The square integrable representations on generalized Weyl-Heisenberg groups
aa r X i v : . [ m a t h . R T ] F e b THE SQUARE INTEGRABLE REPRESENTATIONS ONGENERALIZED WEYL- HEISENBERG GROUPS
FATEMEH ESMAEELZADEH
Abstract.
This paper presents the square integrable representa-tions of generalized Weyl-Heisenberg group. We investigate the quasiregular representation of generalized Weyl-Heisenberg group. More-over, we obtain a concrete form for admissible vector of this represen-tation . Finally, we provide some examples to support our technicalconsiderations. Introduction
Wavelet transform has rich theoretical structures and is extremely use-ful as tools for building signal transforms, adapted to various signal ge-ometries, quantum mechanics, etc. Continuous wavelet transform ad-mits a generalization to locally compact groups. Such a unified approachseems to be useful, since it emphasizes on a clear way to basic featuresof continuous wavelet transform and includes all important cases for ap-plications [2, 3, 5]. It should be mentioned that, the Weyl Heisenberggroup plays a significant designations in various aspects of the connec-tions between the classical harmonic analysis and concrete applicationsof numerical harmonic analysis.
Mathematics Subject Classification.
Primary 43A15, Secondary .
Key words and phrases. generalized Weyl-Heisenberg group, square integrable,admissible wavelet.
This paper contains 4 sections. Section 2 includes the definition ofsemi-direct product of two locally compact groups and generalized WeylHeisenberg group. In Section 3, we study the square integrable represen-tations on the generalized Weyl Heisenberg group and then we obtain thenecessary and sufficient conditions for admissible wavelet on this group.In Section 4, some examples are proved as application of our results.2.
Preliminaries and notation
Let H and K be two locally compact groups with the identity elements e H and e K , respectively and let τ : H → Aut ( K ) be a homomorphismsuch that the map ( h, k ) τ h ( k ) is continuous from H × K onto K ,where H × K equips with the product topology. The semi- direct producttopological group G τ = H × τ K is the locally compact topological space H × K under the product topology, with the group operations:( h , k ) × τ ( h , k ) = ( h h , k τ h ( k ) , ( h, k ) − = ( h − , τ h − ( k − )) . It is worth to note that K = { ( e H , k ); k ∈ K } is a closed normal sub-group and H = { ( h, e K ); h ∈ H } is a closed subgroup of G τ such that G τ = HK . Moreover, the left Haar measure of the locally compactgroup G τ is dµ G τ ( h, k ) = δ H ( h ) dµ H ( h ) dµ K ( k ) , in which dµ H , dµ K are the left Haar measures on H and K , respectivelyand δ H : H → (0 , ∞ ) is a