We define theτ×τ-continuous Gabor transform ofb f ∈ L2(Gτ) with respect to a window function u ∈ L2(K) as a function defined onGτ×bτ

COMPACT GROUPS

ARASH GHAANI FARASHAHI

Abstract. LetHbe a locally compact group,Kbe an LCA group,τ:H→Aut(K) be a continuous homomorphism and Gτ = Hnτ K be the semi-direct product of H and K with respect to the continuous homomorphism τ. In this article we introduce the τ×bτ-time frequency groupGτ×bτ. We define theτ×τ-continuous Gabor transform ofb f ∈ L²(Gτ) with respect to a window function u ∈ L²(K) as a function defined onGτ×bτ. It is also shown that theτ×τ-continuous Gabor transform satisfies the Plancherel Theorem and reconstruction formula. This approach isb tailored for choosing elements ofL²(Gτ) as a window function. Finally, we indicate some possible applications of these methods in the case of some well-known semi-direct product groups.

1. Introduction

In [15] Gabor used translations and modulations of the Gaussian signal to represent one dimensional signals. The Gabor transform, named after Gabor, is a special case of the short-time Fourier transform (STFT). It is used to determine the sinusoidal frequency and phase content of local sections of a signal as it changes over time. The function to be transformed is first multiplied by a Gaussian function, which can be regarded as a window, and the resulting function is then transformed with a Fourier transform to derive the time-frequency analysis. The window function term means that the signal near the time being analyzed will have higher weight. The Gabor transform of a signal x(t) is precisely defined by;

(1.1) G{x}(y, ω) =

Z +∞

−∞

x(t)e^{−π(t−y)2}e^−2πiωtdt.

Due to (1.1) the Gabor transform of a signal x(t) is a function defined on R×Rb called the time-frequency plane.

There is also standard extension of the continuous Gabor transform of a signal x(t) on Rⁿ which is defined for (y,w)∈Rⁿ×Rcⁿ by (see [9, 16])

(1.2) G{x}(y,w) =

Z

Rⁿ

x(t)e^{−πkt−yk2}e^−2πiw.tdt.

Since the theory of Gabor analysis based on the structure of translations and modulations (time-frequency plane), it is also possible to extend concepts of the Gabor theory to other locally compact abelian (LCA) groups. For more explanation, we refer the reader to the monograph of Gr¨ochenig [17] or complete works of Feichtinger and Strohmer [8] and also [7] in the case of finite abelian groups. The continuous Gabor transform for LCA groups is closely related to the Feichtinger-Gr¨ochenig theory (coorbit space theory). In view of voice transform and the coorbit space theory, the continuous Gabor transform for an LCA group Gis precisely the voice transform generated by the Schr¨odinger representation of the Weyl-Heisenberg group associate withG(see [4, 5, 6, 19]).

Many locally compact spaces and locally compact groups which are used in mathematical physics and also various topics of engineering such as the n-dimensional unit sphere, Heisenberg group, affine group or Euclidean groups are non-abelian groups or they are homogeneous spaces of non-abelian groups (see [10, 13, 21]). Although most of those

2000Mathematics Subject Classification. Primary 43A30, Secondary 43A25, 43A15.

Key words and phrases. semi-direct product, time-frequency plane(group), modulation, translation, short time Fourier transform (STFT), continuous Gabor transform, Plancherel Theorem, inversion formula.

E-mail addresses: [email protected] (Arash Ghaani Farashahi).

1

non-abelian locally compact groups can be considered as a semi-direct product of an LCA group with another locally compact group. The theory of harmonic analysis for semi-direct product of locally compact groups is a significant tool in the theory of wavelet analysis (see [1, 12, 18, 23]). We recall that in the classical theory of harmonic analysis for non-abelian locally compact groups (see [3, 10, 24, 26, 27]) we lose many useful results and basic numerical concepts in abelian harmonic analysis of LCA groups (see [10, 25]), which play important roles in the usual Gabor theory of LCA groups. If G is a non-abelain locally compact group via a natural approach, modulation by a character will be replaced by a modulation by an equivalence class of an irreducible representation of G(see [14]) and the natural candidate for the generalization of the time frequency plane will beG×G, whereb Gbstands for the set of all equivalence class of irreducible continuous unitary representations ofG. It is clear that this extension will not be appropriate from the the numerical computational aspects and also application viewpoints. Thus, we need a new approach to find an appropriate generalization of the continuous Gabor transform which be useful and also efficient in application.

This article contains 5 sections. Section 2 is devoted to fix notations including a brief summary about harmonic analysis of semi-direct product of locally compact groups also standard Fourier analysis and Gabor analysis on LCA groups. In section 3 we assume that H is a locally compact group and K is an LCA group, τ : H → Aut(K) is a continuous homomorphism and Gτ = H nτ K. We define the τ×τ-time frequency groupb G_τ×

bτ and the τ×bτ- continuous Gabor transform of f ∈L²(Gτ) with respect to a window functionu∈L²(K). We also prove a Plancherel and inversion formula for theτ×τ-continuous Gabor transform. To choose elements ofb L²(G_τ) as window functions we define the theτ⊗bτ-time frequency groupG_τ⊗

bτ and also theτ⊗bτ-continuous Gabor transform in section 4. As an application, we study this theory on the affine group, Weyl-Heisenberg group and the Euclidean groups in section 5.

2. Preliminaries and notations

LetH andKbe locally compact groups with identity elementseH andeK respectively and left Haar measuresdh anddkrespectively, also letτ:H→Aut(K) be a homomorphism such that the map (h, k)7→τh(k) is continuous from H×K ontoK. There is a natural topology, sometimes called Braconnier topology, turningAut(K) into a Hausdorff topological group(not necessarily locally compact), which is defined by the sub-base of identity neighborhoods (2.1) B(F, U) ={α∈Aut(K) :α(k), α⁻¹(k)∈U k∀k∈F},

where F ⊆K is compact andU ⊆K is an identity neighborhood. Continuity of a homomorphism τ :H →Aut(K) is equivalent with the continuity of the map (h, k)7→τh(k) fromH×KontoK (see [22]).

The semi-direct product G_τ =H nτK is the locally compact topological group with the underlying setH ×K which is equipped by the product topology and also the group operation is defined by

(2.2) (h, k)nτ(h⁰, k⁰) = (hh⁰, kτh(k⁰)) and (h, k)⁻¹= (h⁻¹, τ_h⁻¹(k⁻¹)).

If H₁ = {(h, e_K) : h ∈ H} and K₁ = {(e_H, k) : k ∈ K} then K₁ is a closed normal subgroup and H₁ is a closed subgroup ofG_τ. The left Haar measure ofG_τ isdµ_Gτ(h, k) =δ(h)dhdkand the modular function ∆_Gτ is

∆G_τ(h, k) =δ(h)∆H(h)∆K(k), where the positive continuous homomorphismδ:H →(0,∞) is given by ([21])

(2.3) dk=δ(h)d(τ_h(k)).

From now on, for allp≥1 we denote byL^p(Gτ) the Banach spaceL^p(Gτ, µG_τ) and alsoL^p(K) stands forL^p(K, dk).

Whenf ∈L^p(Gτ), for a.e. h∈H the functionfh defined onK viafh(k) :=f(h, k) belongs toL^p(K) (see [11]).

IfK is an LCA group all irreducible representations ofKare one-dimensional. Thus, ifπis an irreducible unitary representation ofKwe haveHπ=Cand also according to the Schur’s Lemma there exists a continuous homomorphism ω ofK into the circle groupT such that for eachk∈K andz∈Cwe haveπ(k)(z) =ω(k)z. Such homomorphisms are called characters of K and the set of all characters of K denoted by K. Ifb Kb equipped by the topology of compact convergence onKwhich coincides with thew^∗-topology thatKb inherits as a subset ofL^∞(K), thenKb with

respect to the point wise product of characters is an LCA group which is called the dual group ofK. The linear map FK :L¹(K)→ C(K) defined byb v7→ FK(v) via

(2.4) F_K(v)(ω) =bv(ω) =

Z

K

v(k)ω(k)dk,

is called the Fourier transform onK. It is a norm-decreasing∗-homomorphism fromL¹(K) intoC0(K) with a uniformlyb dense range inC0(K) (Proposition 4.13 of [10]). Ifb φ∈L¹(K), the function definedb a.e.onK by

(2.5) φ(x) =˘

Z

Kb

φ(ω)ω(x)dω,

belongs toL^∞(K) and also for allf ∈L¹(K) we have the following orthogonality relation (Parseval formula);

(2.6)

Z

K

f(k) ˘φ(k)dk= Z

Kb

fb(ω)φ(ω)dω.

The Fourier transform (2.4) on L¹(K)∩L²(K) is an isometric transform and it extends uniquely to a unitary iso- morphism fromL²(K) ontoL²(K) (Theorem 4.25 of [10]) also eachb v∈L¹(K) withbv∈L¹(K) satisfies the followingb Fourier inversion formula (Theorem 4.32 of [10]);

(2.7) v(k) =

Z

Kbv(ω)ω(k)dωb for a.e. k∈K.

The fundamental operator in standard Gabor theory is the time-frequency shift operator. If K is an LCA group, the translation (time-shifts) operator is given by Tsv(k) =v(k−s) for all k, s∈K and also the modulation (frequency- shifts) operator is given byMωv(k) =ω(k)v(k) for allω∈K,b k∈K. The time-frequency shift operator is defined on the time-frequency plane (time-frequency group) K×Kb by%(k, ω) =MωTk for all (k, ω)∈K×K.b

Given an appropriate window functionu∈L²(K) onK, the short time Fourier transform (STFT) or the continuous Gabor transform ofv∈L²(K) is given by

(2.8) V_uv(s, ω) =

Z

K

v(k)u(k−s)ω(k)dk=hv, %(s, ω)ui_L2(K). The continuous Gabor transform (2.8) satisfies the following Plancherel formula (2.9)

Z

K×Kb

|Vuv(s, ω)|²dsdω=kuk²_L2(K)kvk²_L2(K),

for all u, v ∈ L²(K) (see [17]). If u, u⁰ ∈ L²(K) with hu, u⁰i_L2(K) 6= 0, then each v ∈ L²(K) satisfies the following inversion formula in the weak sense (see [16])

(2.10) v=hu, u⁰i⁻¹_L2(K)

Z

K×Kb

Vuv(k, ω)%(k, ω)u⁰dkdω.

If a window functionu∈L²(K) has Fourier transform buinL¹(K), then eachb v∈L²(K) withbv∈L¹(K) satisfies theb following inversion formula;

(2.11) v(s) =kuk⁻²_L2(K)

Z

K×Kb

Vuv(k, ω)[%(k, ω)u](s)dkdω, for alls∈K.

3. τ×τb-continuous Gabor transform

Throughout this paper, letH be a locally compact group andK be an LCA group also letτ :H →Aut(K) be a continuous homomorphism and G_τ =H nτ K. For simplicity in notations we use k^h instead of τ_h(k) for all h∈H andk∈K. In this section we introduce theτ×τb-time frequency group and also we define theτ×τ-continuous Gaborb transform off ∈L²(G_τ) with respect to a window function inL²(K).

