G/H)

≤ ‖Wζ‖‖Wξ‖‖x1‖‖x2‖,

By the Little Riesz Representation Theorem (see 12.5, in [17]), we can conclude that, there is acontinuous linear map S : H1 → H2 such that

β(x1, x2) =< S x1, x2 > .

Using the fact that continuous wavelet transform is intertwining, we get

< S$1(gH)x1, x2 > = β($1(gH)x1, x2)

=∫

G/H(Wζ$1(gH)x1)(kH)(Wξx2)(kH)dµ(kH)

=∫

G/H(`(g)Wζx1)(kH)(Wξx2)(kH)dµ(kH)

=∫

G/H( ρ(e)ρ(g) )

1/2(Wζx1)(g−1kH)(Wξx2)(kH)dµ(kH)

=∫

G/H( ρ(e)ρ(g) )

1/2(Wζx1)(kH)(Wξx2)(gkH)(ρ(g)ρ(e) )dµ(kH)

=∫

G/H(ρ(g)ρ(e) )

1/2(Wζx1)(kH)(Wξx2)(gkH)dµ(kH)

=∫

G/H(Wζx1)(kH)(`(g−1)Wξx2)(kH)dµ(kH)

=∫

G/H(Wζx1)(kH)(Wξ$2(g−1H)x2)(kH)dµ(kH)

= β(x1, $2(g−1H)x2)

= < S x1, $2(gH)∗x2 >

= < $2(gH)S x1, x2 >

for all x1 ∈ H1, x2 ∈ H2. It follows that S$1(gH) = $2(gH)S . So by Schur’s Lemma forhomogeneous spaces [2], we conclude that S is zero when $1 is not equivalent to $2. Thus< Wζx1,Wξx2 >= 0, for all x1 ∈ H1, x2 ∈ H2.

3. orthogonality relations for some special homogeneous spaces

In this section G is considered as the semidirect product of a locally compact group K anda compact group H. Let $ be a representation of G/H in which G = K ×τ H. Then it definesa representation of K by letting π(k) = $(gH), where g = kh. In [13] it is shown that ρ(g) =∆H(h)∆G(h) = 1, where g ∈ G, h ∈ H and g = kh for some k ∈ K. Note that in this case the condition


for admissibility is reduced to∫

G/H| < ζ,$(gH)ζ > |2dµ(gH) < ∞. Now, we show that the unique

self adjoint operator C in Theorem 2.1 is a constant multiple of the identity operator if and onlyif K is unimodular. To this end we establish the following lemmas. The next lemma shows thatthe set of all admissible wavelets for $ contains in the set of all admissible wavelets for π and thesecond lemma states that the set of all admissible wavelets is invariant under $.

Lemma 3.1. Let K,H be two locally compact groups and G = K ×τ H and $, π be the represen-tations of G/H and K, respectively on a Hilbert spaceH . If ζ is an admissible wavelet for $ thenζ is an admissible wavelet for π.

Proof. Let $ be an irreducible representation of G/H on H . This representation defines an irre-ducible representation π of K onH by π(k) = $(gH), where g = kh. Using Weil’s formula, (1.2),and the fact that T defined as in (1.1) is onto, we have∫

G/H| < ζ,$(gH)ζ > |2dµ(gH) =

∫G/H

T (ϕ)(gH)dµ(gH)

=∫

Gϕ(g)dg

=∫

K

∫Hϕ(kh)dhdk

=∫

KT (ϕ)(kH)dk

=∫

K| < ζ,$(kH)ζ > |2dk

=∫

K| < ζ, π(k)ζ > |2dk,

for some ϕ ∈ L1(G). Therefore, if ζ is an admissible wavelet for $ then ζ is an admissible waveletfor π.

Lemma 3.2. Let K,H be two locally compact groups and G = K ×τ H and $, H be as inLemma3.1. Then the set of all admissible wavelets is invariant under $.

Proof. Denote the set of all admissible wavelets for $ by A. That is

A = {ζ ∈ H ,

∫G/H| < ζ,$(gH)ζ > |2dµ(gH) < ∞}.

