Continuous Partial Gabor Transform for Semi-Direct Product of Locally Compact Groups

Abstract

Let $H$ be a locally compact group, $K$ be an LCA group, $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism and $G_\tau =H\ltimes _\tau K$ be the semi-direct product of $H$ and $K$ with respect to the continuous homomorphism $\tau$. In this article, we introduce the $\tau \times \widehat{\tau }$-time frequency group $G_{\tau \times \widehat{\tau }}$. We define the $\tau \times \widehat{\tau }$-continuous Gabor transform of $f\in L^2(G_\tau )$ with respect to a window function $u\in L^2(K)$ as a function defined on $G_{\tau \times \widehat{\tau }}$. It is also shown that the $\tau \times \widehat{\tau }$-continuous Gabor transform satisfies the Plancherel Theorem and reconstruction formula. This approach is tailored for choosing elements of $L^2(G_\tau )$ 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

$$\begin{aligned} G\{x\}(y,\omega )=\int _{-\infty }^{+\infty }x(t)e^{-\pi (t-y)^2}e^{-2\pi i\omega t}\mathrm{d}t. \end{aligned}$$

(1.1)

Due to (1.1) the Gabor transform of a signal $x(t)$ is a function defined on $\mathbb {R}\times \widehat{\mathbb {R}}$ called the time–frequency plane. There is also standard extension of the continuous Gabor transform of a signal $x(\mathbf{t})$ on $\mathbb {R}^n$ which is defined for $(\mathbf{y},\mathbf{w})\in \mathbb {R}^n\times \widehat{\mathbb {R}^n}$ by (see [9, 16])

$$\begin{aligned} G\{x\}(\mathbf{y},\mathbf{w})=\int _{\mathbb {R}^n}x(\mathbf{t})e^{-\pi \Vert \mathbf{t}-\mathbf{y}\Vert ^2}e^{-2\pi i\mathbf{w}.\mathbf{t}}\mathrm{d}\mathbf{t}. \end{aligned}$$

(1.2)

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 Gröchenig [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–Gröchenig theory (coorbit space theory). In view of voice transform and the coorbit space theory, the continuous Gabor transform for an LCA group $G$ is precisely the voice transform generated by the Schrödinger representation of the Weyl–Heisenberg group associate with $G$ (see [4–6, 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]). Large class of non-Abelian locally compact groups is the class consist of semi-direct product group 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 be $G\times \widehat{G}$, where $\widehat{G}$ stands for the set of all equivalence class of irreducible continuous unitary representations of $G$. It is clear that this extension will not be appropriate from 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 Sect. 3 we assume that $H$ is a locally compact group and $K$ is an LCA group, $\tau :H\rightarrow Aut(K)$ is a continuous homomorphism and $G_\tau =H\ltimes _\tau {K}$. We define the $\tau \times \widehat{\tau }$-time frequency group $G_{\tau \times \widehat{\tau }}$ and the $\tau \times \widehat{\tau }$-continuous Gabor transform of $f\in L^2(G_\tau )$ with respect to a window function $u\in L^2(K)$. We also prove a Plancherel and inversion formula for the $\tau \times \widehat{\tau }$-continuous Gabor transform. To choose elements of $L^2(G_\tau )$ as window functions we define the $\tau \otimes \widehat{\tau }$-time frequency group $G_{\tau \otimes \widehat{\tau }}$ and also the $\tau \otimes \widehat{\tau }$-continuous Gabor transform in Sect. 4. As an application, we study this theory on the affine group, Weyl–Heisenberg group and the Euclidean groups in Sect. 5.

2 Preliminaries and Notations

Let $H$ and $K$ be locally compact groups with identity elements $e_H$ and $e_K$, respectively, and left Haar measures $\mathrm{d}h$ and $dk$ respectively, also let $\tau :H\rightarrow Aut(K)$ be a homomorphism such that the map $(h,k)\mapsto \tau _h(k)$ is continuous from $H\times K$ onto $K$. There is a natural topology, sometimes called Braconnier topology, turning $Aut(K)$ into a Hausdorff topological group(not necessarily locally compact), which is defined by the sub-base of identity neighborhoods

$$\begin{aligned} \mathcal {B}(F,U)= \{ \alpha \in Aut(K): \alpha (k),\alpha ^{-1}(k)\in Uk\ \forall k\in F\}, \end{aligned}$$

(2.1)

where $F\subseteq K$ is compact and $U\subseteq K$ is an identity neighborhood. Continuity of a homomorphism $\tau :H\rightarrow Aut(K)$ is equivalent with the continuity of the map $(h,k)\mapsto \tau _h(k)$ from $H\times K$ onto $K$ (see [22]).

The semi-direct product $G_\tau =H\ltimes _\tau K$ is the locally compact topological group with the underlying set $H\times K$ which is equipped by the product topology and also the group operation is defined by

$$\begin{aligned} (h,k)\ltimes _\tau (h',k')=(hh',k\tau _h(k'))\quad \mathrm{and}\quad (h,k)^{-1}=(h^{-1},\tau _{h^{-1}}(k^{-1})). \end{aligned}$$

(2.2)

If $H_1=\{(h,e_K):h\in H\}$ and $K_1=\{(e_H,k):k\in K\}$ then $K_1$ is a closed normal subgroup and $H_1$ is a closed subgroup of $G_\tau$. The left Haar measure of $G_\tau$ is $\mathrm{d}\mu _{G_\tau }(h,k)=\delta (h)\mathrm{d}h\mathrm{d}k$ and the modular function $\Delta _{G_\tau }$ is

$$\begin{aligned} \Delta _{G_\tau }(h,k)=\delta (h)\Delta _H(h)\Delta _K(k), \end{aligned}$$

where the positive continuous homomorphism $\delta :H\rightarrow (0,\infty )$ is given by [21]

$$\begin{aligned} \mathrm{d}k=\delta (h)\mathrm{d}(\tau _h(k)). \end{aligned}$$

(2.3)

From now on, for all $p\ge 1$ we denote by $L^p(G_\tau )$ the Banach space $L^p(G_\tau ,\mu _{G_\tau })$ and also $L^p(K)$ stands for $L^p(K,dk)$. When $f\in L^p(G_\tau )$, for a.e. $h\in H$ the function $f_h$ defined on $K$ via $f_h(k):=f(h,k)$ belongs to $L^p(K)$ (see [11]).

If $K$ is an LCA group all irreducible representations of $K$ are one-dimensional. Thus, if $\pi$ is an irreducible unitary representation of $K$ we have $\mathcal {H}_\pi =\mathbb {C}$ and also according to the Schur’s Lemma there exists a continuous homomorphism $\omega$ of $K$ into the circle group $\mathbb {T}$ such that for each $k\in K$ and $z\in \mathbb {C}$ we have $\pi (k)(z)=\omega (k)z$. Such homomorphisms are called characters of $K$ and the set of all characters of $K$ denoted by $\widehat{K}$. If $\widehat{K}$ equipped by the topology of compact convergence on $K$ which coincides with the $w^*$-topology that $\widehat{K}$ inherits as a subset of $L^\infty (K)$, then $\widehat{K}$ with respect to the point-wise product of characters is an LCA group which is called the dual group of $K$. The linear map $\mathcal {F}_K:L^1(K)\rightarrow \mathcal {C}(\widehat{K})$ defined by $v\mapsto \mathcal {F}_K(v)$ via

$$\begin{aligned} \mathcal {F}_K(v)(\omega )=\widehat{v}(\omega )=\int _Kv(k) \overline{\omega (k)}\mathrm{d}k, \end{aligned}$$

(2.4)

is called the Fourier transform on $K$. It is a norm-decreasing $*$-homomorphism from $L^1(K)$ into $\mathcal {C}_0(\widehat{K})$ with a uniformly dense range in $\mathcal {C}_0(\widehat{K})$ (Proposition 4.13 of [10]). If $\phi \in L^1(\widehat{K})$, the function defined $a.e.$ on $K$ by

$$\begin{aligned} \breve{\phi }(x)=\int _{\widehat{K}}\phi (\omega )\omega (x)\mathrm{d}\omega , \end{aligned}$$

(2.5)

belongs to $L^\infty (K)$ and also for all $f\in L^1(K)$ we have the following orthogonality relation (Parseval formula);

$$\begin{aligned} \int _Kf(k)\overline{\breve{\phi }(k)}\mathrm{d}k=\int _{\widehat{K}} \widehat{f}(\omega )\overline{\phi (\omega )}\mathrm{d}\omega . \end{aligned}$$

(2.6)

The Fourier transform (2.4) on $L^1(K)\cap L^2(K)$ is an isometric transform and it extends uniquely to a unitary isomorphism from $L^2(K)$ onto $L^2(\widehat{K})$ (Theorem 4.25 of [10]) also each $v\in L^1(K)$ with $\widehat{v}\in L^1(\widehat{K})$ satisfies the following Fourier inversion formula (Theorem 4.32 of [10]);

$$\begin{aligned} v(k)=\int _{\widehat{K}}\widehat{v}(\omega )\omega (k)\mathrm{d}\omega \ \mathrm{for} \ \mathrm{a.e.}\ k\in K. \end{aligned}$$

(2.7)

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 $T_sv(k)=v(k-s)$ for all $k,s\in K$ and also the modulation (frequency-shifts) operator is given by $M_\omega v(k)=\omega (k)v(k)$ for all $\omega \in \widehat{K}$, $k\in K$. The time–frequency shift operator is defined on the time–frequency plane (time–frequency group) $K\times \widehat{K}$ by $\varrho (k,\omega )=M_{\omega }T_k$ for all $(k,\omega )\in K\times \widehat{K}$.

Given an appropriate window function $u\in L^2(K)$ on $K$, the short time Fourier transform (STFT) or the continuous Gabor transform of $v\in L^2(K)$ is given by

$$\begin{aligned} V_uv(s,\omega )=\int _{K}v(k)\overline{u}(k-s)\overline{\omega (k)}\mathrm{d}k=\langle v,\varrho (s,\omega )u\rangle _{L^2(K)}. \end{aligned}$$

(2.8)

The continuous Gabor transform (2.8) satisfies the following Plancherel formula

$$\begin{aligned} \int _{K\times \widehat{K}}|V_uv(s,\omega )|^2\mathrm{d}s\mathrm{d}\omega = \Vert u\Vert _{L^2(K)}^2\Vert v\Vert _{L^2(K)}^2, \end{aligned}$$

(2.9)

for all $u,v\in L^2(K)$ (see [17]). If $u,u'\in L^2(K)$ with $\langle u,u'\rangle _{L^2(K)}\not =0$, then each $v\in L^2(K)$ satisfies the following inversion formula in the weak sense (see [16])

$$\begin{aligned} v=\langle u,u'\rangle _{L^2(K)}^{-1}\int _{K\times \widehat{K}} V_uv(k,\omega )\varrho (k,\omega )u'\mathrm{d}k\mathrm{d}\omega . \end{aligned}$$

(2.10)

If a window function $u\in L^2(K)$ has Fourier transform $\widehat{u}$ in $L^1(\widehat{K})$, then each $v\in L^2(K)$ with $\widehat{v}\in L^1(\widehat{K})$ satisfies the following inversion formula;

$$\begin{aligned} v(s)=\Vert u\Vert _{L^2(K)}^{-2}\int _{K\times \widehat{K}}V_uv(k,\omega ) [\varrho (k,\omega )u](s)\mathrm{d}k\mathrm{d}\omega , \end{aligned}$$

(2.11)

for all $s\in K$.