Define bτ:H →Aut(K) viab h7→τbh, given by

(3.1) bτ_h(ω) :=ω_h=ω◦τ_h−1

for allω∈K, whereb ωh(k) =ω(τh⁻¹(k)) for allk∈K. Ifω∈Kb andh∈H we haveωh∈K, because for allb k, s∈K we have

ωh(ks) =ω◦τ_h⁻¹(ks)

=ω(τ_h−1(ks))

=ω(τ_h−1(k)τ_h−1(s))

=ω(τ_h⁻¹(k))ω(τ_h⁻¹(s)) =ωh(k)ωh(s).

According to (3.1) for allh∈H we havebτh∈Aut(K). Because, ifb k∈K andh∈H then for allω, η∈Kb we have bτ_h(ω.η)(k) = (ω.η)_h(k)

= (ω.η)◦τ_h⁻¹(k)

=ω.η(τ_h−1(k))

=ω(τ_h−1(k))η(τ_h−1(k))

=ωh(k)ηh(k) =τbh(ω)(k)bτh(η)(k).

Alsoh7→bτhis a homomorphism from H into Aut(K), cause ifb h, t∈H then for allω∈Kb andk∈Kwe get bτ_th(ω)(k) =ω_th(k)

=ω(τ_(th)⁻¹(k))

=ω(τ_h−1τ_t−1(k))

=ω_h(τ_t−1(k))

=bτh(ω)(τ_t⁻¹(k)) =bτt[bτh(ω)](k).

Thus, we can prove the following theorem.

Theorem 3.1. Let H be a locally compact group and K be an LCA group also τ : H → Aut(K) be a continuous homomorphism and let δ : H → (0,∞) be the positive continuous homomorphism satisfying dk = δ(h)dk^h. The semi-direct product G

τb=Hnτ_bKb is a locally compact group with the left Haar measuredµ_G

τb(h, ω) =δ(h)⁻¹dhdω.

Proof. Continuity of the homomorphism bτ :H →Aut(K) given in (3.1) guaranteed by Theorem 26.9 of [21]. Hence,b the semi-direct productG

bτ =HnbτKb is a locally compact group. We also claim that the Plancherel measuredω on Kb for allh∈H satisfies

(3.2) dω_h=δ(h)dω.

Let h ∈H and v ∈L¹(K). Using (2.3) we havev◦τh ∈L¹(K) with kv◦τhkL¹(K) =δ(h)kvkL¹(K). Thus, for all ω∈Kb we achieve

v\◦τ_h(ω) = Z

K

v◦τ_h(k)ω(k)dk

= Z

K

v(k^h)ω(k)dk

= Z

K

v(k)ω_h(k)dk^h−1

=δ(h) Z

K

v(k)ωh(k)dk=δ(h)bv(ωh).

Now letv∈L¹(K)∩L²(K). Due to the Plancherel theorem (Theorem 4.25 of [10]) and also preceding calculation, for allh∈H we get

Z

Kb

|bv(ω)|²dωh= Z

Kb

|bv(ω_h⁻¹)|²dω

=δ(h)² Z

Kb

|v\◦τ_h⁻¹(ω)|²dω

=δ(h)² Z

K

|v◦τ_h⁻¹(k)|²dk

=δ(h)² Z

K

|v(k)|²dk^h

=δ(h) Z

K

|v(k)|²dk= Z

Kb

|bv(ω)|²δ(h)dω, which implies (3.2). Therefore,dµG

τb(h, ω) =δ(h)⁻¹dhdω is a left Haar measure forG

bτ=Hnτ_bK.b Remark 3.2. Due to (3.1) for allk∈K,ω∈Kb andh, t∈H we have

(3.3) k^ht= (k^t)^h, ω_ht= (ω_t)_h.

Now we are in the position to introduce theτ×bτ-time frequency group. Defineτ^×=τ×bτ:H →Aut(K×K) viab h7→τ_h^× given by

(3.4) τ_h^×(k, ω) := (τh(k),τbh(ω)) = (k^h, ωh),

for all (k, ω)∈K×K. Then, for allb h∈H we haveτ_h^× ∈Aut(K×K). Because for all (k, ω),b (k⁰, ω⁰)∈K×Kb we have

τ_h^×((k, ω)(k⁰, ω⁰)) =τ_h^×(kk⁰, ωω⁰)

= (kk⁰)^h,(ωω⁰)h

= k^hk^0h, ωhω_h⁰

= (k^h, ωh)(k^0h, ω_h⁰) =τ_h^×(k, ω)τ_h^×(k⁰, ω⁰).

Also τ^× = τ ×bτ : H → Aut(K×K) defined byb h 7→ τ_h^× is a homomorphism, because for all h, t ∈ H and all (k, ω)∈K×Kb we have

τ_ht^×(k, ω) = (k^ht, ωht)

= (k^t)^h,(ω_t)_h

=τ_h^×(k^t, ω_t) =τ_h^×τ_t^×(k, ω).

In the following proposition we show thatG_τ×

bτ=Hnτ×bτ(K×K) is a locally compact group.b

Proposition 3.3. Let H be a locally compact group and K be an LCA group alsoτ :H →Aut(K)be a continuous homomorphism and let δ : H → (0,∞) be the positive continuous homomorphism satisfying dk = δ(h)dk^h. The semi-direct product G_τ×

τb=Hnτ×τb(K×K)b is a locally compact group with the left Haar measure

(3.5) dµG_τ×bτ(h, k, ω) =dhdkdω.