positive continuous homomorphism that satis-fies dµ K ( k ) = δ H ( h ) dµ ( τ h ( k )) , HE SQUARE INTEGRABLE REPRESENTATION ON H ( G τ ) 3 for h ∈ H, k ∈ K . Moreover, the modular function ∆ G τ is∆ G τ = δ H ( h )∆ H ( h )∆ K ( k ) , where ∆ H , ∆ K are the modular functions of H, K , respectively.When K is also abelian, one can define ˆ τ : H → Aut ( ˆ K ) via h ˆ τ h where ˆ τ h ( ω ) = ω ◦ τ h − , for all ω ∈ ˆ K . We usually denote ω ◦ τ h − by ω h . With this notation, itis easy to see ω h h = ( ω h ) h , where h , h ∈ H and ω ∈ ˆ K . The semi-direct product G ˆ τ = H × ˆ τ ˆ K isa locally compact group with the left Haar measure dµ ˆ G ( h, ω ) = δ H ( h ) − dµ H ( h ) dµ ˆ K ( ω ) , where dµ ˆ K is the Haar measure on ˆ K . Also, for all h ∈ H , dµ ˆ K ( ω h ) = δ H ( h ) dµ ˆ K ( ω ) , for ω ∈ ˆ K, (see more details in [4, 1, 3].)Let G τ = H × τ K , and define θ : G τ → Aut ( ˆ K × T ) via( h, k ) θ ( h,k ) ( ω, z ) = ( ˆ τ h ( ω ) , ˆ τ h ( ω )( k ) z ) = ( ω h , ω h ( k ) z ) , for all ( h, k ) ∈ H × τ K and ( ω, z ) ∈ ˆ K × T . The mapping θ is a contin-uous homomorphism. Thus the semi-direct prodoct G τ × θ ( ˆ K × T ) = ( H × τ K ) × θ ( ˆ K × T ) , is a locally compact group and it is called the generalized Weyl Heisenberggroup associated with the semi direct product group G τ = H × τ K , and F. ESMAEELZADEH denoted by H ( G τ ). It is easy to see that the group operations of H ( G τ )are( h , k , ω , z ) . ( h , k , ω , z ) = ( h h , k τ h ( k ) , ω ω h , ω h ( k ) z z ) , ( h , k , ω , z ) − = ( h − , τ − h ( k − ) , ¯ ω h − , ¯ ω h − ( τ − h ( k − )) z − ) , for ( h , k , ω , z ) , ( h , k , ω , z ) ∈ H ( G τ ) (see [4]) and the left Haar mea-sure of H ( G τ ) is: dµ H ( G τ ) ( h, k, ω, z ) = dµ H ( h ) dµ K ( k ) dµ ˆ K ( ω ) dµ T ( z ) . The square integrable representation of H ( G τ )Throughout this section, we assume that H and K are locally compacttopological groups and that K is abelian, too. We denote the left Haarmeasures of H and K by dµ H , dµ K , respectively. Suppose that h τ h from H to Aut ( K ) is a homomorphism such that ( h, k ) τ h ( k ) from H × K into K is continuous. G τ = H × τ K is the semi-direct productof H and K that is a locally compact topology group with the left Haarmeasure dµ G τ ( h, k ) = δ H ( h ) dµ H (( h ) dµ K ( k ), where δ H : H (0 , ∞ ) isa continuous homomorphism. Consider the homomorphism θ : G τ → Aut ( ˆ K × T ) is defined by(( h, k ) , ( ω, z )) θ ( h,k ) ( ω, z ) , where θ ( h,k ) ( ω, z ) = ( ω ◦ τ h − , ω ◦ τ h − ( k ) .z ) . This makes H ( G τ ) = G τ × θ ( ˆ K × T ) a locally compact topological group where H ( G τ ) is equippedwith the product topology and the group operations as( h , k , ω , z ) . ( h , k , ω , z ) = ( h h , k τ h ( k ) , ω ω h , ω h ( k ) z z ) , ( h , k , ω , z ) − = ( h − , τ − h ( k − ) , ¯ ω h − , ¯ ω h − ( τ − h ( k − )) z − ) , HE SQUARE INTEGRABLE REPRESENTATION ON H ( G τ ) 5 for ( h , k , ω , z ) , ( h , k , ω , z ) ∈ H ( G τ ) . The left Haar measure of H ( G τ )is dµ H ( G τ ) ( h, k, ω, z ) = dµ H ( h ) dµ K ( k ) dµ ˆ K ( ω ) dµ T ( z ) . Now, we are going to define a square integrable representation on H ( G τ ). With the above notations define π : H ( G τ ) → U ( L ( ˆ K )) by(3.1) π ( h, k, ω, z ) f ( ξ ) = δ − / H ( h ) zξ ( k ) ω ( k ) f (( ξω ) h − ) , then π is a homomorphism. Indeed, π (( h , k , ω , z )( h , k , ω , z ) ) f ( ξ ) = π ( h h , k τ h ( k ) , ω ( ω ) h , ( ω ) h ( k ) z z ) f ( ξ )= δ − / H ( h h )( ω ) h ( k ) z z ξ ( k τ h ( k )) ω ( ω ) h ( k τ h ( k )) f (( ξω ( ω ) h ) ( h h ) − = δ − / H ( h h )( ω ) h ( k ) z z ξ ( k ) ξ h − ( k ) ω ( k )( ω ) h − ( k ) ω ( k ) f ( ξ h − h − ( ω ) h − h − ( ω ) h − ) . Also, π ( h , k , ω , z ) π ( h , k , ω , z ) f ( ξ )= δ − / H ( h ) z ξ ( k ) ω ( k ) π ( h , k , ω , z ) f (( ξω ) h − = δ − / H ( h ) δ − / H ( h ) z z ξ ( k ) ω ( k ) ω ( k )( ξω ) h − ( k ) f (( ξω ) h − ( ω ) h − )= δ − / H ( h h ) z z ξ ( k ) ξ h − ( k ) ω ( k )( ω ) h − ( k ) ω ( k ) f ( ξ h − h − ( ω ) h − h − ( ω ) h − ) . .Moreover, π is unitary. In fact we have, k π ( h, k, ω, z ) f k = R ˆ K | π ( h, k, ω, z ) f ( ξ ) | dµ ˆ K ( ξ )= R ˆ K δ − H ( h ) | f (( ξω ) h − | dµ ˆ K ( ξ )= R ˆ K δ − H ( h ) | f (( ξ ) h − | dµ ˆ K ( ξ )= R ˆ K f (( ξ ) | dµ ˆ K ( ξ )= k f k . .And it is easy to check that π is continuous and onto. So, π is a continuousunitary representation of group H ( G τ ) to the Hilbert space L ( ˆ K ). In the F. ESMAEELZADEH sequel, we show that π is irreducible when H is compact. Furthermore,it is also shown that π is square integrable if and only if H is compact.Note that when H is a compact group, we normalize the Haar measure µ H such that µ H ( H ) = 1. Theorem 3.1.
Let H ( G τ ) = ( H × τ K ) × θ ( ˆ K × T ) where H is a locallycompact group and K is a locally compact abelian group. Then for ϕ, ψ in L ( ˆ K ) , (3.2) Z H ( G τ ) | ≺ ϕ, π ( h, k, ω, z ) ψ ≻ | dµ H ( G τ ) ( h, k, ω, z ) = k ϕ k k ψ k . if and only if H is compact. HE SQUARE INTEGRABLE REPRESENTATION ON H ( G τ ) 7 Proof.
For ϕ, ψ in L ( ˆ K ) we first consider the following observations: R H ( G τ ) | ≺ ϕ, π ( h, k, ω, z ) ψ ≻ | dµ H ( G τ ) ( h, k, ω, z )= R H ( G τ ) | R ˆ K ϕ ( ξ ) π ( h, k, ω, z ) ψ ( ξ ) dµ ˆ K ( ξ ) | dµ H ( G τ ) ( h, k, ω, z )= R H ( G τ ) | R ˆ K ϕ ( ξ ) δ − / H ( h ) zξ ( k ) ω ( k ) ψ ( ξω ) h − dµ ˆ K ( ξ ) | dµ H ( G τ ) ( h, k, ω, z )= R H ( G τ ) | R ˆ K ϕ ( ξω ) δ − / H ( h ) zξ ( k ) ψ ( ξ ) h − dµ ˆ K ( ξ ) | dµ H ( G τ ) ( h, k, ω, z )= R H ( G τ ) | R ˆ K R ω ϕ ( ξ ) δ − / H ( h ) zξ ( k ) ψ ( ξ ◦ τ h ) dµ ˆ K ( ξ ) | dµ H ( G τ ) ( h, k, ω, z )= R H ( G τ ) | R ˆ K R ω ϕ ( ξ ◦ τ h − ) δ − / H ( h ) zξ ◦ τ h − ( k ) ψ ( ξ ) dµ ˆ K ( ξ h ) | dµ H ( G τ ) ( h, k, ω, z )= R H ( G τ ) | R ˆ K R ω ϕ ( ξ ◦ τ h − ) δ / H ( h ) zξ ( τ h − ( k )) ψ ( ξ ) dµ ˆ K ( ξ ) | dµ H ( G τ ) ( h, k, ω, z )= R H ( G τ ) δ H ( h ) | R ˆ K ( R ω ϕ ( . ◦ τ h − ) .ψ )( ξ ) ξ ( τ h − ( k )) dµ ˆ K ( ξ ) | dµ H ( G τ ) ( h, k, ω, z )= R H ( G τ ) δ H ( h ) \ | ( R ω ϕ ( . ◦ τ h − ) .ψ )( τ h − ( k )) | dµ H ( G τ ) ( h, k, ω, z )= R H δ H ( h ) R ˆ K R K | \ ( R ω ϕ ( . ◦ τ h − ) .ψ )( τ h − ( k )) | dµ K ( k ) dµ ˆ K ( ω ) dµ H ( h )= R H R ˆ K R K | \ ( R ω ϕ ( . ◦ τ h − ) .ψ )( k ) | dµ K ( k ) dµ ˆ K ( ω ) dµ H ( h )= R H R ˆ K R ˆ K | ( R ω ϕ ( . ◦ τ h − ) .ψ )( ξ ) | dµ ˆ K ( ξ ) dµ ˆ K ( ω ) dµ H ( h )= R H R ˆ K R ˆ K | R ω ϕ ( ξ ◦ τ h − ) .ψ ( ξ ) | dµ ˆ K ( ξ ) dµ ˆ K ( ω ) dµ H ( h )= R H R ˆ K R ˆ K δ H ( h ) | R ω ϕ ( ξ ) .ψ ( ξ ◦ τ h ) | dµ ˆ K ( ξ ) dµ ˆ K ( ω ) dµ H ( h )= R H R ˆ K k ϕ k δ H ( h ) | ψ ( ξ ◦ τ h ) | dµ ˆ K ( ξ ) dµ H ( h )= k ϕ k k ψ k µ H ( H ) .Now, if H is compact, then µ H ( H ) = 1. So, (3.2) holds. Conversely, if(3.2) holds,the above observation implies that µ H ( H ) = 1 . So, we canconclude that H is compact. (cid:3) Corollary 3.2.
With notation as above, the representation π of H ( G τ ) on L ( ˆ K ) is irreducible if H is compact. F. ESMAEELZADEH
Proof. If H is compact, then (3.2) in Theorem 3.1 holds. Now, supposethat M is a closed subspace of the Hilbert space L ( ˆ K ) that is invariantunder π . Then for any ϕ ∈ M we have, { π ( h, k, ω, z ) ϕ ; ( h, k, ω, z ) ∈ H ( G τ ) } ⊆ M. Let ψ ∈ L ( ˆ K ) be orthogonal to M , that is ≺ ψ, π ( h, k, ω, z ) ϕ ≻ = 0 , forall ( h, k, ω, z ) ∈ H ( G τ ). Thus by (3.2), k ϕ k k ψ k = 0, and hence ψ = 0.So, M ⊥ = { } , that is, M = L ( ˆ K ). Namely, π is irreducible. (cid:3) We remind the reader that, an irreducible representation π of H ( G τ )on L ( ˆ K ) is called square integrable if there exists a non zero element ψ in L ( ˆ K ) such that(3.3) ≺ π ( ., ., ., . ) ψ, f ≻∈ L ( H ( G τ )) , for all f ∈ L ( ˆ K ). A unit vector ψ satisfying (3.3) is said to be anadmissible wavelet for π , and the constant c ψ = Z H ( G τ ) | ≺ π ( h, k, ω, z ) ψ, ψ ≻ | dµ H ( G τ ) , is called the wavelet constant associated to the admissible wavelet ψ .Also, for the wavelet vector ψ , the continuous wavelet transform is definedby W ψ f ( h, k, ω, z ) = ≺ f, π ( h, k, ω, z ) ψ ≻ . It is easy to see that ( h, k, ω, z ) W ψ f ( h, k, ω, z ) is a continuous functionon H ( G τ ) . Moreover, W ψ intertwines π and the left regular representationon H ( G τ ) . Corollary 3.3.