3 $\tau \times \widehat{\tau }$-Continuous Gabor Transform

Throughout this paper, let $H$ be a locally compact group and $K$ be an LCA group also let $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism and $G_\tau =H\ltimes _\tau K$. For simplicity in notations we use $k^h$ instead of $\tau _h(k)$ for all $h\in H$ and $k\in K$. In this section we introduce the $\tau \times \widehat{\tau }$-time frequency group and also we define the $\tau \times \widehat{\tau }$-continuous Gabor transform of $f\in L^2(G_\tau )$ with respect to a window function in $L^2(K)$.

Define $\widehat{\tau }:H\rightarrow Aut(\widehat{K})$ via $h\mapsto \widehat{\tau }_h$, given by

$$\begin{aligned} \widehat{\tau }_h(\omega ):=\omega _h=\omega \circ \tau _{h^{-1}} \end{aligned}$$

(3.1)

for all $\omega \in \widehat{K}$, where $\omega _h(k)=\omega (\tau _{h^{-1}}(k))$ for all $k\in K$. If $\omega \in \widehat{K}$ and $h\in H$ we have $\omega _h\in \widehat{K}$, because for all $k,s\in K$ we have

$$\begin{aligned} \omega _h(ks)&=\omega \circ \tau _{h^{-1}}(ks)\\&=\omega (\tau _{h^{-1}}(ks))\\&=\omega (\tau _{h^{-1}}(k)\tau _{h^{-1}}(s))\\&=\omega (\tau _{h^{-1}}(k))\omega (\tau _{h^{-1}}(s))=\omega _h(k)\omega _h(s). \end{aligned}$$

According to (3.1) for all $h\in H$ we have $\widehat{\tau }_h\in Aut(\widehat{K})$. Because, if $k\in K$ and $h\in H$ then for all $\omega ,\eta \in \widehat{K}$ we have

$$\begin{aligned} \widehat{\tau }_h(\omega .\eta )(k)&=(\omega .\eta )_h(k)\\&=(\omega .\eta )\circ \tau _{h^{-1}}(k)\\&=\omega .\eta (\tau _{h^{-1}}(k))\\&=\omega (\tau _{h^{-1}}(k))\eta (\tau _{h^{-1}}(k))\\&=\omega _h(k)\eta _h(k)=\widehat{\tau }_h(\omega )(k) \widehat{\tau }_h(\eta )(k). \end{aligned}$$

Also $h\mapsto \widehat{\tau }_h$ is a homomorphism from $H$ into $Aut(\widehat{K})$, cause if $h,t\in H$ then for all $\omega \in \widehat{K}$ and $k\in K$ we get

$$\begin{aligned} \widehat{\tau }_{th}(\omega )(k)&=\omega _{th}(k)\\&=\omega (\tau _{(th)^{-1}}(k))\\&=\omega (\tau _{h^{-1}}\tau _{t^{-1}}(k))\\&=\omega _h(\tau _{t^{-1}}(k))\\&=\widehat{\tau }_h(\omega )(\tau _{t^{-1}}(k)) =\widehat{\tau }_t[\widehat{\tau }_h(\omega )](k). \end{aligned}$$

Thus, we can prove the following theorem.

Theorem 3.1

Let $H$ be a locally compact group and $K$ be an LCA group also $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism and let $\delta :H\rightarrow (0,\infty )$ be the positive continuous homomorphism satisfying $\mathrm{d}k=\delta (h)\mathrm{d}k^h$. The semi-direct product $G_{\widehat{\tau }}=H\ltimes _{\widehat{\tau }}\widehat{K}$ is a locally compact group with the left Haar measure $\mathrm{d}\mu _{G_{\widehat{\tau }}}(h,\omega )=\delta (h)^{-1}\mathrm{d}h\mathrm{d}\omega$.

Proof

Continuity of the homomorphism $\widehat{\tau }:H\rightarrow Aut(\widehat{K})$ given in (3.1) guaranteed by Theorem 26.9 of [21]. Hence, the semi-direct product $G_{\widehat{\tau }}=H\ltimes _{\widehat{\tau }}\widehat{K}$ is a locally compact group. We also claim that the Plancherel measure $\mathrm{d}\omega$ on $\widehat{K}$ for all $h\in H$ satisfies

$$\begin{aligned} \mathrm{d}\omega _h=\delta (h)\mathrm{d}\omega . \end{aligned}$$

(3.2)

Let $h\in H$ and $v\in L^1(K)$. Using (2.3) we have $v\circ \tau _h\in L^1(K)$ with $\Vert v\circ \tau _h\Vert _{L^1(K)}=\delta (h)\Vert v\Vert _{L^1(K)}$. Thus, for all $\omega \in \widehat{K}$ we achieve

$$\begin{aligned} \widehat{v\circ \tau _h}(\omega )&=\int _Kv \circ \tau _h(k)\overline{\omega (k)}\mathrm{d}k\\&=\int _Kv(k^h)\overline{\omega (k)}\mathrm{d}k\\&=\int _Kv(k)\overline{\omega _h(k)}\mathrm{d}k^{h^{-1}}\\&=\delta (h)\int _Kv(k)\overline{\omega _h(k)}\mathrm{d}k =\delta (h)\widehat{v}(\omega _h). \end{aligned}$$

Now let $v\in L^1(K)\cap L^2(K)$. Due to the Plancherel theorem (Theorem 4.25 of [10]) and also preceding calculation, for all $h\in H$ we get

$$\begin{aligned} \int _{\widehat{K}}|\widehat{v}(\omega )|^2\mathrm{d}\omega _h&= \int _{\widehat{K}}|\widehat{v}(\omega _{h^{-1}})|^2\mathrm{d}\omega \\&=\delta (h)^2\int _{\widehat{K}}|\widehat{v\circ \tau _{h^{-1}}} (\omega )|^2\mathrm{d}\omega \\&=\delta (h)^2\int _{{K}}|{v\circ \tau _{h^{-1}}}(k)|^2\mathrm{d}k\\&=\delta (h)^2\int _{{K}}|v(k)|^2\mathrm{d}k^h\\&=\delta (h)\int _{{K}}|v(k)|^2\mathrm{d}k=\int _{\widehat{K}} |\widehat{v}(\omega )|^2\delta (h)\mathrm{d}\omega , \end{aligned}$$

which implies (3.2). Therefore, $\mathrm{d}\mu _{G_{\widehat{\tau }}}(h,\omega )=\delta (h)^{-1}\mathrm{d}h\mathrm{d}\omega$ is a left Haar measure for $G_{\widehat{\tau }}=H\ltimes _{\widehat{\tau }}\widehat{K}$. $\square$

Remark 3.2

Due to (3.1) for all $k\in K$, $\omega \in \widehat{K}$ and $h,t\in H$ we have

$$\begin{aligned} k^{ht}=(k^t)^h,\omega _{ht}=(\omega _t)_h. \end{aligned}$$

(3.3)

Now we are in the position to introduce the $\tau \times \widehat{\tau }$-time frequency group. Define $\tau ^{\times }=\tau \times \widehat{\tau }:H\rightarrow Aut(K\times \widehat{K})$ via $h\mapsto \tau ^{\times }_h$ given by

$$\begin{aligned} \tau _h^\times (k,\omega ):=\left( \tau _h(k),\widehat{\tau }_h(\omega )\right) =(k^h,\omega _h), \end{aligned}$$

(3.4)

for all $(k,\omega )\in K\times \widehat{K}$. Then, for all $h\in H$ we have $\tau _h^\times \in Aut(K\times \widehat{K})$. Because for all $(k,\omega ),(k',\omega ')\in K\times \widehat{K}$ we have

$$\begin{aligned} \tau ^\times _{h}\left( (k,\omega )(k',\omega ')\right)&=\tau ^\times _{h}(kk',\omega \omega ')\\&=\left( (kk')^h,(\omega \omega ')_h\right) \\&=\left( k^hk'^h,\omega _h\omega '_h\right) \\&=(k^h,\omega _h)(k'^h,\omega '_h)=\tau ^\times _{h}(k,\omega ) \tau ^\times _{h}(k',\omega '). \end{aligned}$$

Also $\tau ^\times =\tau \times \widehat{\tau }:H\rightarrow Aut(K\times \widehat{K})$ defined by $h\mapsto \tau ^\times _{h}$ is a homomorphism, because for all $h,t\in H$ and all $(k,\omega )\in K\times \widehat{K}$ we have

$$\begin{aligned} \tau ^\times _{ht}(k,\omega )&=(k^{ht},\omega _{ht})\\&=\left( (k^{t})^h,(\omega _{t})_h\right) \\&=\tau ^\times _{h}(k^{t},\omega _{t})=\tau ^\times _{h} \tau ^\times _{t}(k,\omega ). \end{aligned}$$

In the following proposition we show that $G_{\tau \times \widehat{\tau }} =H\ltimes _{\tau \times \widehat{\tau }}(K\times \widehat{K})$ is a locally compact group.

Proposition 3.3

Let $H$ be a locally compact group and $K$ be an LCA group also $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism and let $\delta :H\rightarrow (0,\infty )$ be the positive continuous homomorphism satisfying $\mathrm{d}k=\delta (h)\mathrm{d}k^h$. The semi-direct product $G_{\tau \times \widehat{\tau }} =H\ltimes _{\tau \times \widehat{\tau }}(K\times \widehat{K})$ is a locally compact group with the left Haar measure

$$\begin{aligned} \mathrm{d}\mu _{G_{\tau \times \widehat{\tau }}}(h,k,\omega )=\mathrm{d}h\mathrm{d}k\mathrm{d}\omega . \end{aligned}$$

(3.5)

Proof

Continuity of the homomorphism $\tau \times \widehat{\tau }:H\rightarrow Aut(K\times \widehat{K})$ given in (3.4) guaranteed by Theorem 26.9 of [21]. Thus, the semi-direct product $G_{\tau \times \widehat{\tau }}=H\ltimes _{\tau \times \widehat{\tau }}(K\times \widehat{K})$ is a locally compact group. Due to (2.3), (3.2) and also (3.4), for all $h\in H$ we have

$$\begin{aligned} \mathrm{d}\tau ^{\times }_h(k,\omega )&=\mathrm{d}\left( k^h,\omega _h\right) \\&=\mathrm{d}k^h\mathrm{d}\omega _h\\&=\delta (h)^{-1}\mathrm{d}k\delta (h)\mathrm{d}\omega =\mathrm{d}k\mathrm{d}\omega =d(k,\omega ), \end{aligned}$$

which implies that $G_{\tau \times \widehat{\tau }}$ is a locally compact group with the left Haar measure $\mathrm{d}\mu _{G_{\tau \times \widehat{\tau }}}(h,k,\omega )=\mathrm{d}h\mathrm{d}k\mathrm{d}\omega$. $\square$

We call the semi-direct product $G_{\tau \times \widehat{\tau }}$ as the $\tau \times \widehat{\tau }$-time frequency group associated to $G_\tau$. According to (3.4) for each $(h,k,\omega ),(h',k',\omega ')\in G_{\tau \times \widehat{\tau }}$ we have

$$\begin{aligned} (h,k,\omega )\ltimes _{\tau \times \widehat{\tau }}(h',k',\omega ')&=\left( hh',(k,\omega )\tau _h^{\times }(k',\omega ')\right) \\&=\left( hh',(k,\omega )(\tau _h(k'),\omega '_h)\right) = (hh',k+k'^h,\omega \omega '_h). \end{aligned}$$

Let $u\in L^2(K)$ be a window function and $f\in L^2(G_\tau )$. The $\tau \times \widehat{\tau }$ -continuous Gabor transform of $f$ with respect to the window function $u$ is define by

$$\begin{aligned} \mathcal {V}_uf(h,k,\omega ):=\delta (h)^{1/2}V_{u}f_h(k,\omega )=\delta (h)^{1/2}\langle f_h,\varrho (k,\omega )u\rangle _{L^2(K)}. \end{aligned}$$

(3.6)

In the following theorem we prove a Plancherel formula for the $\tau \times \widehat{\tau }$-continuous Gabor transform defined in (3.6).