Proof. Continuity of the homomorphismτ×bτ:H→Aut(K×K) given in (3.4) guaranteed by Theorem 26.9 of [21].b Thus, the semi-direct productGτ×bτ =Hnτ×bτ(K×K) is a locally compact group. Due to (2.3), (3.2) and also (3.4),b for allh∈H we have

dτ_h^×(k, ω) =d k^h, ωh

=dk^hdωh

=δ(h)⁻¹dkδ(h)dω=dkdω=d(k, ω), which implies thatG_τ×

bτ is a locally compact group with the left Haar measuredµG_τ×bτ(h, k, ω) =dhdkdω.

We call the semi-direct productG_τ×

bτ as theτ×τ-time frequency group associated tob G_τ. According to (3.4) for each (h, k, ω),(h⁰, k⁰, ω⁰)∈G_τ×

bτ we have

(h, k, ω)nτ×bτ(h⁰, k⁰, ω⁰) = hh⁰,(k, ω)τ_h^×(k⁰, ω⁰)

= (hh⁰,(k, ω)(τh(k⁰), ω⁰_h)) = (hh⁰, k+k^0h, ωω⁰_h).

Letu∈L²(K) be a window function andf ∈L²(Gτ). Theτ×τb-continuous Gabor transformoff with respect to the window functionuis define by

(3.6) Vuf(h, k, ω) :=δ(h)^1/2Vufh(k, ω) =δ(h)^1/2hfh, %(k, ω)uiL²(K).

In the following theorem we prove a Plancherel formula for theτ×bτ-continuous Gabor transform defined in (3.6).

Theorem 3.4. Let H be a locally compact group and K be an LCA group also τ : H → Aut(K) be a continuous homomorphism and let u ∈ L²(K) be a window function. The τ ×bτ-continuous Gabor transform Vu : L²(Gτ) → L²(G_τ×

bτ) is a multiple of an isometric transform which mapsL²(Gτ)onto a closed subspace ofL²(G_τ×

bτ).

Proof. Let u ∈ L²(K) be a window function and also f ∈ L²(G_τ). Using Fubini’s Theorem and also Plancherel formula (2.9) we have

kVufk²_L2(G_τ×bτ)= Z

Gτ×bτ

|Vuf(h, k, ω)|²dµG_τ×bτ(h, k, ω)

= Z

H

Z

K

Z

Kb

|Vuf(h, k, ω)|²dhdkdω

= Z

H

Z

K×Kb

|hf_h, %(k, ω)ui_L2(K)|²dkdω

δ(h)dh

=kuk²_L2(K)

Z

H

kfhk²_L2(K)δ(h)dh=kuk²_L2(K)kfk²_L2(Gτ). Therefore,kuk⁻²_L2(K)Vu:L²(Gτ)→L²(Gτ×G

bτ) is an isometric transform with a closed range inL²(Gτ×G

τb).

Corollary 3.5. Theτ×τ-continuous Gabor transform defined in (3.6), for allb f, g∈L²(G_τ)and window functions u, v ∈L²(K)satisfies the following orthogonality relation;

(3.7) hVuf,VvgiL²(G_τ×bτ)=hv, uiL²(K)hf, giL²(G_τ). Theτ×bτ-continuous Gabor transform (3.6) satisfies the following inversion formula.

Proposition 3.6. LetH be a locally compact group andKbe an LCA group also letτ :H →Aut(K)be a continuous homomorphism and u∈L²(K) with ub∈ L¹(K). Everyb f ∈L²(Gτ) with cfh ∈L¹(K)b for a.e. h∈ H, satisfies the following reconstruction formula;

(3.8) f(h, k) =δ(h)^−1/2kuk⁻²_L2(K)

Z

K×Kb

Vuf(h, s, ω)[%(s, ω)u](k)dsdω.

Proof. Using (2.11) for a.e. h∈H we have fh(k) =kuk⁻²_L2(K)

Z

K×Kb

Vufh(s, ω)[%(s, ω)u](k)dsdω

=δ(h)^−1/2kuk⁻²_L2(K)

Z

K×Kb

Vuf(h, s, ω)[%(s, ω)u](k)dsdω.

We can also define the generalized form of the τ ×τ-continuous Gabor transform. Letb u ∈ L²(K) be a window function andf ∈L²(G_τ). Thegeneralized τ×bτ-continuous Gabor transformoff with respect to the window function uis define by

(3.9) V_u^†f(h, k, ω) :=δ(h)^1/2V_uf_h(k^h, ω_h) =δ(h)^1/2hf_h, %(k^h, ω_h)ui_L2(K).

The generalizedτ×bτ-continuous Gabor transform given in (3.9) satisfies the following Plancherel Theorem.

Theorem 3.7. Let H be a locally compact group and K be an LCA group also τ : H → Aut(K) be a continuous homomorphism and let u ∈ L²(K) be a window function. The generalized τ ×bτ-continuous Gabor transform V_u^† : L²(G_τ)→L²(G_τ×

bτ)is a multiple of an isometric transform which mapsL²(G_τ)onto a closed subspace ofL²(G_τ×

bτ).

