Zak transform for semidirect product of locally compact groups

Abstract

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 $G_\tau =H\ltimes _\tau K$ be the semidirect product of $H$ and $K$ with respect to $\tau$. In this article we define the Zak transform $\mathcal{Z }_L$ on $L^2(G_\tau )$ with respect to a $\tau$-invariant uniform lattice $L$ of $K$ and we also show that the Zak transform satisfies the Plancherel formula. As an application we analyze how these technique apply for the semidirect product group $\mathrm SL (2,\mathbb{Z })\ltimes _\tau \mathbb{R }^2$ and also the Weyl-Heisenberg groups.

1 Introduction

The Zak transform has been used successfully in various applications in physics, such as studying coherent states representations in quantum field theory [7, 9], decomposing the Hamiltonian and rewriting it as magnetic Weyl quantization of an operator-valued function [13] and also as a discrete Fourier transform performed on the signal blocks. It can be also considered as the polyphase representation of periodic signals and an important tool in the analysis of Gabor system and signal theory [8]. Furthermore, the Zak transforms of B-splines, appear as a base for the integral representation of discrete splines. This integral representation is similar to the Fourier integral, and the Zak splines play the part of the Fourier exponentials [14].

The Zak transform on $\mathbb{R }$ was introduced in 1950 by Gelfand and it was rediscovered in quantum mechanic representation by Zak [16]. Let $f\in L^{2}(\mathbb{R })$, the function $Zf:\mathbb{R }\times \mathbb{R }\rightarrow \mathbb{C }$ defined by

$$\begin{aligned} Zf(x,y)=\sum _{k=-\infty }^{+\infty }f(x+k)e^{2\pi iyk}, \end{aligned}$$

(1.1)

is called the Zak transform of $f$. The Zak transform on locally compact abelian (LCA) groups was introduced by Weil [15] and developed by many authors [4, 11, 12]. Recently, applications of the Zak transform for the circle and $\mathbb{Z }$ have been rediscovered in mathematical physics to study representations of finite quantum systems (see [17]). An approach to the Zak transform on certain non-abelian locally compact groups can be found in [1, 10]. Many groups which appear in mathematical physics and quantum mechanics are non-abelian although they can be considered as semi-direct products of locally compact groups.

In this paper which contains 4 sections, we introduce an approach to define the Zak transform on semidirect product groups of the form $G_\tau =H\ltimes _\tau K$ where $K$ is an LCA group and $\tau :H\rightarrow Aut(K)$ is a continuous homomorphism. This article is organized as follows; Sect. 2 is devoted to fix notation containing a summary of the standard harmonic analysis of semi-direct product groups and also an overview of the Zak transform on LCA groups. In order to define the Zak transform on $G_\tau$, we first introduce the continuous homomorphisms $\widehat{\tau }:H\rightarrow Aut(\widehat{K})$ and $\tau ^L:H\rightarrow Aut(K/L)$, for a $\tau$-invariant closed subgroup $L$ of $K$ in Sect. 3. Then we construct the semidirect products $H\ltimes _\tau \widehat{K}$ and $H\ltimes _\tau K/L$ and study the basic properties of them. Applying these continuous homomorphisms for a given $\tau$-invariant closed subgroup $L$ of $K$, we define the continuous homomorphism $\tau ^{\times ,L}=\tau ^L\times \widehat{\tau }^{L^\perp }:H\rightarrow Aut(K/L\times \widehat{K}/L^\perp )$ which induces the locally compact group $G_{\tau ^{\times ,L}}=H\ltimes _{\tau ^{\times ,L}}\left( K/L\times \widehat{K}/L^\perp \right)$. The $\tau$-Zak transform $\mathcal{Z }_Lf$ of a function $f\in L^2(G_\tau )$ with respect to a $\tau$-invariant uniform lattice $L$ of $K$ is defined as a function on the semi-direct product $G_{\tau ^{\times ,L}}$. It is also proved that $\mathcal{Z }_L:L^2(G_{\tau })\rightarrow L^2(G_{\tau ^{\times ,L}})$ is an isometric transform.

Finally, we study these methods for the semidirect group $\mathrm SL (2,\mathbb{Z })\ltimes _\tau \mathbb{R }^2$ and also the Weyl-Heisenberg groups.

2 Preliminaries and notations

Let $H$ and $K$ be locally compact groups with identity elements $e_H$ and $e_K$ and left Haar measures $dh$ 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 $K$. For simplicity in notation we often use $k^h$ instead of $\tau _h(k)$ for all $h\in H$ and $k\in K$.

There is also 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 neighbourhoods

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

(2.1)

where $F\subseteq K$ is compact and $U\subseteq K$ is an identity neighbourhood. The continuity of a homomorphism $\tau :H\rightarrow Aut(K)$ is equivalent to the continuity of the map $(h,k)\mapsto \tau _h(k)$ from $H\times K$ onto $K$ (see [6]).