Theorem 3.4

Let $H$ be a locally compact group and $K$ be an LCA group also $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism and let $u\in L^2(K)$ be a window function. The $\tau \times \widehat{\tau }$-continuous Gabor transform $\mathcal {V}_u:L^2(G_\tau )\rightarrow L^2(G_{\tau \times \widehat{\tau }})$ is a multiple of an isometric transform which maps $L^2(G_\tau )$ onto a closed subspace of $L^2(G_{\tau \times \widehat{\tau }})$.

Proof

Let $u\in L^2(K)$ be a window function and also $f\in L^2(G_\tau )$. Using Fubini’s Theorem and also Plancherel formula (2.9) we have

$$\begin{aligned} \Vert \mathcal {V}_uf\Vert _{L^2(G_{\tau \times \widehat{\tau }})}^2&=\int _{G_{\tau \times \widehat{\tau }}}|\mathcal {V}_uf(h,k,\omega ) |^2\mathrm{d}\mu _{G_{\tau \times \widehat{\tau }}}(h,k,\omega )\\&=\int _H\int _K\int _{\widehat{K}}|\mathcal {V}_uf(h,k,\omega )|^2\mathrm{d}h\mathrm{d}k\mathrm{d}\omega \\&=\int _H\left( \int _{K\times \widehat{K}}|\langle f_h, \varrho (k,\omega )u\rangle _{L^2(K)}|^2\mathrm{d}k\mathrm{d}\omega \right) \delta (h)\mathrm{d}h\\&=\Vert u\Vert _{L^2(K)}^2\int _H\Vert f_h\Vert _{L^2(K)}^2\delta (h)\mathrm{d}h= \Vert u\Vert _{L^2(K)}^2\Vert f\Vert _{L^2(G_\tau )}^2. \end{aligned}$$

Therefore, $\Vert u\Vert _{L^2(K)}^{-2}\mathcal {V}_u:L^2(G_\tau )\rightarrow L^2(G_\tau \times G_{\widehat{\tau }})$ is an isometric transform with a closed range in $L^2(G_{\tau }\times G_{\widehat{\tau }})$. $\square$

Corollary 3.5

The $\tau \times \widehat{\tau }$-continuous Gabor transform defined in (3.6), for all $f,g\in L^2(G_\tau )$ and window functions $u,v\in L^2(K)$ satisfies the following orthogonality relation;

$$\begin{aligned} \langle \mathcal {V}_uf,\mathcal {V}_{v}g \rangle _{L^2(G_{\tau \times \widehat{\tau }})}=\langle v,u\rangle _{L^2(K)}\langle f,g\rangle _{L^2(G_\tau )}. \end{aligned}$$

(3.7)

The $\tau \times \widehat{\tau }$-continuous Gabor transform (3.6) satisfies the following inversion formula.

Proposition 3.6

Let $H$ be a locally compact group and $K$ be an LCA group also let $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism and $u\in L^2(K)$ with $\widehat{u}\in L^1(\widehat{K})$. Every $f\in L^2(G_\tau )$ with $\widehat{f_h}\in L^1(\widehat{K})$ for a.e. $h\in H$, satisfies the following reconstruction formula;

$$\begin{aligned} f(h,k)=\delta (h)^{-1/2}\Vert u\Vert ^{-2}_{L^2(K)} \int _{K\times \widehat{K}}\mathcal {V}_uf(h,s,\omega ) [\varrho (s,\omega )u](k)\mathrm{d}s\mathrm{d}\omega . \end{aligned}$$

(3.8)

Proof

Using (2.11) for a.e. $h\in H$ we have

$$\begin{aligned} f_h(k)&=\Vert u\Vert ^{-2}_{L^2(K)}\int _{K\times \widehat{K}} V_{u}f_h(s,\omega )[\varrho (s,\omega )u](k)\mathrm{d}s\mathrm{d}\omega \\&=\delta (h)^{-1/2}\Vert u\Vert ^{-2}_{L^2(K)}\int _{K\times \widehat{K}} \mathcal {V}_{u}f(h,s,\omega )[\varrho (s,\omega )u](k)\mathrm{d}s\mathrm{d}\omega . \end{aligned}$$

$\square$

We can also define the generalized form of the $\tau \times \widehat{\tau }$-continuous Gabor transform. Let $u\in L^2(K)$ be a window function and $f\in L^2(G_\tau )$. The generalized $\tau \times \widehat{\tau }$ -continuous Gabor transform of $f$ with respect to the window function $u$ is define by

$$\begin{aligned} \mathcal {V}^\dagger _uf(h,k,\omega ):=\delta (h)^{1/2}V_{u}f_h(k^h,\omega _h) =\delta (h)^{1/2}\langle f_h,\varrho (k^h,\omega _h)u\rangle _{L^2(K)}. \end{aligned}$$

(3.9)

The generalized $\tau \times \widehat{\tau }$-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 $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism and let $u\in L^2(K)$ be a window function. The generalized $\tau \times \widehat{\tau }$-continuous Gabor transform $\mathcal {V}^\dagger _u:L^2(G_\tau )\rightarrow L^2(G_{\tau \times \widehat{\tau }})$ is a multiple of an isometric transform which maps $L^2(G_\tau )$ onto a closed subspace of $L^2(G_{\tau \times \widehat{\tau }})$.

Proof

Let $u\in L^2(K)$ be a window function and also $f\in L^2(G_\tau )$. Using Fubini’s Theorem, Plancherel formula (2.9) and also (2.3), (3.2) we have

$$\begin{aligned} \Vert \mathcal {V}^\dagger _uf\Vert _{L^2(G_{\tau \times \widehat{\tau }})}^2&=\int _{G_{\tau \times \widehat{\tau }}}|\mathcal {V}^\dagger _uf(h,k,\omega ) |^2\mathrm{d}\mu _{G_{\tau \times \widehat{\tau }}}(h,k,\omega )\\&=\int _H\int _K\int _{\widehat{K}}|\mathcal {V}^\dagger _uf(h,k,\omega ) |^2\mathrm{d}h\mathrm{d}k\mathrm{d}\omega \\&=\int _H\left( \int _{K\times \widehat{K}}|\langle f_h, \varrho (k^h,\omega _h)u\rangle _{L^2(K)}|^2\mathrm{d}k\mathrm{d}\omega \right) \delta (h)\mathrm{d}h\\&=\int _H\left( \int _{K\times \widehat{K}}|\langle f_h,\varrho (k,\omega )u \rangle _{L^2(K)}|^2\mathrm{d}k^{h^{-1}}\mathrm{d}\omega _{h^{-1}}\right) \delta (h)\mathrm{d}h\\&=\int _H\left( \int _{K\times \widehat{K}}|\langle f_h,\varrho (k,\omega ) u\rangle _{L^2(K)}|^2\mathrm{d}k\mathrm{d}\omega \right) \delta (h)\mathrm{d}h\\&=\Vert u\Vert _{L^2(K)}^2\int _H\Vert f_h\Vert _{L^2(K)}^2\delta (h) \mathrm{d}h=\Vert u\Vert _{L^2(K)}^2\Vert f\Vert _{L^2(G_\tau )}^2. \end{aligned}$$

Thus, $\Vert u\Vert _{L^2(K)}^{-2}\mathcal {V}^\dagger _u:L^2(G_\tau )\rightarrow L^2(G_{\tau \times \widehat{\tau }})$ is an isometric transform with a closed range in $L^2(G_{\tau \times \widehat{\tau }})$. $\square$

Corollary 3.8

The generalized $\tau \times \widehat{\tau }$-continuous Gabor transform defined in (3.9), for all $f,g\in L^2(G_\tau )$ and window functions $u,v\in L^2(K)$ satisfies the following orthogonality relation;

$$\begin{aligned} \langle \mathcal {V}_u^\dagger f,\mathcal {V}_{v}^\dagger g\rangle _{L^2(G_{\tau \times \widehat{\tau }})}=\langle v,u\rangle _{L^2(K)}\langle f,g\rangle _{L^2(G_\tau )}. \end{aligned}$$

(3.10)

In the next proposition we prove an inversion formula for the generalized $\tau \times \widehat{\tau }$-continuous Gabor transform given in (3.9).

Proposition 3.9

Let $H$ be a locally compact group and $K$ be an LCA group also let $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism and $u\in L^2(K)$ with $\widehat{u}\in L^1(\widehat{K})$. Every $f\in L^2(G_\tau )$ with $\widehat{f_h}\in L^1(\widehat{K})$ for a.e. $h\in H$, satisfies the following reconstruction formula;

$$\begin{aligned} f(h,k)=\int _{K\times \widehat{K}}\mathcal {V}_u^\dagger f(h,s,\omega )[\varrho (s^h,\omega _h)u](k)\mathrm{d}s\mathrm{d}\omega \end{aligned}$$

(3.11)

Proof

Using (2.11) for a.e. $h\in H$ we have

$$\begin{aligned} f_h(k)&=\int _{K\times \widehat{K}}V_{u}f_h(s,\omega ) [\varrho (s,\omega )u](k)\mathrm{d}s\mathrm{d}\omega \\&=\int _{K}\left( \int _{\widehat{K}}V_{u}f_h(s,\omega _h) [\varrho (s,\omega _h)u](k)\mathrm{d}\omega _h\right) \mathrm{d}s\\&=\delta (h)\int _{\widehat{K}}\left( \int _KV_{u}f_h(s,\omega _h) [\varrho (s,\omega _h)u](k)\mathrm{d}s\right) \mathrm{d}\omega \\&=\delta (h)\int _{\widehat{K}}\left( \int _KV_{u}f_h(s^h,\omega _h) [\varrho (s^h,\omega _h)u](k)\mathrm{d}s^h\right) \mathrm{d}\omega \\&=\int _{K\times \widehat{K}}\mathcal {V}_u^\dagger f(h,s,\omega )[\varrho (s^h,\omega _h)u](k)\mathrm{d}s\mathrm{d}\omega . \end{aligned}$$

$\square$

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 transforms $A_u$ and $B_u$ for $f\in L^2(G_\tau )$ be given by

$$\begin{aligned} A_uf(h,k,\omega )=V_uf_h(k^h,\omega )\qquad B_uf(h,k,\omega )=\delta (h)V_uf_h(k,\omega _h). \end{aligned}$$

(3.12)

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

$$\begin{aligned} f(h,k)&= \delta (h)^{-1}\int _{K\times \widehat{K}}A_uf(h,k,\omega ) [\varrho (s^h,\omega )](k)\mathrm{d}s\mathrm{d}\omega ,\nonumber \\ f(h,k)&= \int _{K\times \widehat{K}}B_uf(h,k,\omega ) [\varrho (s,\omega _h)](k)\mathrm{d}s\mathrm{d}\omega . \end{aligned}$$

(3.13)

4 $\tau \otimes \widehat{\tau }$-Continuous Gabor Transform

In this section we introduce another Gabor transform which we call it the $\tau \otimes \widehat{\tau }$-continuous Gabor transform. In the $\tau \otimes \widehat{\tau }$-Gabor theory we can choose elements of $L^2(G_\tau )$ as window functions.