Proof. Letu∈L²(K) be a window function and alsof ∈L²(G_τ). Using Fubini’s Theorem, Plancherel formula (2.9) and also (2.3), (3.2) we have

kV_u^†fk²_L2(Gτ×τb)= Z

G_τ×bτ

|V_u^†f(h, k, ω)|²dµGτ×τb(h, k, ω)

= Z

H

Z

K

Z

Kb

|V_u^†f(h, k, ω)|²dhdkdω

= Z

H

Z

K×Kb

|hfh, %(k^h, ωh)ui_L2(K)|²dkdω

δ(h)dh

= Z

H

Z

K×Kb

|hfh, %(k, ω)uiL²(K)|²dk^h−1dω_h−1

δ(h)dh

= Z

H

Z

K×Kb

|hfh, %(k, ω)ui_L2(K)|²dkdω

δ(h)dh

=kuk²_L2(K)

Z

H

kfhk²_L2(K)δ(h)dh=kuk²_L2(K)kfk²_L2(G_τ). Thus,kuk⁻²_L2(K)V_u^† :L²(Gτ)→L²(G_τ×

bτ) is an isometric transform with a closed range inL²(G_τ×

bτ).

Corollary 3.8. The generalizedτ×bτ-continuous Gabor transform defined in (3.9), for allf, g∈L²(Gτ)and window functionsu, v ∈L²(K)satisfies the following orthogonality relation;

(3.10) hV_u^†f,V_v^†gi_L2(Gτ×τb)=hv, ui_L2(K)hf, gi_L2(Gτ).

In the next proposition we prove an inversion formula for the generalizedτ×bτ-continuous Gabor transform given in (3.9).

Proposition 3.9. LetH be a locally compact group andKbe an LCA group also letτ :H →Aut(K)be a continuous homomorphism and u∈L²(K) with ub∈ L¹(K). Everyb f ∈L²(Gτ) with cfh ∈L¹(K)b for a.e. h∈ H, satisfies the following reconstruction formula;

(3.11) f(h, k) =

Z

K×Kb

V_u^†f(h, s, ω)[%(s^h, ωh)u](k)dsdω Proof. Using (2.11) for a.e. h∈H we have

f_h(k) = Z

K×Kb

V_uf_h(s, ω)[%(s, ω)u](k)dsdω

= Z

K

Z

Kb

Vufh(s, ωh)[%(s, ωh)u](k)dωh

ds

=δ(h) Z

Kb

Z

K

Vufh(s, ωh)[%(s, ωh)u](k)ds

dω

=δ(h) Z

Kb

Z

K

Vufh(s^h, ωh)[%(s^h, ωh)u](k)ds^h

dω= Z

K×Kb

V_u^†f(h, s, ω)[%(s^h, ωh)u](k)dsdω.

Remark 3.10. It is also possible to define different variants of the Gabor transform as we defined in (3.6) and (3.9), with similar properties. Let transformsAu andBu forf ∈L²(Gτ) be given by

(3.12) Auf(h, k, ω) =Vufh(k^h, ω) Buf(h, k, ω) =δ(h)Vufh(k, ωh).

It can be checked that transforms given in (3.12) satisfy the Plancherel theorem and the following inversion formulas;

(3.13) f(h, k) =δ(h)⁻¹ Z

K×Kb

Auf(h, k, ω)[%(s^h, ω)](k)dsdω, f(h, k) = Z

K×Kb

Buf(h, k, ω)[%(s, ωh)](k)dsdω.

4. τ⊗τb-continuous Gabor transform

In this section we introduce another Gabor transform which we call it the τ⊗bτ-continuous Gabor transform. In theτ⊗bτ-Gabor theory we can choose elements ofL²(Gτ) as window functions.

Again let H be a locally compact group and K be an LCA group also let τ : H → Aut(K) be a continuous homomorphism. Defineτ^⊗ =τ⊗bτ:H×H →Aut(K×K) via (h, t)b 7→τ_(h,t)^⊗ given by

(4.1) τ_(h,t)^⊗ (k, ω) := (τh(k),bτt(ω)) = (k^h, ωt),

for all (k, ω)∈K×K. Then, for all (h, t)b ∈H×H we getτ_(h,t)^⊗ ∈Aut(K×K). Because for all (k, ω),b (k⁰, ω⁰)∈K×Kb we have

τ_(h,t)^⊗ ((k, ω)(k⁰, ω⁰)) =τ_(h,t)^⊗ (k+k⁰, ωω⁰)

= (k+k⁰)^h,(ωω⁰)t

= k^h+k^0h, ω_tω⁰_t

= (k^h, ωt)(k^0h, ω⁰_t) =τ_(h,t)^⊗ (k, ω)τ_(h,t)^⊗ (k⁰, ω⁰).

As well as τ^⊗ = τ ⊗bτ : H ×H → Aut(K×K) defined by (h, t)b 7→ τ_(h,t)^⊗ is a homomorphism, because for all (h, t),(h⁰, t⁰)∈H×H and also all (k, ω)∈K×Kb we have

τ_(h,t)(h^⊗ 0,t⁰)(k, ω) =τ_(hh^⊗ 0,tt⁰)(k, ω)

= (k^hh0, ωtt⁰)

=

(k^h0)^h,(ωt⁰)t

=τ_(h,t)^⊗ (k^h0, ωt⁰) =τ_(h,t)^⊗ τ_(h^⊗0,t⁰)(k, ω).

Hence, we can prove the following interesting theorem.

Theorem 4.1. Let H be a locally compact group and K be an LCA group also letτ :H →Aut(K)be a continuous homomorphism. The semi-direct product G_τ⊗

τb = (H×H)nτ⊗bτ

K×Kb

is a locally compact group with the left Haar measure

(4.2) dµG_τ⊗bτ(h, t, k, ω) =δ(h)δ(t)⁻¹dhdtdkdω, and also Φ :Gτ×G

bτ→G_τ⊗

bτ given by

(4.3) (h, k, t, ω)7→Φ(h, k, t, ω) := (h, t, k, ω) is a topological group isomorphism.