The representation π of the GW H group H ( G τ ) = ( H × τ K ) × θ ( ˆ K × T ) on L ( ˆ K ) is square integrable if and only if H is compact. HE SQUARE INTEGRABLE REPRESENTATION ON H ( G τ ) 9 Proof. If H is compact, then by Theorem 3.1 and Corollary 3.2, π issquare integrable. For the inverse, if π is square integrable, then thereexists a non zero element ϕ ∈ L ( ˆ K ) such that ≺ π ( ., ., ., . ) ϕ, ψ ≻∈ L ( H ( G τ )) , for all ψ ∈ L ( ˆ K ). On the other hand, Z H ( G τ ) | ≺ ϕ, π ( h, k, ω, z ) ψ ≻ | dµ H ( G τ ) ( h, k, ω, z ) = k ϕ k k ψ k µ H ( H ) . So µ H ( H ) < ∞ . That is H is compact. (cid:3) Remark . There is another irreducible representation of H ( G τ ) onHilbert space L ( K ). Indeed, consider˜ π : H ( G τ ) → U ( L ( K )) , ˜ π ( h, k, ω, z ) f ( k ′ ) = δ H ( h ) / zω ( k ′ ) f ( τ h − ( k ′ k )) , for all ( h, k, ω, z ) ∈ H ( G τ ) , f ∈ L ( K ). ˜ π is homomorphism and unitary.In fact we have˜ π (( h , k , ω , z )( h , k , ω , z )) f ( k ′ )= ˜ π ( h h , k τ h ( k ) , ω ( ω ) h , ( ω ) h ( k ) z z ) f ( k ′ )= δ / H ( h h )( ω ) h ( k ) z z ω ( ω ) h ( k ′ ) f ( τ ( h h ) − ( k ′ ( k τ h ( k ))= δ / H ( h h ) z z ω ( k ′ )( ω ) h ( k ′ k ) f ( τ h − h − ( k ′ k τ h ( k ))= δ / H ( h h ) z z ω ( k ′ ) ω ) h ( k ′ k ) f ( τ h − h − ( k ′ k ) τ h − ( k )) , and˜ π ( h , k , ω , z )˜ π ( h , k , ω , z ) f ( k ′ )= δ / H ( h ) z ω ( k ′ )˜ π ( h , k , ω , z ) f ( τ h − ( k ′ k ))= δ / H ( h ) δ / H ( h ) z z ω ( k ′ ) ω ( τ h − ( k ′ k )) f ( τ h − ( τ h − ( k ′ k ) k )= δ / H ( h h ) z z ω ( k ′ )( ω ) h ( k ′ k ) f ( τ h − h − ( k ′ k ) τ h − ( k )) . .Also, k ˜ π ( h, k, ω, z ) f k = R K | ˜ π ( h, k, ω, z ) f ( k ′ ) | dµ K ( k ′ )= R K δ H ( h ) | f ( τ h − ( k ′ k )) | dµ K ( k ′ )= R K δ H ( h ) | f ( k ′ ) | dµ K ( τ h ( k ′ ))= R K | f ( k ′ ) | dµ K ( k ′ )= k f k . .Using the Plancherel theorem, π, ˜ π are unitarily equivalent. So, ˜ π issquare integrable if and only if π is square integrable. Remark . The inverse of Corollary 3.2 does not hold, generally. Anobvious example is when H is a non compact group and K is the trivialgroup { e } . Then the representation π : H ( H × τ { e } ) → U ( C ) is anirreducible representation. Here we give a non trivial example in which π is an irreducible representation, but H is not compact. Let H = R + , K = R . Define the representation π of H ( R + × τ R ) as follows: π : H ( R + × τ R ) → U ( L ( R )); π ( a, x, ω, z ) f ( ξ ) = a / ze πix ( ξ − ω ) f (( ξ ¯ ω ) a − ) , in which ( ξ ¯ ω ) a − = ( ξ ¯ ω ) ◦ τ a , τ a ( x ) = a.x and δ H ( a ) = a − . This rep-resentation is irreducible. Indeed, let M be a closed invariant subspaceof L ( R ) under π . Then for any f ∈ M , we have π ( h, k, ω, z ) f ∈ M .Consider 0 = g ∈ M ⊥ , so that ≺ g, π ( h, k, ω, z ) f ≻ = 0. Then0 = Z R g ( ξ ) e − πixξ ¯ f (( ξ ¯ ω ) a − ) dξ = Z R g ( ξ a ω ) e − πixξ a ω ¯ f ( ξ ) dξ. Thus, g ( ξ a ω ) ¯ f ( ξ ) = 0, for almost all ξ ∈ R . Suppose that ¯ f ( ξ ) = 0, forall ξ in a set A with positive measure. Then for all ξ ∈ A, g ( ξ a ω ) = 0,for all ω ∈ R , a ∈ R + . Thus g = 0. This is a contradiction. So, π is anirreducible representation, but H is not compact. HE SQUARE INTEGRABLE REPRESENTATION ON H ( G τ ) 11 In the sequel, we define the quasi regular representation and we ob-tain a concrete form for an admissible vector. Note that H ( G τ ) acts onthe Hilbert space L ( ˆ K × T ) and this action induces the quasi regularrepresentation { ρ, L ( ˆ K × T ) } as follows:(3.4) ρ : ( H × τ K ) × θ ( ˆ K × T ) → U ( L ( ˆ K × T )) , where ρ ( h, k, ω, z ) f ( ξ, t ) = δ / H × τ K ( h, k ) f ( θ ( h,k ) − ( ξ, t )( ω, z ) − )= δ − / H ( h ) f ( θ ( h − ,τ h − ( k − ) ( ξ ¯ ω, tz − ))= δ − / H ( h ) f (( ξ ¯ ω ) h − , ( ξ ¯ ω ) h − ( τ h − ( k − )) .tz − ) . Note that δ H × τ K ( h, k ) = δ H ( h ) − . (see Corollary 3.3 in [4])A type of the Fourier transform of the quasi regular representation ρ obtains as follows: \ ρ ( h, k, ω, z ) f ( k ′ , n ′ )= R ˆ K × T ρ ( h, k, ω, z ) f ( ξ, t )( k ′ , n ′ )( ξ, t ) dµ ˆ K ( ξ ) dµ T ( t )= δ H ( h ) − / R ˆ K × T f (( ξω ) h − , ( ξω ) h − ( τ h − ( k − )) tz − ) ξ ( k ′ ) t n ′ dµ ˆ K ( ξ ) dµ T ( t )= δ ( h ) − / z n ′ R ˆ K × T f ( ξ ) h − , ξ h − ( τ h − ( k − )) t ) ω ( k ′ ) ¯ ξ ( k ′ ) t n ′ dµ ˆ K ( ξ ) dµ T ( t )= δ H ( h ) − / ¯ z n ′ ¯ ω ( k ′ ) R ˆ K × T f ( θ ( h − ,τ h − ) ( ξ, t ) ¯ ξ ( k ′ ) ¯ t n ′ dµ ˆ K ( ξ ) dµ T ( t )= δ H ( h ) − / ¯ z n ′ ¯ ω ( k ′ ) R ˆ K × T f ◦ θ ( h,k ) − ( ξ, t )( k ′ , n ′ )( ξ, t ) dµ ˆ K ( ξ ) dµ T ( t )= δ H ( h ) − / ¯ z n ′ ¯ ω ( k ′ ) \ ( f ◦ θ ( h,k ) − )( k ′ , n ′ ) , for all ( k ′ , n ′ ) ∈ K × Z = \ ( ˆ K × T ) . So,(3.5) \ ρ ( h, k, ω, z ) f ( k ′ , n ′ ) = δ H ( h ) − / ¯ z n ′ ¯ ω ( k ′ ) \ ( f ◦ θ ( h,k ) − )( k ′ , n ′ ) . Theorem 3.6.