Again let $H$ be a locally compact group and $K$ be an LCA group also let $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism. Define $\tau ^\otimes =\tau \otimes \widehat{\tau }:H\times H\rightarrow Aut(K\times \widehat{K})$ via $(h,t)\mapsto \tau ^\otimes _{(h,t)}$ given by

$$\begin{aligned} \tau ^\otimes _{(h,t)}(k,\omega ):=(\tau _h(k),\widehat{\tau }_t(\omega )) =(k^h,\omega _t), \end{aligned}$$

(4.1)

for all $(k,\omega )\in K\times \widehat{K}$. Then, for all $(h,t)\in H\times H$ we get $\tau ^\otimes _{(h,t)}\in Aut(K\times \widehat{K})$. Because for all $(k,\omega ),(k',\omega ')\in K\times \widehat{K}$ we have

$$\begin{aligned} \tau ^\otimes _{(h,t)}\left( (k,\omega )(k',\omega ')\right)&=\tau ^\otimes _{(h,t)}(k+k',\omega \omega ')\\&=\left( (k+k')^h,(\omega \omega ')_t\right) \\&=\left( k^h+k'^h,\omega _t\omega '_t\right) \\&=(k^h,\omega _t)(k'^h,\omega '_t)= \tau ^\otimes _{(h,t)}(k,\omega )\tau ^\otimes _{(h,t)}(k',\omega '). \end{aligned}$$

As well as $\tau ^\otimes =\tau \otimes \widehat{\tau }:H\times H\rightarrow Aut(K\times \widehat{K})$ defined by $(h,t)\mapsto \tau ^\otimes _{(h,t)}$ is a homomorphism, because for all $(h,t),(h',t')\in H\times H$ and also all $(k,\omega )\in K\times \widehat{K}$ we have

$$\begin{aligned} \tau ^\otimes _{(h,t)(h',t')}(k,\omega )&=\tau ^\otimes _{(hh',tt')}(k,\omega )\\&=(k^{hh'},\omega _{tt'})\\&=\left( (k^{h'})^h,(\omega _{t'})_t\right) \\&=\tau ^\otimes _{(h,t)}(k^{h'},\omega _{t'})= \tau ^\otimes _{(h,t)}\tau ^\otimes _{(h',t')}(k,\omega ). \end{aligned}$$

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 let $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism. The semi-direct product $G_{\tau \otimes \widehat{\tau }}=\left( H\times H\right) \ltimes _{\tau \otimes \widehat{\tau }}\left( K\times \widehat{K}\right)$ is a locally compact group with the left Haar measure

$$\begin{aligned} \mathrm{d}\mu _{G_{\tau \otimes \widehat{\tau }}}(h,t,k,\omega )= \delta (h)\delta (t)^{-1}\mathrm{d}h\mathrm{d}t\mathrm{d}k\mathrm{d}\omega , \end{aligned}$$

(4.2)

and also $\Phi :G_\tau \times G_{\widehat{\tau }}\rightarrow G_{\tau \otimes \widehat{\tau }}$ given by

$$\begin{aligned} (h,k,t,\omega )\mapsto \Phi (h,k,t,\omega ):=(h,t,k,\omega ) \end{aligned}$$

(4.3)

is a topological group isomorphism.

Proof

Using Theorem 26.9 of [21], homomorphism $\tau \otimes \widehat{\tau }:H\times H\rightarrow Aut(K\times \widehat{K})$ given in (4.1) is continuous. Therefore, $G_{\tau \otimes \widehat{\tau }}=\left( H\times H\right) \ltimes _{\tau \otimes \widehat{\tau }}\left( K\times \widehat{K}\right)$ is a locally compact group. Also, $\mathrm{d}\mu _{G_{\tau \otimes \widehat{\tau }}} (h,t,k,\omega )=\delta (h)\delta (t)^{-1}\mathrm{d}h\mathrm{d}t\mathrm{d}k\mathrm{d}\omega$ is a left Haar measure for $G_{\tau \otimes \widehat{\tau }}$. Indeed, due to (2.3) and (3.2) for all $(h,t)\in H\times H$ we have

$$\begin{aligned} \mathrm{d}\tau ^\otimes _{(h,t)}(k,\omega )&=\mathrm{d}(k^h,\omega _t)\\&=\mathrm{d}k^h\mathrm{d}\omega _t\\&=\delta (h)^{-1}\mathrm{d}k\delta (t)\mathrm{d}\omega = \delta (h)^{-1}\delta (t)d(k,\omega ). \end{aligned}$$

The $\tau \otimes \widehat{\tau }$-group law for all $(h,t,k,\omega ),(h',t',k',\omega ')\in G_{\tau \otimes \widehat{\tau }}$ is

$$\begin{aligned} (h,t,k,\omega )\ltimes _{\tau \otimes \widehat{\tau }}(h',t',k',\omega ')&=\left( (hh',tt'),(k,\omega )\tau ^\otimes _{(h,t)}(k',\omega ')\right) \\&=\left( (hh',tt'),(k,\omega )(k'^h,\omega '_t)\right) \\&=(hh',tt',k+k'^h,\omega \omega _t'). \end{aligned}$$

It is clear that $\Phi :G_\tau \times G_{\widehat{\tau }}\rightarrow G_{\tau \otimes \widehat{\tau }}$ is a homeomorphism. It is also a group homomorphism, because for all $(h,k,t,\omega ),(h',k',t',\omega ')$ in $G_\tau \times G_{\widehat{\tau }}$ we get

$$\begin{aligned} \Phi [(h,k,t,\omega )(h',k',t',\omega ')]&= \Phi [(h,k)\ltimes _\tau (h',k'),(t,\omega ) \ltimes _{\widehat{\tau }}(t',\omega ')]\\&=\Phi [(hh',k+k'^h),(tt',\omega \omega _t')]\\&=(hh',tt',k+k'^h,\omega \omega _t')\\&=(h,t,k,\omega )\ltimes _{\tau \otimes \widehat{\tau }}(h',t',k',\omega '). \end{aligned}$$

$\square$

We call the semi-direct product $G_{\tau \otimes \widehat{\tau }}$ as the $\tau \otimes \widehat{\tau }$-time frequency group associated to $G_\tau$ which is precisely $G_\tau \times G_{\widehat{\tau }}$. Thus, form now on we use the locally compact group $G_\tau \times G_{\widehat{\tau }}$ instead of the semi-direct product $G_{\tau \otimes \widehat{\tau }}$.

Let $g\in L^2(G_\tau )$ be a window function and $f\in L^2(G_\tau )$. The $\tau \otimes \widehat{\tau }$ -continuous Gabor transform of $f$ with respect to the window function $g$ is defined by

$$\begin{aligned} \mathcal {G}_gf(h,k,t,\omega ):=\delta (t)V_{g_h}f_t(k,\omega )=\delta (t)\langle f_t,\varrho (k,\omega )g_h\rangle _{L^2(K)}. \end{aligned}$$

(4.4)

The $\tau \otimes \widehat{\tau }$-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 $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism also $G_\tau =H\ltimes _\tau K$ and let $g\in L^2(G_\tau )$ be a window function. The continuous Gabor transform $\mathcal {G}_g:L^2(G_\tau )\rightarrow L^2(G_\tau \times G_{\widehat{\tau }})$ is a multiple of an isometric transform which maps $L^2(G_\tau )$ onto a closed subspace of $L^2(G_{\tau }\times G_{\widehat{\tau }})$.

Proof

Let $g\in L^2(G_\tau )$ be a window function and also let $f\in L^2(G_\tau )$.

$$\begin{aligned} \Vert \mathcal {G}_gf\Vert _{L^2(G_\tau \times G_{\widehat{\tau }})}^2&=\int _{G_\tau \times G_{\widehat{\tau }}}|\mathcal {G}_gf(h,k,t,\omega ) |^2\mathrm{d}\mu _{G_\tau \times G_{\widehat{\tau }}}(h,k,t,\omega )\\&=\int _{G_\tau }\int _{G_{\widehat{\tau }}}|\mathcal {G}_gf(h,k,t,\omega ) |^2\mathrm{d}\mu _{G_\tau }(h,k) \mathrm{d}\mu _{G_{\widehat{\tau }}}(t,\omega )\\&=\int _H\int _K\int _H\int _{\widehat{K}}|\mathcal {G}_gf(h,k,t,\omega ) |^2\delta (h)\mathrm{d}h\mathrm{d}k\delta (t)^{-1}\mathrm{d}t\mathrm{d}\omega \\&=\int _H\int _H\left( \int _{K\times \widehat{K}}|\langle f_t, \varrho (k,\omega )g_h\rangle _{L^2(K)}|^2\mathrm{d}k\mathrm{d}\omega \right) \delta (h)\mathrm{d}h\delta (t)\mathrm{d}t\\&=\int _H\int _H\Vert f_t\Vert _{L^2(K)}^2\Vert g_h\Vert _{L^2(K)}^2\delta (h)\mathrm{d}h \delta (t)\mathrm{d}t\!=\!\Vert f\Vert _{L^2(G_\tau )}^2\Vert g\Vert _{L^2(G_\tau )}^2 \end{aligned}$$

Thus, $\Vert g\Vert _{L^2(G_\tau )}^{-2}\mathcal {G}_g:L^2(G_\tau )\rightarrow L^2(G_\tau \times G_{\widehat{\tau }})$ is an isometric transform with a closed range in $L^2(G_{\tau }\times G_{\widehat{\tau }})$. $\square$

Corollary 4.3

The $\tau \times \widehat{\tau }$-continuous Gabor transform defined in (4.4), for all $f,f'\in L^2(G_\tau )$ and window functions $g,g'\in L^2(G_\tau )$ satisfies the following orthogonality relation;

$$\begin{aligned} \langle \mathcal {G}_gf,\mathcal {G}_{g'}f'\rangle _{L^2(G_\tau \times G_{\widehat{\tau }})}=\langle g',g\rangle _{L^2(G_\tau )}\langle f,f'\rangle _{L^2(G_\tau )}. \end{aligned}$$

(4.5)

In the following proposition we also prove an inversion formula.