Proof. Using Theorem 26.9 of [21], homomorphismτ⊗τb:H×H →Aut(K×K) given in (4.1) is continuous. Therefore,b G_τ⊗

τb= (H×H)nτ⊗bτ

K×Kb

is a locally compact group. Also,dµ_Gτ⊗bτ(h, t, k, ω) =δ(h)δ(t)⁻¹dhdtdkdω is a left Haar measure for G_τ⊗

τb. Indeed, due to (2.3) and (3.2) for all (h, t)∈H×H we have dτ_(h,t)^⊗ (k, ω) =d(k^h, ωt)

=dk^hdω_t

=δ(h)⁻¹dkδ(t)dω=δ(h)⁻¹δ(t)d(k, ω).

Theτ⊗bτ-group law for all (h, t, k, ω),(h⁰, t⁰, k⁰, ω⁰)∈G_τ⊗

bτ is (h, t, k, ω)nτ⊗bτ(h⁰, t⁰, k⁰, ω⁰) =

(hh⁰, tt⁰),(k, ω)τ_(h,t)^⊗ (k⁰, ω⁰)

= (hh⁰, tt⁰),(k, ω)(k^0h, ω⁰_t)

= (hh⁰, tt⁰, k+k^0h, ωω⁰_t).

It is clear that Φ : Gτ ×G

τb → G_τ⊗

bτ is a homeomorphism. It is also a group homomorphism, because for all (h, k, t, ω),(h⁰, k⁰, t⁰, ω⁰) in Gτ×G

bτ we get

Φ[(h, k, t, ω)(h⁰, k⁰, t⁰, ω⁰)] = Φ[(h, k)nτ(h⁰, k⁰),(t, ω)n_bτ(t⁰, ω⁰)]

= Φ[(hh⁰, k+k^0h),(tt⁰, ωω_t⁰)]

= (hh⁰, tt⁰, k+k^0h, ωω⁰_t) = (h, t, k, ω)nτ⊗τb(h⁰, t⁰, k⁰, ω⁰).

We call the semi-direct productG_τ⊗

bτas theτ⊗τb-time frequency group associated toG_τwhich is preciselyG_τ×G

τb. Thus, form now on we use the locally compact group Gτ×G

bτ instead of the semi-direct productG_τ⊗

bτ.

Letg∈L²(Gτ) be a window function andf ∈L²(Gτ). Theτ⊗bτ-continuous Gabor transformoff with respect to the window functiong is defined by

(4.4) Ggf(h, k, t, ω) :=δ(t)Vg_hft(k, ω) =δ(t)hft, %(k, ω)ghiL²(K).

Theτ⊗bτ-continuous Gabor transform given in (4.4) satisfies the following Plancherel Theorem.

Theorem 4.2. Let H be a locally compact group, K be an LCA group and τ : H → Aut(K) be a continuous homomorphism also Gτ = H nτ K and let g ∈ L²(Gτ) be a window function. The continuous Gabor transform Gg : L²(Gτ)→ L²(Gτ×G

bτ) is a multiple of an isometric transform which maps L²(Gτ) onto a closed subspace of L²(G_τ×G

τb).

Proof. Letg∈L²(G_τ) be a window function and also letf ∈L²(G_τ).

kG_gfk²_L2(G_τ×Gτ_b)= Z

Gτ×G

τb

|G_gf(h, k, t, ω)|²dµ_Gτ×G

τb(h, k, t, ω)

= Z

Gτ

Z

Gτ_b

|Ggf(h, k, t, ω)|²dµG_τ(h, k)dµG

τb(t, ω)

= Z

H

Z

K

Z

H

Z

Kb

|Ggf(h, k, t, ω)|²δ(h)dhdkδ(t)⁻¹dtdω

= Z

H

Z

H

Z

K×Kb

|hft, %(k, ω)ghi_L2(K)|²dkdω

δ(h)dhδ(t)dt

= Z

H

Z

H

kftk²_L2(K)kghk²_L2(K)δ(h)dhδ(t)dt=kfk²_L2(G_τ)kgk²_L2(G_τ)

Thus,kgk⁻²_L2(G_τ)Gg:L²(Gτ)→L²(Gτ×G

τb) is an isometric transform with a closed range inL²(Gτ×G

bτ).

Corollary 4.3. The τ×bτ-continuous Gabor transform defined in (4.4), for allf, f⁰∈L²(Gτ)and window functions g, g⁰ ∈L²(Gτ)satisfies the following orthogonality relation;

(4.5) hGgf,Gg⁰f⁰i_L2(Gτ×G_bτ)=hg⁰, gi_L2(Gτ)hf, f⁰i_L2(Gτ). In the following proposition we also prove an inversion formula.

Proposition 4.4. LetH be a locally compact group andKbe an LCA group also letτ :H →Aut(K)be a continuous homomorphism. Everyf, g∈L²(Gτ)withcfh,gbh∈L¹(K)b for a.e. h∈H, satisfy the following reconstruction formula;

(4.6) f(t, k) =hgh, ghi⁻¹_L2(K)δ(t)⁻¹ Z

K×Kb

Ggf(h, s, t, ω)[%(s, ω)gh](k)dsdω, for a.e. h, t∈H andk∈K. In particular, for a.e. h∈H we have

(4.7) f(h, k) =hgh, ghi⁻¹_L2(K)δ(h)⁻¹ Z

K×Kb

Ggf(h, s, h, ω)[%(s, ω)gh](k)dsdω.