Consider the representation π of K as π(k) = $(gH), in which g = kh. Let the Haar measure on Hbe normalized. Since π is trivial on H, we have

1∆(k1)

∫K| < ζ, ω(k)ζ > |2dk[1ex] =

∫K| < ζ, ω(k−1

1 kk1)ζ > |2dk

=∫

K| < ζ, ω(k−1

1 )ω(k)ω(k1)ζ > |2dk

=∫

K| < ω(k1)ζ, ω(k)ω(k1)ζ > |2dk

=∫

K

∫H| < ω(k1h1)ζ, ω(kh)ω(k1h1)ζ > |2dhdk

=∫

K

∫H| < ω(g1)ζ, ω(g)ω(g1)ζ > |2dhdk

=∫

Gϕ(g)dg

=∫

G/H

∫Hϕ(gη)dηdµ(gH)

=∫

G/H

∫H| < ω(k1h1)ζ, ω(khη)ω(k1h1)ζ > |2dηdµ(gH)

=∫

G/H| < $(g1H)ζ,$(gH)$(g1H)ζ > |2dµ(gH),


for some ϕ ∈ L1(G), where g1 = k1h1. Thus the set A is invariant under $.

Now we are ready to prove that the unique self adjoint operator C in Theorem 2.1 is a constantmultiple of the identity operator if and only if K is unimodular.

Theorem 3.3. Let K,H be two locally compact groups and G = K ×τ H. Then the unique selfadjoint positive operator C in Theorem 2.1 is a constant multiple of the identity operator if andonly if K is unimodular.

Proof. In Theorem 2.1, we have shown that there exists a unique self adjoint positive operator Csuch that q(ζ, ξ) = cζ,ξ =< Cζ,Cξ >, where cζ,ξ is as in (2.4), for admissible wavelets ζ, ξ . So wehave

q($(gH)ζ,$(gH)ξ) = q(π(k)ζ, π(k)ξ)

= 1‖υ‖2

∫K< π(k1k)ξ, υ > < π(k1k)ζ, υ > dk1

=∆(k−1)‖υ‖2

∫K< π(k1)ξ, υ > < π(k1)ζ, υ > dk1.

Thus< C$(gH)ζ,C$(gH)ξ >=

1∆(k)

q(ζ, ξ) =1

∆(k)< Cζ,Cξ > . (3.1)

Also, domain C2 is invariant. Indeed, for ζ ∈ D(C2), we have ζ ∈ D(C) such that Cζ ∈ D(C) andby invariance of D(C) we get $(gH)ζ ∈ D(C). Then by (3.1)

< C$(gH)ζ,C$(gH)ξ >=1

∆(k)q(ζ, ξ) =

1∆(k)

< ζ,C2ξ >, (3.2)

which implies that C$(gH)ξ ∈ D(C). That is $(gH)ξ ∈ D(C2) and C2 is densely defined. Also,(3.2) implies that

C2$(gH) =1

∆(k)$(gH)C2.

By using Shur’s Lemma [2], K is unimodular, i.e. ∆(k) = 1 if and only if C = λI.

Now, we conclude that if ζ is an admissible wavelet for $, then every vector in H is anadmissible wavelet provided that K is unimodular.

Proposition 3.4. Let G = K ×τ H such that K is a unimodular, H is a compact subgroup of G and$ is a representation of G/H. If there exists an admissible vector in H for $, then all vectors ofH are admissible.

Proof. By Lemma 3.2 and irreducibility$, it is enough to show that A is closed. By unimodularityof K we have

< C$(gH)ζ,C$(gH)ξ >=< Cζ,Cξ >,

for ζ, ξ ∈ A. Then C$(gH) = $(gH)C. Let A be the Hilbert space equipped with the innerproduct of which the induced norm ‖.‖q defined as in (2.8). By the extended Schur’s Lemma [2]there exists a positive number α such that

‖Cζ‖2 = α‖ζ‖2q = α‖ζ‖2 + α‖Cζ‖2, ζ ∈ A.


It implies that α < 1 and‖Cζ‖2 =

α

1 − α‖ζ‖2, ζ ∈ A. (3.3)

Using (3.3) and density of A in H , we can extend C : A → H to a bounded linear operatorC : H → H . Now let {ζk}k∈N is a sequence of elements in A such that ζk → ζ in H as k → ∞.Then Cζk → Cζ in L2(G/H) as k → ∞. Since C is a closed linear operator with domain A, itfollows that ζ ∈ A. Therefore A is a closed subspace.

Here we intend to support our technical considerations developed in this note by giving someexamples.

Example 3.5. Let N be an even number and G be the group with two generators a and b satisfying|a| = N, |b| = 3 and ba = ab2 i.e.

G =< a, b; aN = b3 = e, ba = ab2 > .