Proposition 4.4

Let $H$ be a locally compact group and $K$ be an LCA group also let $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism. Every $f,g\in L^2(G_\tau )$ with $\widehat{f_h},\widehat{g_h}\in L^1(\widehat{K})$ for a.e. $h\in H$, satisfy the following reconstruction formula;

$$\begin{aligned} f(t,k)=\langle g_h,g_h\rangle ^{-1}_{L^2(K)}\delta (t)^{-1} \int _{K\times \widehat{K}}\mathcal {G}_gf(h,s,t,\omega ) [\varrho (s,\omega )g_h](k)\mathrm{d}s\mathrm{d}\omega , \end{aligned}$$

(4.6)

for a.e. $h,t\in H$ and $k\in K$. In particular, for a.e. $h\in H$ we have

$$\begin{aligned} f(h,k)=\langle g_h,g_h\rangle ^{-1}_{L^2(K)}\delta (h)^{-1} \int _{K\times \widehat{K}}\mathcal {G}_gf(h,s,h,\omega ) [\varrho (s,\omega )g_h](k)\mathrm{d}s\mathrm{d}\omega . \end{aligned}$$

(4.7)

Proof

Using (2.11) for a.e. $h,t\in H$ we have

$$\begin{aligned} f_t(k)&=\langle g_h,g_h\rangle _{L^2(K)}^{-1}\int _{K\times \widehat{K}}V_{g_h}f_t(s,\omega )[\varrho (s,\omega )g_h](k)\mathrm{d}s\mathrm{d}\omega \\&=\langle g_h,g_h\rangle ^{-1}_{L^2(K)}\delta (t)^{-1} \int _{K\times \widehat{K}}\mathcal {G}_gf(h,s,t,\omega ) [\varrho (s,\omega )g_h](k)\mathrm{d}s\mathrm{d}\omega . \end{aligned}$$

$\square$

Let $g\in L^2(G_\tau )$ be a window function and $f\in L^2(G_\tau )$. The generalized $\tau \otimes \widehat{\tau }$ -continuous Gabor transform of $f$ with respect to the window function $g$ is defined by

$$\begin{aligned} \mathcal {G}_g^\dagger f(h,k,t,\omega )&:= \delta (h)^{-1/2} \delta (t)^{3/2}V_{g_h}f_t(k^h,\omega _t)\nonumber \\&= \delta (h)^{-1/2}\delta (t)^{3/2}\langle f_t,\varrho (k^h,\omega _t)g_h\rangle _{L^2(K)}. \end{aligned}$$

(4.8)

In the next theorem, a Plancherel formula for the generalized $\tau \otimes \widehat{\tau }$-continuous Gabor transform defined in (4.8) proved.

Theorem 4.5

Let $H$ be a locally compact group, $K$ be an LCA group and $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism also $G_\tau =H\ltimes _\tau K$ and also let $g\in L^2(G_\tau )$ be a window function. The generalized continuous Gabor transform $\mathcal {G}_g^\dagger :L^2(G_\tau )\rightarrow L^2(G_\tau \times G_{\widehat{\tau }})$ is a multiple of an isometric transform which maps $L^2(G_\tau )$ onto a closed subspace of $L^2(G_{\tau }\times G_{\widehat{\tau }})$.

Proof

Let $g\in L^2(G_\tau )$ be a window function and also let $f\in L^2(G_\tau )$. Using Fubini’s theorem and also Theorem we achieve

$$\begin{aligned} \Vert \mathcal {G}_gf\Vert _{L^2(G_\tau \times G_{\widehat{\tau }})}^2&= \int _{G_\tau \times G_{\widehat{\tau }}}|\mathcal {G}_g^\dagger f(h,k,t,\omega ) |^2\mathrm{d}\mu _{G_\tau \times G_{\widehat{\tau }}}(h,k,t,\omega )\\&=\int _{G_\tau }\int _{G_{\widehat{\tau }}}|\mathcal {G}_g^\dagger f(h,k,t,\omega ) |^2\mathrm{d}\mu _{G_\tau }(h,k) \mathrm{d}\mu _{G_{\widehat{\tau }}}(t,\omega )\\&=\int _H\int _K\int _H\int _{\widehat{K}}|\mathcal {G}_g^\dagger f(h,k,t,\omega ) |^2\delta (h)\mathrm{d}h\mathrm{d}k\delta (t)^{-1}\mathrm{d}t\mathrm{d}\omega \\&=\int _H\int _H\left( \int _{K\times \widehat{K}}|\langle f_t, \varrho (k^h,\omega _t)g_h\rangle _{L^2(K)}|^2\mathrm{d}k\mathrm{d}\omega \right) \mathrm{d}h\delta (t)^{2}\mathrm{d}t\\&=\int _H\int _H\left( \int _{K\times \widehat{K}}|\langle f_t, \varrho (k,\omega )g_h\rangle _{L^2(K)}|^2\mathrm{d}k^{h^{-1}}\mathrm{d}\omega _{t^{-1}}\right) \mathrm{d}h\delta (t)^{2}\mathrm{d}t\\&=\int _H\int _H\left( \int _{K\times \widehat{K}}|\langle f_t, \varrho (k,\omega )g_h\rangle _{L^2(K)}|^2\mathrm{d}k\mathrm{d}\omega \right) \delta (h)\mathrm{d}h\delta (t)\mathrm{d}t\\&=\int _H\int _H\Vert f_t\Vert _{L^2(K)}^2\Vert g_h\Vert _{L^2(K)}^2 \delta (h)\mathrm{d}h\delta (t)\mathrm{d}t\!=\!\Vert f\Vert _{L^2(G_\tau )}^2\Vert g\Vert _{L^2(G_\tau )}^2 \end{aligned}$$

Thus, $\Vert g\Vert _{L^2(G_\tau )}^{-2}\mathcal {G}_g^\dagger :L^2(G_\tau )\rightarrow L^2(G_\tau \times G_{\widehat{\tau }})$ is an isometric transform with a closed range in $L^2(G_{\tau }\times G_{\widehat{\tau }})$. $\square$

Corollary 4.6

The $\tau \times \widehat{\tau }$-continuous Gabor transform defined in (4.8), for all $f,f'\in L^2(G_\tau )$ and window functions $g,g'\in L^2(G_\tau )$ satisfies the following orthogonality relation;

$$\begin{aligned} \langle \mathcal {G}_g^\dagger f,\mathcal {G}_{g'}^\dagger f'\rangle _{L^2(G_\tau \times G_{\widehat{\tau }})}=\langle g',g\rangle _{L^2(G_\tau )}\langle f,f'\rangle _{L^2(G_\tau )}. \end{aligned}$$

(4.9)

Also, the generalized $\tau \otimes \widehat{\tau }$-continuous Gabor transform satisfies the following inversion formula.

Proposition 4.7

Let $H$ be a locally compact group and $K$ be an LCA group also let $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism. Every $f,g\in L^2(G_\tau )$ with $\widehat{f_h},\widehat{g_h}\in L^1(\widehat{K})$ for a.e. $h,t\in H$, satisfy the following reconstruction formula;

$$\begin{aligned} f(t,k)\!=\!\langle g_h,g_h\rangle ^{-1}_{L^2(K)}\delta (h)^{-1/2} \delta (t)^{-1/2}\int _{K\times \widehat{K}}\mathcal {G}_g^\dagger f(h,s,t,\omega )[\varrho (k^h,\omega _t)g_h](k)\mathrm{d}s\mathrm{d}\omega , \end{aligned}$$

(4.10)

for a.e. $h,t\in H$ and $k\in K$. In particular for a.e. $h\in H$ we have

$$\begin{aligned} f(h,k)=\langle g_h,g_h\rangle ^{-1}_{L^2(K)}\delta (h)^{-1} \int _{K\times \widehat{K}}\mathcal {G}_g^\dagger f(h,s,h,\omega )[\varrho (s^h,\omega _h)g_h](k)\mathrm{d}s\mathrm{d}\omega . \end{aligned}$$

(4.11)

Proof

Using (2.11) for a.e. $h,t\in H$ we have

$$\begin{aligned} f_t(k)&=\langle g_h,g_h\rangle ^{-1}_{L^2(K)}\int _{K\times \widehat{K}} V_{g_h}f_t(s,\omega )[\varrho (s,\omega )g_h](k)\mathrm{d}s\mathrm{d}\omega \\&=\langle g_h,g_h\rangle ^{-1}_{L^2(K)}\int _K\int _{\widehat{K}}V_{g_h} f_t(s,\omega )[\varrho (s,\omega )g_h](k)\mathrm{d}\omega \mathrm{d}s\\&=\langle g_h,g_h\rangle ^{-1}_{L^2(K)}\int _K\left( \int _{\widehat{K}} V_{g_h}f_t(s,\omega _t)[\varrho (s,\omega _t)g_h](k)\mathrm{d}\omega _t\right) \mathrm{d}s\\&=\langle g_h,g_h\rangle ^{-1}_{L^2(K)}\delta (t)\int _{\widehat{K}} \left( \int _KV_{g_h}f_t(s^h,\omega _t)[\varrho (s^h,\omega _t)g_h] (k)\mathrm{d}s^h\right) \mathrm{d}\omega \\&=\langle g_h,g_h\rangle ^{-1}_{L^2(K)}\delta (h)^{-1}\delta (t) \int _{\widehat{K}}\left( \int _KV_{g_h}f_t(s^h,\omega _t) [\varrho (s^h,\omega _t)g_h](k)\mathrm{d}s\right) \mathrm{d}\omega \\&=\langle g_h,g_h\rangle ^{-1}_{L^2(K)}\delta (h)^{-1/2}\delta (t)^{-1/2} \int _{K\times \widehat{K}}\mathcal {G}_gf(h,s,t,\omega ) [\varrho (s^h,\omega _t)g_h](k)\mathrm{d}s\mathrm{d}\omega . \end{aligned}$$

$\square$

5 Examples and Applications

5.1 The Affine Group $\mathbf {a}\mathbf {x}+\mathbf {b}$

Let $H=\mathbb {R}^*_+=(0,+\infty )$ and $K=\mathbb {R}$. The affine group $a\mathbf{x}+b$ is the semi-direct product $H\ltimes _\tau K$ with respect to the homomorphism $\tau :H\rightarrow Aut(K)$ given by $a\mapsto \tau _a$, where $\tau _a(x)=ax$ for all $x\in \mathbb {R}$. Hence, the underlying manifold of the affine group is $(0,\infty )\times \mathbb {R}$ and also the group law is

$$\begin{aligned} (a,x)\ltimes _\tau (a',x')=(aa',x+ax'). \end{aligned}$$

(5.1)