Proof. Using (2.11) for a.e. h, t∈H we have f_t(k) =hgh, g_hi⁻¹_L2(K)

Z

K×Kb

V_ghf_t(s, ω)[%(s, ω)g_h](k)dsdω

=hgh, ghi⁻¹_L2(K)δ(t)⁻¹ Z

K×Kb

Ggf(h, s, t, ω)[%(s, ω)gh](k)dsdω.

Let g ∈ L²(Gτ) be a window function and f ∈ L²(Gτ). Thegeneralized τ⊗bτ-continuous Gabor transform of f with respect to the window function gis defined by

(4.8) G_g^†f(h, k, t, ω) :=δ(h)^−1/2δ(t)^3/2Vg_hft(k^h, ωt) =δ(h)^−1/2δ(t)^3/2hft, %(k^h, ωt)ghiL²(K).

In the next theorem, a Plancherel formula for the generalized τ⊗bτ-continuous Gabor transform defined in (4.8) proved.

Theorem 4.5. Let H be a locally compact group, K be an LCA group and τ : H → Aut(K) be a continuous homomorphism also Gτ =HnτK and also let g∈L²(Gτ)be a window function. The generalized continuous Gabor transform G_g^† : L²(Gτ) → L²(Gτ ×G

bτ) is a multiple of an isometric transform which maps L²(Gτ) onto a closed subspace of L²(Gτ×G

bτ).

Proof. Letg∈L²(G_τ) be a window function and also letf ∈L²(G_τ). Using Fubini’s theorem and also Theorem we achieve

kGgfk²_L2(Gτ×G_τb)= Z

G_τ×Gτ_b

|G_g^†f(h, k, t, ω)|²dµGτ×G

τb(h, k, t, ω)

= Z

G_τ

Z

G

τb

|G_g^†f(h, k, t, ω)|²dµ_Gτ(h, k)dµ_G

bτ(t, ω)

= Z

H

Z

K

Z

H

Z

Kb

|G_g^†f(h, k, t, ω)|²δ(h)dhdkδ(t)⁻¹dtdω

= Z

H

Z

H

Z

K×Kb

|hf_t, %(k^h, ω_t)g_hi_L2(K)|²dkdω

dhδ(t)²dt

= Z

H

Z

H

Z

K×Kb

|hft, %(k, ω)ghi_L2(K)|²dk^h−1dω_t⁻¹

dhδ(t)²dt

= Z

H

Z

H

Z

K×Kb

|hf_t, %(k, ω)g_hi_L2(K)|²dkdω

δ(h)dhδ(t)dt

= Z

H

Z

H

kftk²_L2(K)kghk²_L2(K)δ(h)dhδ(t)dt=kfk²_L2(Gτ)kgk²_L2(Gτ)

Thus,kgk⁻²_L2(G

τ)G_g^†:L²(G_τ)→L²(G_τ×G

bτ) is an isometric transform with a closed range inL²(G_τ×G

bτ).

Corollary 4.6. The τ×bτ-continuous Gabor transform defined in (4.8), for allf, f⁰∈L²(Gτ)and window functions g, g⁰ ∈L²(Gτ)satisfies the following orthogonality relation;

(4.9) hG_g^†f,G_g^†0f⁰i_L2(Gτ×G_bτ)=hg⁰, gi_L2(Gτ)hf, f⁰i_L2(Gτ).

Also, the generalizedτ⊗bτ-continuous Gabor transform satisfies the following inversion formula.

Proposition 4.7. LetH be a locally compact group andKbe an LCA group also letτ :H →Aut(K)be a continuous homomorphism. Every f, g ∈ L²(Gτ) with cfh,gbh ∈ L¹(K)b for a.e. h, t ∈ H, satisfy the following reconstruction formula;

(4.10) f(t, k) =hgh, ghi⁻¹_L2(K)δ(h)^−1/2δ(t)^−1/2 Z

K×Kb

G^†_gf(h, s, t, ω)[%(k^h, ωt)gh](k)dsdω,

for a.e. h, t∈H andk∈K. In particular for a.e. h∈H we have

(4.11) f(h, k) =hgh, ghi⁻¹_L2(K)δ(h)⁻¹ Z

K×Kb

G_g^†f(h, s, h, ω)[%(s^h, ωh)gh](k)dsdω.

Proof. Using (2.11) for a.e. h, t∈H we have ft(k) =hgh, ghi⁻¹_L2(K)

Z

K×Kb

Vg_hft(s, ω)[%(s, ω)gh](k)dsdω

=hgh, ghi⁻¹_L2(K)

Z

K

Z

Kb

Vg_hft(s, ω)[%(s, ω)gh](k)dωds

=hg_h, g_hi⁻¹_L2(K) Z

K

Z

Kb

V_ghf_t(s, ω_t)[%(s, ω_t)g_h](k)dω_t

ds

=hgh, ghi⁻¹_L2(K)δ(t) Z

Kb

Z

K

Vg_hft(s^h, ωt)[%(s^h, ωt)gh](k)ds^h

dω

=hgh, ghi⁻¹_L2(K)δ(h)⁻¹δ(t) Z

Kb

Z

K

Vg_hft(s^h, ωt)[%(s^h, ωt)gh](k)ds

dω

=hgh, g_hi⁻¹_L2(K)δ(h)^−1/2δ(t)^−1/2 Z

K×Kb

Ggf(h, s, t, ω)[%(s^h, ω_t)g_h](k)dsdω.