With the notation as above, let ρ be the quasi regularrepresentation on H ( G τ ) , and ψ, f ∈ L ( ˆ K × T ) . (i) If ψ is a wavelet vector, then W ψ f ( h, k, ω, z ) = δ − / H ( h ) Z K X n ′ ∈ Z ˆ f ( k ′ , n ′ ) z n ′ ω ( k ′ ) \ ( ψ ◦ θ ) ( h,k ) − ( k ′ , n ′ ) dµ K ( k ′ ) . (ii) The vector ψ is wavelet if Z H × τ K | ˆ ψ ( k ′ , n ′ ) ◦ θ ( h,k ) − | dµ H × K ( h, k ) < ∞ . Proof.
For ( k ′ , n ′ ) ∈ K × Z , (i) By the Plancherel’s theorem and (3.5), we have W ψ f ( h, k, ω, z ) = ≺ f, ρ ( h, k, ω, z ) ψ ≻ = ≺ ˆ f , \ ρ ( h, k, ω, z ) ψ ≻ = δ − / H ( h ) Z K X n ′ ∈ Z ˆ f ( k ′ , n ′ ) z n ′ ω ( k ′ ) \ ( ψ ◦ θ ) ( h,k ) − ( k ′ , n ′ ) dµ K ( k ′ ) . (ii) By applying the part (i), for f ∈ L ( ˆ K × T ), we get R ˆ K × T | W ψ f ( h, k, ω, z ) | dµ ˆ K × T ( ω, z )= R ˆ K × T W ψ f ( h, k, ω, z ) W ψ f ( h, k, ω, z ) dµ ˆ K × T ( ω, z )= δ − H ( h ) R ˆ K × T [( R K P n ′ ∈ Z ˆ f ( k ′ , n ′ ) z n ′ ω ( k ′ ) \ ( ψ ◦ θ ) ( h,k ) − ( k ′ , n ′ ) dµ K ( k ′ )) × ( R K P n ′′ ∈ Z ˆ f ( k ′ , n ′ ) z n ′′ ω ( k ′′ ) \ ( ψ ◦ θ ) ( h,k ) − ( k ′′ , n ′′ ) dµ K ( k ′′ ))]= δ − H ( h ) R ˆ K × T | ˆ F ( ω, z ) | dµ ˆ K × T = δ − H ( h ) R ˆ K × T | F ( k ′ , n ′ ) | dµ K × Z = δ − H ( h ) R K P n ′ ∈ Z | ˆ f ( k ′ , n ′ ) | | \ ( ψ ◦ θ )( k ′ , n ′ ) | dµ K ( k ′ ) , HE SQUARE INTEGRABLE REPRESENTATION ON H ( G τ ) 13 where ˆ F = ˆ f ˆ( ψ ◦ θ ) ∈ L ( K × Z ) . It is easy to see that \ ( ψ ◦ θ )(( k ′ , n ′ ) = δ − H ( h ) ˆ ψ ( k ′ , n ′ ) ◦ θ ( h,k ) − . Then(3.6) Z ˆ K × T | W ψ f ( h, k, ω, z ) | dµ ˆ K × T ( ω, z ) = δ − H ( h ) Z K X n ′ ∈ Z | ˆ f ( k ′ , n ′ ) | | \ ( ψ ( k ′ , n ′ ) ◦ θ ( h,k ) − | dµ K ( k ′ ) . Now, by using (3.6) we have k W ψ f k = R H ( G τ ) | W ψ f ( h, k, ω, z ) | dµ H ( G τ ) ( h, k, ω, z )= R H × τ K R ˆ K × T | W ψ f ( h, k, ω, z ) | δ − H ( h ) dµ ˆ K × T ( ω, z ) dµ H × τ K ( h, k )= R H × τ K R K P n ′ ∈ Z | ˆ f ( k ′ , n ′ ) | | \ ( ψ ( k ′ , n ′ ) ◦ θ ( h,k ) − | dµ K ( k ′ ) dµ H × τ K ( h, k )= k f k R H × τ K | \ ( ψ ( k ′ , n ′ ) ◦ θ ( h,k ) − | dµ H × τ K ( h, k ) , and then the proof of part ( ii ) is complete. (cid:3) Examples and applications
Example 4.1.