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 given by

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

(2.2)

The left Haar measure of $G_\tau$ is $d\mu _{G_\tau }(h,k)=\delta _K(h)dhdk$ and the modular function of $G_\tau$ is $\Delta _{G_\tau }(h,k)=\delta _K(h)\Delta _H(h)\Delta _K(k)$, where the positive and continuous homomorphism $\delta _K:H\rightarrow (0,\infty )$ is given by (15.29 of [5])

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

(2.3)

If $L$ is a closed subgroup of $K$ which is $\tau$-invariant [i.e. $\tau _h(L)\subseteq L$ for all $h\in H$] with the left Haar measure $dl$, then $\tau :H\rightarrow Aut(L)$ is a well-defined continuous homomorphism and $H\ltimes _{\tau }L$ is a locally compact group with the left Haar measure $\delta _L(h)dhdl$, where $\delta _L:H\rightarrow (0,\infty )$ is given by

$$\begin{aligned} dl=\delta _L(h)d(\tau _h(l)). \end{aligned}$$

(2.4)

It is clear that if $H$ is a compact group we have $\delta _L=\delta _K=1$ and also if $L$ is an open subgroup of $K$ we get $\delta _L=\delta _K$. 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 [3]).

If $L$ is a closed subgroup of an LCA group $K$ then $K/L$ is a locally compact group with left Haar measure of $\sigma _{K/L}$ and also $L^1(K/L,\sigma _{K/L})$ is precisely the set of all functions of the form $T_Lv$ with $v\in L^1(K)$ and

$$\begin{aligned} T_Lv(k+L)=\int _Lf(k+l)dl. \end{aligned}$$

(2.5)

In fact, $T_L:L^1(K)\rightarrow L^1(K/L,\sigma _{K/L})$ given by $v\mapsto T_Lv$ is a surjective bounded linear map. As well as, all $v\in L^1(K)$ satisfy the following Weil’s formula(see [2]);

$$\begin{aligned} \int _Kv(k)dk=\int _{K/L}T_Lv(k+L)d\sigma _{K/L}(k+L). \end{aligned}$$

(2.6)

If $K$ is an LCA group all irreducible representations of $K$ are one-dimensional. Thus, if $\pi$ be an irreducible unitary representation of $K$ we have $\mathcal{H }_\pi =\mathbb{C }$ and also according to the Shur’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 $\widehat{K}$. It is a usual notation to use $\langle k,\omega \rangle$ instead of $\omega (k)$. 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 dot 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

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

(2.7)

is called the Fourier transform on $K$, where $\mathcal{C }(K)$ is the set of all continuous functions on $K$. The Fourier transform (2.7) on $L^1(K)\cap L^2(K)$ is an isometric transform and it extends uniquely to a unitary isomorphism from $L^2(K)$ to $L^2(\widehat{K})$ (Theorem 4.25 of [2]) 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 [2]);

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

(2.8)

If $L$ is a closed subgroup of an LCA group $K$ the annihilator of $L$ in $K$ is defined by

$$\begin{aligned} L^{\perp }=\left\{ \omega \in \widehat{K}:\ \omega (l)=1~ \text{ for } \text{ all }~ l\in L \right\} , \end{aligned}$$

(2.9)

which is a closed subgroup of $\widehat{K}$. Then, $(L^{\perp })^{\perp }=L$ and $\widehat{K}/L^{\perp }=\widehat{L}$ also $\widehat{K/L}=L^{\perp }$ (see Theorem 4.39 of [2]). By a uniform lattice $L$ of $K$ we mean a discrete and co-compact (i.e $K/L$ is compact) subgroup $L$ of $K$. If $K$ is second countable it is always guaranteed that $K$ possesses a uniform lattice.

The Zak transform associated to a uniform lattice $L$ of $v\in \mathcal{C }_c(K)$ is defined on $K\times \widehat{K}$ by

$$\begin{aligned} Z_Lv(k,\omega )=\sum _{l\in L}f(k+l)\omega (l), \end{aligned}$$

(2.10)

where $\mathcal{C }_c(K)$ stands for the function space of all continuous functions on $K$ with compact support. It is shown that $Z_L:\mathcal{C }_c(K)\rightarrow \mathcal{C }_c(K/L\times \widehat{K}/L^\perp )$ is an isometry in $L^2$-norms and so that it can be uniquely extended into the Zak transform $Z_L:L^2(K)\rightarrow L^2(K/L\times \widehat{K}/L^\perp )$ which is still an isometry (see [4, 11]).

3 $\tau$-Zak transform

Throughout this article, let $H$ be locally compact group, K an LCA group and $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism also let $G_\tau =H\ltimes _\tau L$. 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 $k\in K$. According to (3.1) for all $h\in H$ we have $\widehat{\tau }_h\in Aut(\widehat{K})$ and also $h\mapsto \widehat{\tau }_h$ is a homomorphism from $H$ into $Aut(\widehat{K})$. Because if $h,t\in H$ then for all $\omega \in \widehat{K}$ and also $k\in K$ we have

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

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 $dk=\delta (h)dk^h$. The semi-direct product $G_{\widehat{\tau }}=H\ltimes _{\widehat{\tau }}\widehat{K}$ is a locally compact group with the left Haar measure $d\mu _{G_{\widehat{\tau }}}(h,\omega )=\delta (h)^{-1}dhd\omega$.