The continuous homomorphism $\delta :H\rightarrow (0,\infty )$ is given by $\delta (a)=a^{-1}$ and so that the left Haar measure is in fact $\mathrm{d}\mu _{G_\tau }(a,x)=a^{-2}\mathrm{d}a\mathrm{d}x$. Due to Theorem 4.5 of [10] we can identify $\widehat{\mathbb {R}}$ with $\mathbb {R}$ via $\omega (x)=\langle x,\omega \rangle =e^{2\pi i\omega x}$ for each $\omega \in \widehat{\mathbb {R}}$ and so we can consider the continuous homomorphism $\widehat{\tau }:H\rightarrow Aut(\widehat{K})$ given by $a\mapsto \widehat{\tau }_a$ via

$$\begin{aligned} \langle x,\omega _a\rangle&=\langle x,\widehat{\tau }_a(\omega )\rangle \\&=\langle \tau _{a^{-1}}(x),\omega \rangle = \langle a^{-1}x,\omega \rangle =e^{2\pi i\omega a^{-1}x}. \end{aligned}$$

Thus, $G_{\widehat{\tau }}$ has the underlying manifold $(0,\infty )\times \mathbb {R}$, with the group law given by

$$\begin{aligned} (a,\omega )\ltimes _{\widehat{\tau }}(a',\omega ')=(aa',\omega \omega '_a), \end{aligned}$$

(5.2)

Due to Theorem 3.1 the left Haar measure $\mathrm{d}\mu _{G_{\widehat{\tau }}}(a,\omega )$ is precisely $\mathrm{d}a\mathrm{d}\omega$. The $\tau \times \widehat{\tau }$-time frequency group $G_{\tau \times \widehat{\tau }}$ has the underlying manifold $(0,\infty )\times \mathbb {R}\times \widehat{\mathbb {R}}$ and the group law is

$$\begin{aligned} (a,x,\omega )\ltimes _{\tau \times \widehat{\tau }}(a',x',\omega ') =(aa',x+ax',\omega \omega '_a), \end{aligned}$$

(5.3)

with the left Haar measure $\mathrm{d}\mu _{G_{\tau \times \widehat{\tau }}}(a,x,\omega )=a^{-1}\mathrm{d}a\mathrm{d}x\mathrm{d}\omega$. The geometry of this locally compact group and also the wave packet approaches of this locally compact group was studied in [2, 20]. If $u\in L^2(\mathbb {R})$ is a window function and also $f\in L^2(G_\tau )$. According to (3.6) we have

$$\begin{aligned} \mathcal {V}_uf(a,x,\omega )&=\delta (a)^{1/2}V_uf_a(x,\omega )\\&=a^{-1/2}\langle f_a,\varrho (x,\omega )u\rangle _{L^2(\mathbb {R})}\\&=a^{-1/2}\int _{-\infty }^{\infty }f(a,y)\overline{[\varrho (x,\omega )u](y)}\mathrm{d}y\\&=a^{-1/2}\int _{-\infty }^{\infty }f(a,y)\overline{u}(y-x)\overline{\omega (y)}\mathrm{d}y\\&=a^{-1/2}\int _{-\infty }^{\infty }f(a,y)\overline{u}(y-x)e^{-2\pi i\omega y}\mathrm{d}y. \end{aligned}$$

Using Theorem 3.4, if $\Vert u\Vert _{L^2(\mathbb {R})}=1$ we get

$$\begin{aligned} \int _{0}^\infty \int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \frac{|\mathcal {V}_uf(a,x,\omega )|^2}{a}\mathrm{d}a\mathrm{d}x\mathrm{d}\omega =\int _0^\infty \int _{-\infty }^{\infty }\frac{|f(a,x)|^2}{a^2}\mathrm{d}a\mathrm{d}x. \end{aligned}$$

(5.4)

Due to the reconstruction formula (3.8) if for a.e. $a\in (0,\infty )$ we have $\widehat{f_a}\in L^1(\mathbb {R})$, then for a.e. $x\in \mathbb {R}$ we get

$$\begin{aligned} f(a,x)&=\delta (a)^{-1/2}\Vert u\Vert _{L^2(K)}^{-2}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\mathcal {V}_uf(a,y,\omega )[\varrho (y,\omega )u] (x)\mathrm{d}y\mathrm{d}\omega \\&=a^{1/2}\Vert u\Vert _{L^2(K)}^{-2}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \mathcal {V}_uf(a,y,\omega )u(x-y)e^{2\pi i\omega x}\mathrm{d}y\mathrm{d}\omega . \end{aligned}$$

As well as according to (3.9) we have

$$\begin{aligned} \mathcal {V}_u^\dagger f(a,x,\omega )&=\delta (a)^{1/2}V_uf_a(x^a,\omega _a)\\&=a^{-1/2}\langle f_a,\varrho (x^a,\omega _a)u\rangle _{L^2(\mathbb {R})}\\&=a^{-1/2}\int _{-\infty }^{\infty }f(a,y)\overline{[\varrho (x^a,\omega _a)u] (y)}\mathrm{d}y\\&=a^{-1/2}\int _{-\infty }^{\infty }f(a,y)\overline{u}(y-ax) \overline{\omega _a(y)}\mathrm{d}y\\&=a^{-1/2}\int _{-\infty }^{\infty } f(a,y)\overline{u}(y-ax)e^{-2\pi i\omega a^{-1}y}\mathrm{d}y. \end{aligned}$$

Using Theorem 3.7, if $\Vert u\Vert _{L^2(\mathbb {R})}=1$ we get

$$\begin{aligned} \int _{0}^\infty \int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \frac{|\mathcal {V}_u^\dagger f(a,x,\omega )|^2}{a}\mathrm{d}a\mathrm{d}x\mathrm{d}\omega =\int _0^\infty \int _{-\infty }^{\infty }\frac{|f(a,x)|^2}{a^2}\mathrm{d}a\mathrm{d}x. \end{aligned}$$

(5.5)

Due to the reconstruction formula (3.11) if for a.e. $a\in (0,\infty )$ we have $\widehat{f_a}\in L^1(\mathbb {R})$, then for $x\in \mathbb {R}$ we achieve

$$\begin{aligned} f(a,x)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\mathcal {V}_u^\dagger f(a,y,\omega )[\varrho (y^a,\omega _a)u](x)\mathrm{d}y\mathrm{d}\omega \\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \mathcal {V}_u^\dagger f(a,y,\omega )u(x-ay)e^{2\pi i\omega a^{-1}x}\mathrm{d}y\mathrm{d}\omega . \end{aligned}$$

Example 5.1

Let $N>0$ and also $u_N=\chi _{[-N,N]}$ be a window function with compact support and $\Vert u_N\Vert _{L^2(\mathbb {R})}=2N$. Then, for all $f\in L^2(G_\tau )$ and $(a,x,\omega )\in G_{\tau \times \widehat{\tau }}$ we have

$$\begin{aligned} \mathcal {V}_{u_N}f(a,x,\omega )&=a^{-1/2}\int _{-\infty }^{\infty } f(a,y)u_N(y-x)\overline{\omega (y)}\mathrm{d}y\\&=a^{-1/2}\overline{\omega (x)}\int _{-\infty }^{\infty }f(a,y+x) u_N(y)\overline{\omega (y)}\mathrm{d}y\\&=a^{-1/2}\overline{\omega (x)}\int _{-N}^{N}f(a,y+x)\overline{\omega (y)}\mathrm{d}y\\&=a^{-1/2}e^{-2\pi i\omega x}\int _{-N}^{N}f(a,y+x)e^{-2\pi i\omega y}\mathrm{d}y. \end{aligned}$$

If we set $x=0$, then we get

$$\begin{aligned} \mathcal {V}_{u_N}f(a,0,\omega )=a^{-1/2}\int _{-N}^Nf(a,y)e^{-2\pi i\omega y}\mathrm{d}y. \end{aligned}$$

Similarly, for the generalized $\tau \times \widehat{\tau }$-continuous Gabor transform we have

$$\begin{aligned} \mathcal {V}_{u_N}^\dagger f(a,x,\omega )&=a^{-1/2}\int _{-\infty }^{\infty }f(a,y)u_N(y-ax)e^{-2\pi i\omega a^{-1}y}\mathrm{d}y\\&=a^{-1/2}e^{-2\pi i\omega x}\int _{-\infty }^{\infty }f(a,y+ax)u_N(y)e^{-2\pi i\omega a^{-1}y}\mathrm{d}y\\&=a^{-1/2}e^{-2\pi i\omega x}\int _{-N}^{N}f(a,y+ax)e^{-2\pi i\omega a^{-1}y}\mathrm{d}y. \end{aligned}$$

and also if we set $x=0$, then we get

$$\begin{aligned} \mathcal {V}_{u_N}^\dagger f(a,0,\omega )=a^{-1/2}\int _{-N}^Nf(a,y)e^{-2\pi i\omega a^{-1}y}\mathrm{d}y. \end{aligned}$$

Example 5.2

Let $u(x)=e^{-\pi x^2}$ be the one-dimensional Gaussian window function with $\widehat{u}=u$ and $\Vert u\Vert _{L^2(\mathbb {R})}=2^{-1/4}$. Then, for all $f\in L^2(G_\tau )$ and $(a,x,\omega )\in G_{\tau \times \widehat{\tau }}$ we have

$$\begin{aligned} \mathcal {V}_uf(x,a,\omega )&=a^{-1/2}\int _{-\infty }^{\infty }f(a,y) u(y-x)e^{-2\pi i\omega y}\mathrm{d}y\\&=a^{-1/2}\int _{-\infty }^{\infty }f(a,y)e^{-\pi (y-x)^2}e^{-2\pi i\omega y}\mathrm{d}y. \end{aligned}$$

If $f$ for a.e. $a\in (0,\infty )$ satisfies $\widehat{f_a}\in L^1(\mathbb {R})$, then we can reconstruct $f$ via

$$\begin{aligned} f(a,x)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \mathcal {V}_{u}f(a,y,\omega )u(x-y)e^{2\pi i\omega y}\mathrm{d}y\mathrm{d}\omega \\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \mathcal {V}_{u}f(a,y,\omega )e^{-\pi (x-y)^2}e^{2\pi i\omega x}\mathrm{d}y\mathrm{d}\omega . \end{aligned}$$

As well as, we can compute the generalized $\tau \times \widehat{\tau }$-continuous Gabor transform by

$$\begin{aligned} \mathcal {V}_u^\dagger f(x,a,\omega )&=a^{-1/2}\int _{-\infty }^{\infty } f(a,y)\overline{u}(y-ax)e^{-2\pi i\omega a^{-1}y}\mathrm{d}y\\&=a^{-1/2}\int _{-\infty }^{\infty }f(a,y)e^{-\pi (y-ax)^2} e^{-2\pi i\omega a^{-1}y}\mathrm{d}y. \end{aligned}$$

If $f$ for a.e. $a\in (0,\infty )$ satisfies $\widehat{f_a}\in L^1(\mathbb {R})$, then we can reconstruct $f$ via

$$\begin{aligned} f(a,x)&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \mathcal {V}_u^\dagger f(a,y,\omega )u(x-ay)e^{2\pi i\omega a^{-1}x}\mathrm{d}y\mathrm{d}\omega \\&=\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\mathcal {V}_{u}^\dagger f(a,y,\omega )e^{-\pi (x-ay)^2}e^{2\pi i\omega a^{-1}x}\mathrm{d}y\mathrm{d}\omega . \end{aligned}$$