Let K be an abelian locally compact group and H = { e } (the trivial group). In this case the generalized weyl Heisenberggroup H ( G τ ) coincides with the standard weyl Heisenberg group G := K × θ ( ˆ K × T ). In this case the square integrable representation of G = K × θ ( ˆ K × T ) on L ( ˆ K ) is as follows:(4.1) π ( k, ω, z ) f ( ξ ) = zξ ( k ) ω ( k ) f ( ξω ) . Example 4.2.
Let E ( n ) be the Euclidean group which is the semi-directproduct of So ( n ) × τ R n where the continuous homomorphism τ : So ( n ) → Aut ( R n ) given by σ τ σ via τ σ ( x ) = σx , for all x ∈ R n . The groupoperation for E ( n ) is( σ , x ) × τ ( σ , x ) = ( σ σ , x + σ x ) . Consider the continuous homomorphism ˆ τ : So ( n ) → Aut ( R n ) via σ ˆ τ σ which is given by ˆ τ σ (( ω ) = ω σ = ω ◦ τ σ − . Thus the generalized WeylHeisenberg group of E ( n ), is the set H ( E ( n )) = ( So ( n ) × τ R n ) × θ ( R n × T )with the group operation( σ , x , ω , z )( σ , x , ω , z ) = ( σ σ , x + σ x , ω ( ω ) σ , ( ω ) σ ( x ) z z ) , for all ( σ , x , ω , z )( σ , x , ω , z ) ∈ H ( E ( n )) and with the producttopology. Then the square integrable representation π of H ( E ( n )) onto L ( R n ) is π ( σ, x, ω, z ) f ( ξ ) = e πix ( ξ − ω ) f (( ξ − ω ) σ − ) . Note that H is compact and δ H ( h ) = 1. Example 4.3.
Let H ( R n ) = R n × θ ( R n × T ) be the classical Heisenberggroup on R n , in which the continuous homomorphism x θ x from R n into Aut ( R n × T ) is defined by θ x ( y, z ) = ( y, ze πix.y ) . Then the squareintegrable representation π of H ( R n ) onto L ( R n ) is π ( x, ω, z ) f ( ξ ) = z.e πix ( ξ − ω ) f ( ξ − ω ) . References
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Continuous wavelet transforms from semidirect products:Cyclic representations and Plancherel measure , J. Fourier Anal. Appl., Vol. 8,375-398, 2002.4. A. Ghaani Farashahi,
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HE SQUARE INTEGRABLE REPRESENTATION ON H ( G τ ) 15
5. M.W. Wong,
Wavelet Transform and Localization Operator , Verlag, Basel-Boston- Berlin, 2002. Department of Mathematics, Bojnourd Branch, Islamic Azad Univer-sity, Bojnourd, Iran.
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