Proof

Continuity of the homomorphism $\widehat{\tau }:H\rightarrow Aut(\widehat{K})$ given in (3.1) guaranteed by Theorem 26.9 of [5]. 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 $d\omega$ on $\widehat{K}$ for all $h\in H$ satisfies

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

(3.2)

Let $h\in H$ and also $v\in L^1(K)$. 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

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

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

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

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

If $L$ is a closed $\tau$-invariant subgroup of $K$, let $\tau ^L:H\rightarrow Aut(K/L)$ via $h\mapsto \tau ^L_h$ be given by

$$\begin{aligned} \tau ^L_h(k+L):=\tau _h(k)+L=k^h+L. \end{aligned}$$

(3.3)

For all $h\in H$ we have $\tau ^L_h\in Aut(K/L)$ and $\tau ^L:H\rightarrow Aut(K/L)$ is a well-defined homomorphism. For simplicity in notations, from now on we use $(k+L)^h$ instead of $\tau ^L_h(k+L)$. Thus, via algebraic structures we can consider the semi-direct product $G_{\tau ^L}=H\ltimes _{\tau ^L} K/L$. The group operation for $(h,k+L),(h^{\prime },k^{\prime }+L)\in G_{\tau ^L}$ is

$$\begin{aligned} (h,k+L)\ltimes _{\tau ^L}(h^{\prime },k^{\prime }+L)=(hh^{\prime },k+k^{\prime h}+L). \end{aligned}$$

(3.4)

In the next theorem we show that $G_{\tau ^L}=H\ltimes _{\tau ^L} K/L$ is a locally compact group and we also identify the left Haar measure.

Theorem 3.2

Let $H$ be a locally compact group, $K$ be an LCA group and $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism also let $L$ be a closed $\tau$-invariant subgroup of $K$ and $\delta _K,\delta _L:H\rightarrow (0,\infty )$ be the continuous homomorphisms satisfying (2.3) and (2.4) respectively. The semi-direct product $G_{\tau ^L}=H\ltimes _{\tau ^L} K/L$ is a locally compact group with the left Haar measure $d\mu _{G_{\tau ^L}}(h,k+L)=\delta _K(h)\delta _L(h^{-1})dhd\sigma _{K/L}(k+L)$, where $\sigma _{K/L}$ is the left Haar measure of $K/L$.

Proof

Continuity of $\tau ^L:H\rightarrow Aut(K/L)$ is an immediate consequence of continuity of $\tau :H\rightarrow Aut(K)$. Hence, the semi-direct product $G_{\tau ^L}=H\ltimes _{\tau ^L} K/L$ is a locally compact group. Let $h\in H$ and $v\in L^1(K)$ be given. Then, $v\circ \tau _h\in L^1(K)$ and also for $k+L\in K/L$ we have

$$\begin{aligned} \begin{aligned} T_L(v\circ \tau _h)(k+L)&=\int _Lv\circ \tau _h(k+l)dl\\&=\int _Lv(k^h+l^h)dl\\&=\int _Lv(k^h+l)dl^{h^{-1}} =\delta _L(h)\int _Lv(k^h+l)dl\\&=\delta _L(h)T_L(v)(k^h+L). \end{aligned} \end{aligned}$$

Now, due to the Weil’s formula we get

$$\begin{aligned} \begin{aligned} \int _{K/L}T_L(v)(k+L)d\sigma _{K/L}\left( (k+L)^h\right)&= \int _{K/L}T_L(v)(k+L)d\sigma _{K/L}\left( k^h+L\right) \\&=\int _{K/L}T_L(v)(k^{h^{-1}}+L)d\sigma _{K/L}\left( k+L\right) \\&=\delta _L(h)\int _{K/L}T_L(v\circ \tau _{h^{-1}})\\&\qquad (k+L)d\sigma _{K/L}\left( k+L\right) \\&=\delta _L(h)\int _Kv\circ \tau _{h^{-1}}(k)dk\\&=\delta _L(h)\int _Kv(k)dk^h\\&=\delta _L(h)\delta _K(h^{-1})\int _Kv(k)dk\\&=\int _{K/L}T_L(v)(k+L)d\sigma _{K/L}(k+L). \end{aligned} \end{aligned}$$

Thus, $d\mu _{G{\tau ^L}}(h,k+L)=\delta _K(h)\delta _L(h^{-1})dhd\sigma _{K/L}(k+L)$ is the left Haar measure of $G_{\tau ^L}=H\ltimes _{\tau ^L} K/L$.$\square$

Applying Theorem 3.2 for $\widehat{K}$ and $L^\perp$ and also using Theorem 3.1 we achieve the following corollary.