The $\tau \otimes \widehat{\tau }$-time frequency group $G_{\tau \otimes \widehat{\tau }}$ has the underlying manifold $(0,\infty )\times (0,\infty )\times \mathbb {R}\times \widehat{\mathbb {R}}$ and the group law is

$$\begin{aligned} (a,b,x,\omega )\ltimes _{\tau \otimes \widehat{\tau }}(a',b',x',\omega ') =(aa',bb',x+ax',\omega \omega '_b), \end{aligned}$$

(5.6)

with the left Haar measure $\mathrm{d}\mu _{G_{\tau \otimes \widehat{\tau }}}(a,b,x,\omega )=a^{-2}\mathrm{d}a\mathrm{d}b\mathrm{d}x\mathrm{d}\omega$. If $g\in L^2(G_\tau )$ is a window function and also $f\in L^2(G_\tau )$. According to (4.4) we have

$$\begin{aligned} \mathcal {G}_gf(a,x,b,\omega )&=\delta (b)V_{g_a}f_b(x,\omega )\\&=b^{-1}\langle f_b,\varrho (x,\omega )g_a\rangle _{L^2(\mathbb {R})}\\&=b^{-1}\int _{-\infty }^{\infty }f(b,y)\overline{[\varrho (x,\omega )g_a](y)}\mathrm{d}y\\&=b^{-1}\int _{-\infty }^{\infty }f(b,y)\overline{g}(a,y-x)e^{-2\pi i\omega y}\mathrm{d}y. \end{aligned}$$

Using Theorem 4.2, if $\Vert g\Vert _{L^2(G_\tau )}=1$ we get

$$\begin{aligned} \int _0^{\infty }\int _{0}^\infty \int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\frac{|\mathcal {G}_gf(a,x,b,\omega )|^2}{a^2}\mathrm{d}a\mathrm{d}b\mathrm{d}x\mathrm{d}\omega =\int _0^\infty \int _{-\infty }^{\infty } \frac{|f(a,x)|^2}{a^2}\mathrm{d}a\mathrm{d}x.\nonumber \\ \end{aligned}$$

(5.7)

Using the reconstruction formula (4.7), if for a.e. $a\in (0,\infty )$ we have $\widehat{f_a},\widehat{g_a}\in L^1(\mathbb {R})$, then for $x\in \mathbb {R}$ we can write

$$\begin{aligned} f(b,x)&=b\Vert g_a\Vert _{L^2(\mathbb {R})}^{-2}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\mathcal {G}_gf(a,y,b,\omega ) [\varrho (y,\omega )g_a](x)\mathrm{d}y\mathrm{d}\omega \\&=b\Vert g_a\Vert _{L^2(\mathbb {R})}^{-2}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\mathcal {G}_gf(a,y,b,\omega )g(a,x-y) e^{-2\pi i \omega x}\mathrm{d}y\mathrm{d}\omega , \end{aligned}$$

and also in particular we get

$$\begin{aligned} f(a,x)&=a\Vert g_a\Vert _{L^2(\mathbb {R})}^{-2}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\mathcal {G}_gf(a,y,a,\omega )g(a,x-y)e^{-2\pi i \omega x}\mathrm{d}y\mathrm{d}\omega . \end{aligned}$$

As well as according to (4.8) we have

$$\begin{aligned} \mathcal {G}_g^\dagger f(a,x,b,\omega )&=\delta (a)^{-1/2} \delta (b)^{3/2}V_{g_a}f_b(x^a,\omega _b)\\&=a^{1/2}b^{-3/2}\langle f_b,\varrho (x^a,\omega _b) g_a\rangle _{L^2(\mathbb {R})}\\&=a^{1/2}b^{-3/2}\int _{-\infty }^{\infty }f(b,y) \overline{[\varrho (x^a,\omega _b)g_a](y)}\mathrm{d}y\\&=a^{1/2}b^{-3/2}\int _{-\infty }^{\infty }f(b,y)\overline{g}(a,y-ax) \overline{\omega _b(y)}\mathrm{d}y\\&=a^{1/2}b^{-3/2}\int _{-\infty }^{\infty }f(b,y) \overline{g}(a,y-ax)e^{-2\pi i\omega b^{-1}y}\mathrm{d}y. \end{aligned}$$

Using Theorem 4.5, if $\Vert g\Vert _{L^2(G_\tau )}=1$ we get

$$\begin{aligned} \int _0^{\infty }\int _{0}^\infty \int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\frac{|\mathcal {G}_g^\dagger f(a,x,b,\omega )|^2}{a^2}\mathrm{d}a\mathrm{d}b\mathrm{d}x\mathrm{d}\omega =\int _0^\infty \int _{-\infty }^{\infty }\frac{|f(a,x)|^2}{a^2}\mathrm{d}a\mathrm{d}x. \end{aligned}$$

(5.8)

Due to the reconstruction formula (4.7) if for a.e. $a\in (0,\infty )$ we have $\widehat{f_a}\in L^1(\mathbb {R})$, then for all $x\in \mathbb {R}$ and a.e. $a,b\in (0,\infty )$ we get

$$\begin{aligned} f(b,x)&=\delta (a)^{-1/2}\delta (b)^{-1/2}\Vert g_a\Vert _{L^2(\mathbb {R})}^{-2} \int _{-\infty }^{\infty }\int _{-\infty }^{\infty }\mathcal {G}_g^\dagger f(a,g,b,\omega )[\varrho (g^a,\omega _b)g_a](x)\mathrm{d}g\mathrm{d}\omega \\&=a^{1/2}b^{1/2}\Vert g_a\Vert _{L^2(\mathbb {R})}^{-2}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\mathcal {G}_g^\dagger f(a,g,b,\omega )g(a,x-ag) e^{2\pi i b^{-1}\omega }\mathrm{d}g\mathrm{d}\omega , \end{aligned}$$

and also in particular we have

$$\begin{aligned} f(a,x)&=a\Vert g_a\Vert _{L^2(\mathbb {R})}^{-2}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\mathcal {G}_g^\dagger f(a,g,a,\omega ) [\varrho (g^a,\omega _a)g_a](x)\mathrm{d}g\mathrm{d}\omega \\&=a\Vert g_a\Vert _{L^2(\mathbb {R})}^{-2}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\mathcal {G}_g^\dagger f(a,g,a,\omega ) g(a,x-ag)e^{2\pi i a^{-1}\omega }\mathrm{d}g\mathrm{d}\omega . \end{aligned}$$

Example 5.3

Let $N>0$ and also $g_N(a,x)=\chi _{[1/N,N]\times [-N,N]}(a,x)$ be a window function which has a compact support. Then, for all $f\in L^2(G_\tau )$ and $(a,x,b,\omega )\in G_{\tau \otimes \widehat{\tau }}$ we have

$$\begin{aligned} \mathcal {G}_{g_N}f(a,x,b,\omega )&=b^{-1}\int _{-\infty }^{\infty }f(b,y)g_N(a,y-x)e^{-2\pi i\omega y}\mathrm{d}y\\&=b^{-1}\chi _{[1/N,N]}(a)e^{-2\pi i\omega x}\int _{-N}^{N}f(b,y+x) e^{-2\pi i\omega y}\mathrm{d}y. \end{aligned}$$

If we set $x=0$ and $a=1$ we achieve

$$\begin{aligned} \mathcal {G}_{g_N}f(1,0,b,\omega )=b^{-1}\int _{-N}^{N}f(b,y)e^{-2\pi i\omega y}\mathrm{d}y. \end{aligned}$$

(5.9)

Also, for $(a,x,b,\omega )$ we have

$$\begin{aligned} \mathcal {G}_{g_N}^\dagger f(a,x,b,\omega )&=a^{1/2} b^{-3/2}\int _{-\infty }^{\infty }f(b,y)g_N(a,y-ax)e^{-2\pi i\omega b^{-1}y}\mathrm{d}y\\&=a^{1/2}b^{-3/2}\chi _{[1/N,N]}(a)e^{-2\pi i\omega b^{-1}ax} \int _{-N}^{N}f(b,y\!+\!ax)e^{-2\pi i\omega b^{-1}y}\mathrm{d}y. \end{aligned}$$

If we set $x=0$ and $a=1$ then

$$\begin{aligned} \mathcal {G}_{g_N}^\dagger f(1,0,b,\omega )=b^{-3/2}\int _{-N}^{N}f(b,y)e^{-2\pi i\omega b^{-1}y}\mathrm{d}y. \end{aligned}$$

(5.10)

Example 5.4

Let $g(a,x)=ae^{-\pi (a^2+x^2)}$ be the Gaussian type window function in $L^2(G_\tau )$ with $\Vert g\Vert _{L^2(G_\tau )}=2^{-1}$. For a.e. $a\in (0,\infty )$ we have $\widehat{g_a}=g_a$ and $\Vert g_a\Vert _{L^1(\mathbb {R})}=ae^{-\pi a^2}$ also $\Vert g_a\Vert _{L^2(\mathbb {R})}=2^{-1/4}ae^{-\pi a^2}$. It is also separable i.e. $g(a,x)=au(a)u(x)$. Then, for all $f\in L^2(G_\tau )$ and also $(a,x,b,\omega )\in G_{\tau \otimes \widehat{\tau }}$ we have

$$\begin{aligned} \mathcal {G}_gf(a,x,b,\omega )&=b^{-1}\int _{-\infty }^{\infty }f(b,y) g(a,y-x)e^{-2\pi i\omega y}\mathrm{d}y\\&=b^{-1}ae^{-\pi a^2}\int _{-\infty }^{\infty }f(b,y)e^{-\pi (y-x)^2} e^{-2\pi i\omega y}\mathrm{d}y. \end{aligned}$$

Using the reconstruction formula if $\widehat{f_a}\in L^1(\mathbb {R})$ for a.e. $a\in (0,\infty )$ we have

$$\begin{aligned} f(a,x)&=a\Vert g_a\Vert _{L^2(\mathbb {R})}^{-2}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\mathcal {G}_{g}f(a,y,a,\omega )g(a,x-y) e^{2\pi i\omega x}\mathrm{d}y\mathrm{d}\omega \\&=2^{1/2}a^{-1}e^{\pi a^2}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\mathcal {G}_{g}f(a,y,a,\omega ) u(x-y)e^{2\pi i\omega x}\mathrm{d}y\mathrm{d}\omega \\&=2^{1/2}a^{-1}e^{\pi a^2}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \mathcal {G}_{g}f(a,y,a,\omega )e^{-\pi (x-y)^2}e^{2\pi i\omega x}\mathrm{d}y\mathrm{d}\omega . \end{aligned}$$

As well as, for all $(a,x,b,\omega )\in G_{\tau \otimes \widehat{\tau }}$ we have

$$\begin{aligned} \mathcal {G}_g^\dagger f(a,x,b,\omega )&=a^{1/2}b^{-3/2} \int _{-\infty }^{\infty }f(b,y)g(a,y-ax)e^{-2\pi i\omega b^{-1}y}\mathrm{d}y\\&=e^{-\pi a^2}a^{3/2}b^{-3/2}\int _{-\infty }^{\infty }f(b,y)e^{-\pi (y-ax)^2} e^{-2\pi i\omega b^{-1}y}\mathrm{d}y. \end{aligned}$$

Due to the reconstruction formula if $\widehat{f_a}\in L^1(\mathbb {R})$ for a.e. $a\in (0,\infty )$ we have

$$\begin{aligned} f(a,x)&=a\Vert g_a\Vert _{L^2(\mathbb {R})}^{-2}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\mathcal {G}_g^\dagger f(a,y,a,\omega ) g(a,x-ay)e^{-2\pi i a^{-1}\omega }\mathrm{d}y\mathrm{d}\omega \\&=2^{1/2}a^{-1}e^{\pi a^2}\int _{-\infty }^{\infty }\int _{-\infty }^{\infty } \mathcal {G}_g^\dagger f(a,y,a,\omega )u(x-ay)e^{-2\pi i a^{-1}\omega } \mathrm{d}y\mathrm{d}\omega \\&=2^{1/2}a^{-1}e^{\pi a^2}\int _{-\infty }^{\infty } \int _{-\infty }^{\infty }\mathcal {G}_g^\dagger f(a,y,a,\omega ) e^{-\pi (x-ay)^2}e^{-2\pi i a^{-1}\omega }\mathrm{d}y\mathrm{d}\omega . \end{aligned}$$

In the sequel we compute $\tau \times \widehat{\tau }$-time frequency group $G_{\tau \times \widehat{\tau }}$ and $\tau \otimes \widehat{\tau }$-time frequency group $G_{\tau \otimes \widehat{\tau }}$ associate to other semi-direct products.