5. Examples and applications

5.1. The Affine group ax+b. Let H =R^∗+ = (0,+∞) and K =R. The affine group ax+b is the semi direct product Hnτ K with respect to the homomorphism τ : H → Aut(K) given bya 7→ τa, where τa(x) = ax for all x∈R. Hence, the underlying manifold of the affine group is (0,∞)×Rand also the group law is

(5.1) (a, x)nτ(a⁰, x⁰) = (aa⁰, x+ax⁰).

The continuous homomorphism δ:H →(0,∞) is given by δ(a) =a⁻¹ and so that the left Haar measure is in fact dµ_Gτ(a, x) =a⁻²dadx. Due to Theorem 4.5 of [10] we can identifyRb withRviaω(x) =hx, ωi=e^2πiωxfor eachω∈Rb and so we can consider the continuous homomorphism bτ:H →Aut(K) given byb a7→bτ_a via

hx, ωai=hx,τba(ω)i

=hτ_a⁻¹(x), ωi=ha⁻¹x, ωi=e^2πiωa−1x. Thus,G

τbhas the underlying manifold (0,∞)×R, with the group law given by (5.2) (a, ω)n_bτ(a⁰, ω⁰) = (aa⁰, ωω_a⁰), Due to Theorem 3.1 the left Haar measure dµ_G

bτ(a, ω) is precisely dadω. Theτ×bτ-time frequency groupG_τ×

τbhas the underlying manifold (0,∞)×R×Rb and the group law is

(5.3) (a, x, ω)nτ×bτ(a⁰, x⁰, ω⁰) = (aa⁰, x+ax⁰, ωω⁰_a), with the left Haar measure dµ_Gτ×

τb(a, x, ω) =a⁻¹dadxdω. The geometry of this locally compact group and also the wave packet approaches of this locally compact group was studied in [2, 20]. If u∈L²(R) is a window function and alsof ∈L²(G_τ). According to (3.6) we have

V_uf(a, x, ω) =δ(a)^1/2V_uf_a(x, ω)

=a^−1/2hf_a, %(x, ω)ui_L2(R)

=a^−1/2 Z ∞

−∞

f(a, y)[%(x, ω)u](y)dy

=a^−1/2 Z ∞

−∞

f(a, y)u(y−x)ω(y)dy=a^−1/2 Z ∞

−∞

f(a, y)u(y−x)e^−2πiωydy.

Using Theorem 3.4, if kukL²(R)= 1 we get (5.4)

Z ∞

0

Z ∞

−∞

Z ∞

−∞

|Vuf(a, x, ω)|²

a dadxdω=

Z ∞

0

Z ∞

−∞

|f(a, x)|² a² dadx.

Due to the reconstruction formula (3.8) if for a.e. a∈(0,∞) we havefba∈L¹(R), then for a.e. x∈Rwe get f(a, x) =δ(a)^−1/2kuk⁻²_L2(K)

Z ∞

−∞

Z ∞

−∞

V_uf(a, y, ω)[%(y, ω)u](x)dydω

=a^1/2kuk⁻²_L2(K)

Z ∞

−∞

Z ∞

−∞

V_uf(a, y, ω)u(x−y)e^2πiωxdydω.

As well as according to (3.9) we have

V_u^†f(a, x, ω) =δ(a)^1/2Vufa(x^a, ωa)

=a^−1/2hfa, %(x^a, ωa)ui_L2(R)

=a^−1/2 Z ∞

−∞

f(a, y)[%(x^a, ω_a)u](y)dy

=a^−1/2 Z ∞

−∞

f(a, y)u(y−ax)ω_a(y)dy=a^−1/2 Z ∞

−∞

f(a, y)u(y−ax)e^{−2πiωa−1y}dy.

Using Theorem 3.7, if kuk_L2(R)= 1 we get (5.5)

Z ∞

0

Z ∞

−∞

Z ∞

−∞

|V_u^†f(a, x, ω)|²

a dadxdω=

Z ∞

0

Z ∞

−∞

|f(a, x)|² a² dadx.

Due to the reconstruction formula (3.11) if for a.e. a∈(0,∞) we havefba ∈L¹(R), then forx∈Rwe achieve f(a, x) =

Z ∞

−∞

Z ∞

−∞

V_u^†f(a, y, ω)[%(y^a, ωa)u](x)dydω

= Z ∞

−∞

Z ∞

−∞

V_u^†f(a, y, ω)u(x−ay)e^2πiωa−1xdydω.

Example 5.1. LetN >0 and alsouN =χ_[−N,N]be a window function with compact support andkuNk_L2(R)= 2N. Then, for allf ∈L²(Gτ) and (a, x, ω)∈G_τ×

τbwe have V_uNf(a, x, ω) =a^−1/2

Z ∞

−∞

f(a, y)u_N(y−x)ω(y)dy

=a^−1/2ω(x) Z ∞

−∞

f(a, y+x)uN(y)ω(y)dy

=a^−1/2ω(x) Z N

−N

f(a, y+x)ω(y)dy=a^−1/2e^−2πiωx Z N

−N

f(a, y+x)e^−2πiωydy.

If we set x= 0, then we get

VuNf(a,0, ω) =a^−1/2 Z N

−N

f(a, y)e^−2πiωydy.

Similarly, for the generalized τ×τb-continuous Gabor transform we have V_u^†_Nf(a, x, ω) =a^−1/2

Z ∞

−∞

f(a, y)uN(y−ax)e^{−2πiωa−1y}dy

=a^−1/2e^−2πiωx Z ∞

−∞

f(a, y+ax)uN(y)e^{−2πiωa−1y}dy=a^−1/2e^−2πiωx Z N

−N

f(a, y+ax)e^{−2πiωa−1y}dy.