Corollary 3.3

Let $H$ be a locally compact group, $K$ be an LCA group and $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism also let $L$ be a closed $\tau$-invariant subgroup of $K$ and $\delta _K,\delta _L:H\rightarrow (0,\infty )$ be the continuous homomorphisms satisfying (2.3) and (2.4) respectively. The semi-direct product $G_{\widehat{\tau }^{L^\perp }}=H\ltimes _{\widehat{\tau }^{L^\perp }}\widehat{K}/L^\perp$ is a locally compact group with the left Haar measure $d\mu _{\widehat{\tau }^{L^\perp }}(h,\omega L^\perp )=\delta _K(h^{-1})\delta _L(h)dhd\sigma _{\widehat{K}/L^\perp }(\omega L^\perp )$, where $\sigma _{\widehat{K}/L^\perp }$ is the left Haar measure of $\widehat{K}/L^\perp$.

We can also conclude the following propositions.

Proposition 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 $L$ be a closed $\tau$-invariant subgroup of $K$. The semi-direct product group $G_{\widehat{\tau ^L}}$ is a locally compact group and also $\Psi :G_{\widehat{\tau ^L}}\rightarrow H\ltimes _{\widehat{\tau }}L^\perp$ given by $(h,\zeta )\mapsto \Psi (h,\zeta )=(h,[\zeta ])$ where $[\zeta ](k)=\zeta (k+L)$, is a topological group isomorphism.

Proof

For all $\zeta \in \widehat{K/L}$, $h\in H$ and also $k\in K$ we have

$$\begin{aligned} \begin{aligned}{}[\zeta _h](k)&=\zeta _h(k+L)\\&=\zeta \left( (k+L)^{h^{-1}}\right) =\zeta (k^{h^{-1}}+L)=[\zeta ]_h. \end{aligned} \end{aligned}$$

Let $(h,\zeta ),(h^{\prime },\zeta ^{\prime })$ in $G_{\widehat{\tau ^L}}$. Due to Proposition 4.38 of [2] we get

$$\begin{aligned} \begin{aligned} \Psi [(h,\zeta )\ltimes _{\widehat{\tau ^L}}(h^{\prime },\zeta ^{\prime })]&=\Psi (hh^{\prime },\zeta \zeta ^{\prime }_h)\\&=(hh^{\prime },[\zeta \zeta ^{\prime }_h])\\&=(hh^{\prime },[\zeta ][\zeta ^{\prime }_h])\\&=(hh^{\prime },[\zeta ][\zeta ^{\prime }]_h)=\Psi (h,\zeta )\ltimes _{\widehat{\tau }^{L^\perp }}\Psi (h^{\prime },\zeta ^{\prime }). \end{aligned} \end{aligned}$$

Now, again Proposition 4.38 of [2] implies that $\Psi$ is a topological group isomorphism.   $\square$

Also, by similar argument we achieve the following proposition.

Proposition 3.5

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 $L$ be a closed $\tau$-invariant subgroup of $K$. The semi-direct product group $G_{\widehat{\tau }^{L^\perp }}$ is a locally compact group and also $\Phi :G_{\widehat{\tau }^{L^\perp }}\rightarrow H\ltimes _{\widehat{\tau }}\widehat{L}$ given by $(h,\omega L^\perp )\mapsto (h,\omega |_L)$, is a topological group isomorphism.

If $L$ is a $\tau$-invariant closed subgroup of $K$. Define $\tau ^{\times ,L}:=\tau ^L\times \widehat{\tau }^{L^\perp }:H\rightarrow Aut(K/L\times \widehat{K}/L^\perp )$ via $h\mapsto \tau ^{\times ,L}_h$ given by

$$\begin{aligned} \tau ^{\times ,L}_h(k+L,\omega L^\perp )=(k+L,\omega L^\perp )^h:=(k^h+L,\omega _hL^\perp ). \end{aligned}$$

(3.5)

It can be easily checked that $\tau ^{\times ,L}=\tau ^L\times \widehat{\tau }^{L^\perp }:H\rightarrow Aut(K/L\times \widehat{K}/L^\perp )$ is a continuous homomorphism. Thus, we can prove the following theorem.

Theorem 3.6

Let $H$ be a locally compact group, $K$ be an LCA group and $\tau :H\rightarrow Aut(K)$ be a continuous homomorphism also let $L$ be a closed $\tau$-invariant subgroup of $K$ and $\delta _K,\delta _L:H\rightarrow (0,\infty )$ be the continuous homomorphisms satisfying (2.3) and (2.4) respectively. The semi-direct product $G_{\tau ^{\times ,L}}=H\ltimes _{\tau ^{\times ,L}}\left( K/L\times \widehat{K}/L^\perp \right)$ is a locally compact group with the left Haar measure $d\mu _{G_{\tau ^{\times ,L}}}(h,k+L,\omega L^\perp )=dhd\sigma _{K/L}(k+L)d\sigma _{\widehat{K}/L^\perp }(\omega L^\perp )$, where $\sigma _{K/L}$ and $\sigma _{\widehat{K}/L^\perp }$ are the left Haar measures of $K/L$ and $\widehat{K}/L^\perp$ respectively.