5.2 The Weyl–Heisenberg Group

Let $K$ be an LCA group with the Haar measure $\mathrm{d}k$ and $\widehat{K}$ be the dual group of $K$ with the Haar measure $\mathrm{d}\omega$ also $\mathbb {T}$ be the circle group and let the continuous homomorphism $\tau :K\rightarrow Aut(\widehat{K}\times \mathbb {T})$ via $s\mapsto \tau _s$ be given by $\tau _s(\omega ,z)=(\omega ,z\cdot \omega (s))$. The semi-direct product $G_\tau =K\ltimes _\tau (\widehat{K}\times \mathbb {T})$ is called the Weyl–Heisenberg group associated with $K$. The group operation for all $(k,\omega ,z),(k',\omega ',z')\in K\ltimes _\tau (\widehat{K}\times \mathbb {T})$ is

$$\begin{aligned} (k,\omega ,z)\ltimes _\tau (k',\omega ',z')=(k+k',\omega \omega ',zz'\omega '(k)). \end{aligned}$$

(5.11)

If $\mathrm{d}z$ is the Haar measure of the circle group, then $\mathrm{d}k\mathrm{d}\omega \mathrm{d}z$ is a Haar measure for the Weyl–Heisenberg group and also the continuous homomorphism $\delta :K\rightarrow (0,\infty )$ given in (2.3) is the constant function $1$. Thus, using Theorem 4.5, Proposition 4.6 of [10] and also Proposition 3.1 we can obtain the continuous homomorphism $\widehat{\tau }:K\rightarrow Aut(K\times \mathbb {Z})$ via $s\mapsto \widehat{\tau }_s$, where $\widehat{\tau }_s$ is given by $\widehat{\tau }_s(k,n)=(k,n)\circ \tau _{s^{-1}}$ for all $(k,n)\in K\times \mathbb {Z}$ and $s\in K$. Due to Theorem 4.5 of [10], for each $(k,n)\in K\times \mathbb {Z}$ and also for all $(\omega ,z)\in \widehat{K}\times \mathbb {T}$ we have

$$\begin{aligned} \langle (\omega ,z),(k,n)_s\rangle&=\langle (\omega ,z), \widehat{\tau }_s(k,n)\rangle \\&=\langle \tau _{s^{-1}}(\omega ,z),(k,n)\rangle \\&=\langle (\omega ,z\overline{\omega (s)}),(k,n)\rangle \\&=\langle \omega ,k\rangle \langle z\overline{\omega (s)},n\rangle \\&=\omega (k)z^n\overline{\omega (s)}^n\\&=\omega (k-ns)z^n\\&=\langle \omega ,k-ns\rangle \langle z,n\rangle =\langle (\omega ,z), (k-ns,n)\rangle . \end{aligned}$$

Thus, $(k,n)_s=(k-ns,n)$ for all $k,s\in K$ and $n\in \mathbb {Z}$. Therefore, $G_{\widehat{\tau }}$ has the underlying set $K\times K\times \mathbb {Z}$ with the following group operation;

$$\begin{aligned} (s,k,n)\ltimes _{\widehat{\tau }}(s',k',n') =&\, \left( s+s',(k,n) \widehat{\tau }_s(k',n')\right) \\ =&\,\left( s+s',(k,n)(k'-n's,n')\right) \\=&\,(s+s',k+k'-n's,n+n'). \end{aligned}$$

The $G_{\tau \times \widehat{\tau }}$-time frequency group has the underlying set $K\times \widehat{K}\times \mathbb {T}\times K\times \mathbb {Z}$ with the group law

$$\begin{aligned} (k,\omega ,z,s,n)\ltimes _{\tau \times \widehat{\tau }} (k',\omega ',z',s',n')=\big (k+k',\omega \omega ',zz'\omega '(k),\\ \nonumber s+s'-n'k,n+n'\big ). \end{aligned}$$

(5.12)

and the left Haar measure is $\mathrm{d}\mu _{G_{\tau \times \widehat{\tau }}}(k,\omega ,z,s,n)=\mathrm{d}k\mathrm{d}\omega \mathrm{d}z\mathrm{d}s\mathrm{d}n$.

The $G_{\tau \otimes \widehat{\tau }}$-time frequency group has the underlying set $K\times K\times \widehat{K}\times \mathbb {T}\times K\times \mathbb {Z}$ with group operation

$$\begin{aligned} (r,k,\omega ,z,s,n)\ltimes _{\tau \otimes \widehat{\tau }} (r',k',\omega ',z',s',n')&= \big (r+r',k+k',\omega \omega ', zz'\omega '(r),\nonumber \\&\quad s+s'-n'k,n+n'\big ), \end{aligned}$$

(5.13)

and also the left Haar measure is $\mathrm{d}\mu _{G_{\tau \otimes \widehat{\tau }}}(r,k,\omega ,z,s,n)=\mathrm{d}r\mathrm{d}k\mathrm{d}\omega \mathrm{d}z\mathrm{d}s\mathrm{d}n$.

5.3 Euclidean Groups

Let $E(n)$ be the group of rigid motions of $\mathbb {R}^n$, the group generated by rotations and translations. If we put $H=\mathrm{SO}(n)$ and also $K=\mathbb {R}^n$, then $E(n)$ is the semi-direct product of $H$ and $K$ with respect to the continuous homomorphism $\tau :\mathrm{SO}(n)\rightarrow Aut(\mathbb {R}^n)$ given by $\sigma \mapsto \tau _\sigma$ via $\tau _\sigma (\mathbf {x})=\sigma {\mathbf {x}}$ for all $\mathbf {x}\in \mathbb {R}^n$. The group operation for $E(n)$ is

$$\begin{aligned} (\sigma ,\mathbf {x})\ltimes _{\tau }(\sigma ',\mathbf {x}')=(\sigma \sigma ',\mathbf {x} +\tau _\sigma (\mathbf {x}'))=(\sigma \sigma ',\mathbf {x}+\sigma \mathbf {x}'). \end{aligned}$$

(5.14)

Identifying $\widehat{K}$ with $\mathbb {R}^n$, the continuous homomorphism $\widehat{\tau }:\mathrm{SO}(n)\rightarrow Aut(\mathbb {R}^n)$ is given by $\sigma \mapsto \widehat{\tau }_\sigma$ via

$$\begin{aligned} \langle {\mathbf {x}},\widehat{\tau }_\sigma (\mathbf{{w}})\rangle&= \langle \mathbf{{x}},\mathbf{{w}}_\sigma \rangle \\&=\langle \tau _{\sigma ^{-1}}(\mathbf{{x}}),\mathbf{{w}}\rangle \\&=\langle \sigma ^{-1}\mathbf{{x}},\mathbf{{w}}\rangle = e^{-2\pi i (\sigma ^{-1}\mathbf{{x}},\mathbf{{w}})}, \end{aligned}$$

where $(.,.)$ stands for the standard inner product of $\mathbb {R}^n$. Since $H$ is compact we have $\delta \equiv 1$ and therefore $d\sigma d\mathbf{x}$ is a left Haar measure for $E(n)$. Thus, $G_{\widehat{\tau }}$ has the underlying manifold $\mathrm{SO}(n)\times \mathbb {R}^n$ with the group operation

$$\begin{aligned} (\sigma ,\mathbf {w})\ltimes _{\widehat{\tau }}(\sigma ',\mathbf {w}') =(\sigma \sigma ',\mathbf {w}+\mathbf {w}'_\sigma ). \end{aligned}$$

(5.15)

According to Theorem 3.1 the left Haar measure $\mathrm{d}\mu _{G_{\widehat{\tau }}}(\sigma ,\mathbf{w})$ is precisely $\mathrm{d}\sigma \mathrm{d}\mathbf{w}$. The $\tau \times \widehat{\tau }$-time frequency group $G_{\tau \times \widehat{\tau }}$ has the underlying manifold $\mathrm{SO}(n)\times \mathbb {R}^n\times \mathbb {R}^n$ with the group law

$$\begin{aligned} (\sigma ,\mathbf{x},\mathbf{w})\ltimes _{\tau \times \widehat{\tau }} (\sigma ',\mathbf{x}',\mathbf{w}')=(\sigma \sigma ',\mathbf{x} +\sigma \mathbf{x}',\mathbf{w}+\mathbf{w}'_\sigma ). \end{aligned}$$

(5.16)

Due to Proposition 3.3 and also compactness of $H$, $\mathrm{d}\mu _{G_{\tau \times \widehat{\tau }}}(\sigma ,\mathbf{x},\mathbf{w})=d\sigma d\mathbf{x}d\mathbf{w}$ is a Haar measure for $G_{\tau \times \widehat{\tau }}$. The $\tau \otimes \widehat{\tau }$-time frequency group $G_{\tau \otimes \widehat{\tau }}$ has the underlying manifold $\mathrm{SO}(n)\times \mathrm{SO}(n)\times \mathbb {R}^n\times \mathbb {R}^n$ equipped with the group law

$$\begin{aligned} (\sigma ,\varrho ,\mathbf{x},\mathbf{w})\ltimes _{\tau \otimes \widehat{\tau }}(\sigma ',\varrho ',\mathbf{x}',\mathbf{w}') =(\sigma \sigma ',\varrho \varrho ',\mathbf{x}+\sigma \mathbf{x}',\mathbf{w}+\mathbf{w}'_\varrho ), \end{aligned}$$

and also the Haar measure is $\mathrm{d}\mu _{G_{\tau \otimes \widehat{\tau }}}(\sigma ,\varrho ,\mathbf{x},\mathbf{w})=\mathrm{d}\sigma \mathrm{d}\varrho \mathrm{d}\mathbf{x}\mathrm{d}\mathbf{w}$.