Also, let H =< b >, K =< a > and

$ : G/H → U(L2(G/H)), $(aib jH)ψ(anH) = ψ(an−iH)

be a representation for all 0 ≤ i, n ≤ N − 1 and 0 ≤ j ≤ 2. It is easy to check that $ admits anadmissible vector. Therefore we can define a continuous wavelet transform

(Wψϕ)(aib jH) =1√cψ

N−1∑n=0

ϕ(anH)ψ(an−iH).

for all 0 ≤ n ≤ N − 1 and admissible vector ψ in L2(G/H). Since L2(G/H) is isometry isomor-phism with L2(K), the representation π : K → L2(K), π(a j) f (an) = f (an− j) is a square integrablerepresentation. For two admissible wavelets f , g ∈ L2(K), the two-wavelet constant is as follows:

c f ,g =

N−1∑i=0

N−1∑n=0

N−1∑m=0

( f (an)) f (an−i)( f (am)g(am−i)).

Since K is unimodular, by Theorem 3.3 the self adjoint positive operator C on the set of admissiblewavelets is a constant multiple of identity operator. Note that, in this example G , H ×τ K sinceK is not a normal subgroup of G. But H is a closed normal subgroup of G and this causes Wψϕ tobe constant on left cosets of H [12].

Example 3.6. Let G be the Weyl-Heisenberg group (WH)n and H = {(0, 0, t), t ∈ R/2πZ}. TheEuclidean space Rn × Rn can be considered as homogenous space of (WH)n and (WH)n

H = Rn × Rn

which admits the Lebesgue measure. The representation

$ : Rn × Rn → U(L2(Rn)), ($(q, p)ϕ)(x) = ei(px−qp)ϕ(x − q),


where x ∈ Rn, ϕ ∈ L2(Rn) is square integrable (see [18]) and for two admissible wavelets ζ, ξ ∈L2(Rn), two-wavelet constant is as follows

cζ,ξ =∫Rn

∫Rn < ζ,$(q, p)ζ >< $(q, p)ξ, ζ > dqdp

=∫Rn

∫Rn < ζ, ζq,p >< ξq,p, ζ > dqdp,

where ζq,p(x) = eipxζ(x−q), x ∈ Rn. Since Rn×Rn is unimodular, by Theorem 3.3 the self adjointpositive operator C on the set of admissible wavelets is a constant multiple of identity operator.

References

[1] S.T. Ali, J-P. Antoine, J-P. Gazeau, Coherent States, Wavelets and Their Generalizations, Springer-Verlag, NewYork, 2000.

[2] F. Esmaeelzadeh, R. A. Kamyabi Gol, R. Raisi Tousi, On the continuous wavelet transform on homogeneousspaces, Int. J. Wavelets. Multiresolut, 10(4)(2012).

[3] F. Esmaeelzadeh, R. A. Kamyabi Gol, R. Raisi Tousi, Localization operators on Homogeneous spaces, Bull.Iranian Math. Soc, Vol. 39, no. 3 (2013), 455-467.

[4] F. Esmaeelzadeh, R. A. Kamyabi Gol, R. Raisi Tousi, Positive type functions and representations of some ho-mogenous spaces , submitted.

[5] G.B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.[6] J.M.G. Fell, R.S. Doran, Representation of ∗-algebras, Locally Compact Groups, and Banach ∗-algebraic

Bundles, Vol. 1, Academic Press, 1988.[7] H. Fuhr, Abstract Harmonic Analysis of Continuous Wavelet Transform, Springer Lecture Notes in Mathematics,

Nr. 1863, Berlin, 2005.[8] A. Grossmann, J. Morlet, T. Paul, Transform associated to square integrable group representation I, J. Math.

Phys. 26 (1985), 24-79.[9] K.Grochenig, Foundations of Time-Frequency Analysis, Birkhauser Boston, 2001.

[10] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, Springer-Verlag OHG. Gottingen. Heidelberg, 1963.[11] R.A. Kamyabi Gol, N. Tavallaei, Relatively invariant measure and semidirect product groups, submitted.[12] R.A. Kamyabi Gol, N. Tavallaei, Wavelet transforms via generalized quasi regular representations, Appl. Com-

put. Harmon. Anal. 26 (2009), no. 3, 291-300.[13] R.A. Kamyabi Gol, N. Tavallaei, Convolution and homogeneous spaces, Bull. Iran. Math. Soc. 35(2009), 129-

146.[14] T. Kato, Perturbation Theory for Linear Operator, Springer-Verlag, Berlin, 1976.[15] M. Kyed, Square Integrable representations and the continuous wavelet transformation , Ph.D Thesis, 1999.[16] H. Reiter, J. Stegeman, Classical Harmonic Analysis and Locally compact Group, clarendon press, 2000.[17] W. Rudin, Functional Analysis, Mc Graw-Hill, 1974.[18] M. W. Wong, Wavelet Transform and Localization Operators. Birkhauser Verlag, Basel-Boston-Berlin, 2002.

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