Proof

Continuity of the homomorphism $\tau ^{\times ,L}:H\rightarrow Aut(K/L\times \widehat{K}/L^\perp )$ guarantee that the semi-direct product $G_{\tau ^{\times ,L}}=H\ltimes _{\tau ^{\times ,L}}\left( K/L\times \widehat{K}/L^\perp \right)$ is a locally compact group. Now, due to Theorem 3.2 and also Corollary 3.3 for all $h\in H$ we have

$$\begin{aligned} \begin{aligned} d\sigma _{K/L}\times \sigma _{\widehat{K}/L^\perp }\left( \tau ^{\times ,L}_h(k+L,\omega L^\perp )\right)&=d\sigma _{K/L}\times \sigma _{\widehat{K}/L^\perp }\left( (k+L,\omega L^\perp )^h\right) \\&=d\sigma _{K/L}\times \sigma _{\widehat{K}/L^\perp }(k^h+L,\omega _hL^\perp )\\&=d\sigma _{K/L}(k^h+L)d\sigma _{\widehat{K}/L^\perp }(\omega _hL^\perp )\\&=\delta _K(h)\delta _L(h^{-1})d\sigma _{K/L}(k+L)\delta _L(h)\\&\quad \delta _K(h^{-1})d\sigma _{\widehat{K}/L^\perp }(\omega L^\perp )\\&=d\sigma _{K/L}(k+L)d\sigma _{\widehat{K}/L^\perp }(\omega L^\perp )\\&=d\sigma _{K/L}\times \sigma _{\widehat{K}/L^\perp }(k+L,\omega L^\perp ). \end{aligned} \end{aligned}$$

Hence, $d\mu _{G_{\tau ^{\times ,L}}}(h,k+L,\omega L^\perp )=dhd\sigma _{K/L}(k+L)d\sigma _{\widehat{K}/L^\perp }(\omega L^\perp )$ is the left Haar measure.$\square$

Now let $L$ be a $\tau$-invariant uniform lattice in $K$. We define the $\tau$-Zak transform $\mathcal{Z }:L^2(G_\tau )\rightarrow L^2(G_{\tau ^{\times ,L}})$ via

$$\begin{aligned} \mathcal{Z }_Lf(h,k,\omega ):=\delta _K(h)^{1/2}Zf_h(k^h,\omega _h)=\delta _K(h)^{1/2}\sum _{l\in L}f(h,k^h+l)\omega _h(l). \end{aligned}$$

(3.6)

When $f\in L^2(G_\tau )$ then for a.e. $h\in H$ we have $f_h\in L^2(K)$, so that $Z_Lf_h$ and hence $\mathcal{Z }_L$ is well-defined. If $v\in L^2(K)$ and $u\in L^2(H)$ let $u\otimes _\delta v(h,k)=\delta _K(h)^{-1/2}u(h)v(k)$. Then, $u\otimes _\delta v\in L^2(G_\tau )$ and also we have

$$\begin{aligned} \begin{aligned} \mathcal{Z }_L(u\otimes _\delta v)(h,k,\omega )&=\delta _K(h)^{1/2}Z_Lf_h(k^h,\omega _h)\\&=u(h)Z_Lv(k^h,\omega _h). \end{aligned} \end{aligned}$$

The following proposition states the basic concrete properties of the $\tau$-Zak transform.

Proposition 3.7

Let $f\in L^2(G_\tau )$. Then for a.e. $(h,k,\omega )\in G_{\tau ^{\times ,L}}$ and also $(l,\xi )\in L\times L^\perp$ we have

• (1) $\mathcal{Z }_Lf(h,k+l,\omega )=\overline{\omega _h(l)}\mathcal{Z }_Lf(h,k,\omega )$.

• (2) $\mathcal{Z }_Lf(h,k,\omega \xi )=\mathcal{Z }_Lf(h,k,\omega )$.

Proof

• (1) Due to the quasi-periodicity of the Zak transform $Z_L$ we get

  $$\begin{aligned} \begin{aligned} \mathcal{Z }_Lf(h,k+l,\omega )&=\delta _K(h)^{1/2}Z_Lf_h(k^h+l,\omega _h)\\&=\delta _K(h)^{1/2}\overline{\omega _h(l)}Z_Lf_h(k^h,\omega _h) =\overline{\omega _h(l)}\mathcal{Z }_Lf(h,k,\omega ). \end{aligned} \end{aligned}$$

• (2) It is also guaranteed by the quasi-periodicity of the Zak transform $Z_L$.

In the following theorem we show that the $\tau$-Zak transform defined in (3.6) is an isometric transform.

Theorem 3.8

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 $L$ be a $\tau$-invariant uniform lattice in $K$. The $\tau$-Zak transform $\mathcal{Z }:L^2(G_\tau )\rightarrow L^2(G_{\tau ^{\times ,L}})$ is an isometric transform.

Proof

Using Theorem 3.6 and also Fubini’s theorem we have

$$\begin{aligned} \begin{aligned} \Vert \mathcal{Z }f\Vert _{L^2(G_{\tau ^{\times ,L}})}^2&=\int _{G_{\tau ^{\times ,L}}}|\mathcal{Z }f(h,k,\omega )|^2d\mu _{G_{\tau ^{\times ,L}}}(h,k+L,\omega L^\perp )\\&=\int _H\int _{K/L}\int _{\widehat{K}/L^\perp }|\mathcal{Z }f(h,k,\omega )|^2dhd\sigma _{K/L}(k+L)d\sigma _{\widehat{K}/L^\perp }(\omega L^\perp )\\&=\int _H\left( \int _{K/L}\int _{\widehat{K}/L^\perp }|Zf_h(k^h,\omega _h)|^2d\sigma _{K/L}(k+L)d\sigma _{\widehat{K}/L^\perp }(\omega L^\perp )\right) \\&\quad \delta _K(h)dh\\&=\int _H\left( \int _{K/L}\int _{\widehat{K}/L^\perp }|Zf_h(k,\omega )|^2d\sigma _{K/L}(k^{h^{-1}}+L)d\sigma _{\widehat{K}/L^\perp }(\omega _{h^{-1}} L^\perp )\right) \\&\quad \delta _K(h)dh\\&=\int _H\Vert f_h\Vert ^2_{L^2(K)}\delta _K(h)dh=\Vert f\Vert _{L^2(G_\tau )}^2. \end{aligned} \end{aligned}$$

Using linearity of the $\tau$-Zak transform and also the polarization identity we achieve the following orthogonality relation for the $\tau$-Zak transform.

Corollary 3.9

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 $L$ be a $\tau$-invariant uniform lattice in $K$. The $\tau$-Zak transform, for all $f,g\in L^2(G_\tau )$ satisfies the following orthogonality relation;

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

(3.7)

Note that the $\tau$-Zak transform generalizes the Zak transform given by (2.10) when $H$ is the identity group.

4 Examples and applications

In this section via some examples we illustrate how the preceding results apply for semidirect product groups.

4.1 Euclidean group

Let $H=\mathrm SL (2,\mathbb{Z })$ and $K=\mathbb{R }^2$ also for all $\sigma \in H$ let $\tau _\sigma :\mathbb{R }^2\rightarrow \mathbb{R }^2$ be given via $\tau _\sigma (\mathbf x )=\sigma \mathbf x$, for all $\mathbf x \in \mathbb{R }^2$. Then, $G_\tau =\mathrm SL (2,\mathbb{Z })\ltimes _\tau \mathbb{R }^2$ is a locally compact group with left Haar measure $d\mu _{G_\tau }(\sigma ,\mathbf x )=d\sigma d\mathbf {x}$. Then, $G_{\widehat{\tau }}=\mathrm SL (2,\mathbb{Z })\ltimes _{\widehat{\tau }}\widehat{\mathbb{R }^2}=\mathrm SL (2,\mathbb{Z })\ltimes _{\widehat{\tau }}\mathbb{R }^2$ where $\widehat{\tau }:\mathrm SL (2,\mathbb{Z })\rightarrow Aut(\widehat{\mathbb{R }}^2)$ is given by $\sigma \mapsto \widehat{\tau }_\sigma$ via $\widehat{\tau }_\sigma (\mathbf{w})=\mathbf{w}_\sigma$ for all $\mathbf{w}\in \widehat{\mathbb{R }}^2$, where for all $\mathbf x \in \mathbb{R }^2$ we have

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

If $\sigma =\left( \begin{array}{ll} a &{} b \\ c &{} d \\ \end{array} \right) \in \mathrm SL (2,\mathbb{Z })$, then for all $\mathbf w =(w_1,w_2)\in \widehat{\mathbb{R ^2}}$ and $\mathbf{x}=(x_1,x_2)\in \mathbb{R }^2$ we have

$$\begin{aligned} \langle \mathbf{{x}},\mathbf{{w}}_\sigma \rangle =e^{-2\pi i (\sigma ^{-1}\mathbf{{x}},\mathbf{{w}})} =e^{-2\pi i(dw_1x_1-bw_1x_2-cw_2x_1+aw_2x_2)}. \end{aligned}$$

(4.1)

Example 4.1

Let $\alpha ,\beta \in \mathbb{R }$ and $L_{\alpha ,\beta }=\alpha \mathbb{Z }\times \beta \mathbb{Z }$. Then, $L_{(\alpha ,\beta )}$ is a 2D $\tau$-invariant uniform lattice in $\mathbb{R }^2$ with $L_{(\alpha ,\beta )}^\perp =L_{(\alpha ^{-1},\beta ^{-1})}$. The continuous homomorphism $\tau ^{\times ,L_{(\alpha ,\beta )}}:\mathrm SL (2,\mathbb{Z })\rightarrow Aut(\mathbb{R }^2/L_{\alpha ,\beta }\times \widehat{\mathbb{R }}^2/L_{(\alpha ^{-1},\beta ^{-1})})=Aut(\mathbb{T }^4)$ is given by $\sigma \mapsto \tau ^{\times ,L_{(\alpha ,\beta )}}_\sigma$ via

$$\begin{aligned} \tau ^{\times ,L_{(\alpha ,\beta )}}_\sigma \left( \mathbf x +L_{(\alpha ,\beta )},\mathbf w +L_{(\alpha ,\beta )}^\perp \right) =\left( \sigma \mathbf x +L_{(\alpha ,\beta )},\mathbf w \sigma ^{{-1}}+L_{(\alpha ,\beta )}^\perp \right) . \end{aligned}$$

(4.2)

If $f:\mathrm SL (2,\mathbb{Z })\times \mathbb{R }^2\rightarrow \mathbb{C }$ satisfies

$$\begin{aligned} \int _\mathrm{SL (2,\mathbb{Z })}\int _\mathbb{R }|f(\sigma ,\mathbf x )|^2d\sigma d\mathbf x <\infty , \end{aligned}$$

(4.3)

then for $(\sigma ,\mathbf x ,\mathbf w )\in \mathrm SL (2,\mathbb{Z })\times \mathbb{R }^2\times \mathbb{R }^2$ we have

$$\begin{aligned} \begin{aligned} \mathcal{Z }_{L_{(\alpha ,\beta )}}f(\sigma ,\mathbf x ,\mathbf w )&=Z_{L_{(\alpha ,\beta )}}f_\sigma (\sigma \mathbf x ,\mathbf w \sigma ^{-1})\\&=\sum _{n=-\infty }^{\infty }\sum _{m=-\infty }^{\infty }f\left( \sigma ,\sigma \mathbf x +(\alpha n,\beta m)\right) \langle (\alpha ^{-1} n,\beta ^{-1} m),\mathbf w \sigma ^{-1}\rangle . \end{aligned} \end{aligned}$$

4.2 Weyl-Heisenberg group

Let $K$ be an LCA group with the Haar measure $dk$ and $\widehat{K}$ be the dual group of $K$ with the Haar measure $d\omega$ 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 $\tau _s(\omega ,z)=(\omega ,z.\omega (s))$. The semidirect product $G_\tau =K\ltimes _\tau (\widehat{K}\times \mathbb{T })$ is called the Weyl-Heisenberg group associated with $K$ which is also denoted by $\mathbb{H }(K)$. The group operation for $(k,\omega ,z),(k^{\prime },\omega ^{\prime },z^{\prime })\in K\ltimes _\tau (\widehat{K}\times \mathbb{T })$ is

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

(4.4)

If $dz$ is the Haar measure of the circle group, then $dkd\omega dz$ 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 and also Proposition 4.6 of [2] and 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 [2], for each $(k,n)\in K\times \mathbb{Z }$ and also for all $(\omega ,z)\in \widehat{K}\times \mathbb{T }$ we have

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

Thus, for all $k,s\in K$ and $n\in \mathbb{Z }$ we have

$$\begin{aligned} \widehat{\tau }_s(k,n)=(k,n)_s=(k-ns,n). \end{aligned}$$

(4.5)

Therefore, $G_{\widehat{\tau }}$ has the underlying set $K\times K\times \mathbb{Z }$ with the following group operation;

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

Example 4.2

Let $K=\mathbb{Z }$ then $G_{{\tau }}=\mathbb{Z }\ltimes _{{\tau }}\mathbb{T }^2$, where $\tau :\mathbb{Z }\rightarrow Aut(\mathbb{T }^2)$ is defined via $\ell \mapsto \tau _\ell$ given by $\tau _\ell (w,z)=(w,zw^\ell )$ for all $w,z\in \mathbb{T }$ and $\ell \in \mathbb{Z }$. Due to (4.5), $\widehat{\tau }:\mathbb{Z }\rightarrow Aut(\mathbb{Z }^2)$ is given by $\ell \mapsto \widehat{\tau }_{\ell }$ where $\widehat{\tau }_\ell (p,q)=(p-q\ell ,q)$ for all $\ell ,p,q\in \mathbb{Z }$. Now for all $n\in \mathbb{Z }$ let $\mathbb{T }_n=\{z\in \mathbb{T }:z^n=1\}$ and also for $(n,m)\in \mathbb{Z }^2$ let $L_{(n,m)}=\mathbb{T }_n\times \mathbb{T }_m$. Then,

$$\begin{aligned} \begin{aligned} L_{(n,m)}^\perp&=(\mathbb{T }_n\times \mathbb{T }_m)^\perp \\&=\mathbb{T }_n^\perp \times \mathbb{T }_m^\perp =n\mathbb{Z }\times m\mathbb{Z }. \end{aligned} \end{aligned}$$

If $n|m$ then $L_{(n,m)}$ is a $\tau$-invariant uniform lattice in $\mathbb{T }^2$. In this case $\tau ^{\times ,L_{n,m}}:\mathbb{Z }\rightarrow Aut(\mathbb{T }^2/L_{(n,m)}\times \mathbb{Z }^2/L_{(n,m)}^\perp )$ for all $\ell \in \mathbb{Z }$ is given by

$$\begin{aligned} \begin{aligned}&\tau ^{\times ,L_{n,m}}_\ell \left( (w,z)+L_{(n,m)},(p,q)+L_{(n,m)}^\perp \right) \\&\quad =\left( \tau _\ell ^{L_{(n,m)}}(w,z)+L_{(n,m)},\tau _\ell ^{L_{(n,m)}^\perp }(p,q)+L_{(n,m)}^\perp \right) \\&\quad =\left( (w,z)^\ell +L_{(n,m)},(p,q)_\ell +L_{(n,m)}^\perp \right) \\&\quad =\left( (w,zw^\ell )+L_{(n,m)},(p-q\ell ,q)+L_{(n,m)}^\perp \right) . \end{aligned} \end{aligned}$$

Since for all $n\in \mathbb{Z }$ we have $\mathbb{T }/\mathbb{T }_n=\mathbb{T }$ and also $\mathbb{Z }/n\mathbb{Z }=\mathbb{Z }_n$ we can consider

$$\begin{aligned} \tau ^{\times ,L_{n,m}}:\mathbb{Z }\rightarrow Aut(\mathbb{T }^2\times \mathbb{Z }_n\times \mathbb{Z }_m) \end{aligned}$$

(4.6)

via $\tau ^{\times ,L_{n,m}}_\ell (w,z,p,q)=\left( w,zw^\ell ,p-q\ell ,q\right)$ for all $(w,z,p,q)\in \mathbb{T }^2\times \mathbb{Z }_n\times \mathbb{Z }_m$. If $f:\mathbb{Z }\times \mathbb{T }^2\rightarrow \mathbb{C }$ satisfies

$$\begin{aligned} \sum _{\ell =-\infty }^{\infty }\int _{0}^{2\pi }\int _{0}^{2\pi }|f(\ell ,e^{it},e^{i\theta })|^2dtd\theta <\infty , \end{aligned}$$

(4.7)

then for $(\ell ,e^{i\theta },e^{it},p,q)\in \mathbb{Z }\times \mathbb{T }^2\times \mathbb{Z }^2$ we have

$$\begin{aligned} \begin{aligned} \mathcal{Z }_{L_{(n,m)}}f(\ell ,e^{i\theta },e^{it},p,q)&=Z_{L_{(n,m)}}f_\ell \left( \ell ,e^{i\theta },e^{it},p,q\right) \\&=Z_{L_{(n,m)}}f_\ell \left( (e^{it},e^{i(t+\theta \ell )}),(p-q\ell ,q)\right) \\&=\sum _{(w,z)\in L_{(n,m)}}f\left( \ell ,w+e^{it},z+e^{i(t+\theta \ell )}\right) \\&\quad (p-q\ell ,q)(w,z)\\&=\sum _{(w,z)\in L_{(n,m)}}f\left( \ell ,w+e^{it},z+e^{i(t+\theta \ell )}\right) w^{p-q\ell }z^q\\&=\sum _{k=1}^n\sum _{j=1}^mf\left( \ell ,e^{i(2\pi k/n+t)},e^{i(2\pi j/m+t+\theta \ell )}\right) \\&\quad e^{\frac{2\pi ik}{n}(p-q\ell )}e^{\frac{2\pi ij}{m}q}. \end{aligned} \end{aligned}$$

According to Theorem 3.8 we get

$$\begin{aligned} \begin{aligned}&\sum _{\ell =-\infty }^{\infty }\int _0^{2\pi }\int _0^{2\pi }\sum _{p=0}^n\sum _{q=0}^m\left| \sum _{k=1}^n\sum _{j=1}^mf\left( \ell ,e^{i(2\pi k/n+t)},e^{i(2\pi j/m+t+\theta \ell )}\right) e^{\frac{2\pi ik}{n}(p-q\ell )}\right. \\&\left. \quad e^{\frac{2\pi ij}{m}q}\right| ^2dtd\theta \\&\quad \quad =\sum _{\ell =-\infty }^{\infty }\int _0^{2\pi }\int _0^{2\pi }\sum _{p=0}^n\sum _{q=0}^m\left| \mathcal{Z }_{L_{(n,m)}}f(\ell ,e^{it},e^{i\theta },p,q)\right| ^2dtd\theta \\&\quad \quad =\sum _{\ell =-\infty }^{\infty }\int _{0}^{2\pi }\int _{0}^{2\pi }\left| f(\ell ,e^{it},e^{i\theta })\right| ^2dtd\theta . \end{aligned} \end{aligned}$$