Sampling theorem and reconstruction formula for the space of translates on the Heisenberg group

TL;DR

This paper establishes necessary and sufficient conditions for sampling and reconstruction of functions in shift-invariant subspaces of L^2 on the Heisenberg group, extending classical results to a non-commutative setting.

Contribution

It provides a comprehensive framework for sampling and reconstruction in the Heisenberg group, generalizing classical theorems to this non-commutative context.

Findings

Derived conditions for sampling in the Heisenberg group

Established reconstruction formulas for shift-invariant subspaces

Extended classical sampling theorems to a non-commutative setting

Abstract

The paper deals with the necessary and sufficient conditions for obtaining reconstruction formulae and sampling theorems for every function belonging to the principal shift invariant subspace of $L^2(\mathbb{H}^n)$, both in the time domain and a transform domain, where $\mathbb{H}^n$ denotes the Heisenberg group.

Equations

$$f(x)=\sum _{k\in \mathbb{Z}}f(k)\frac{\sin \pi (x-k)}{\pi (x-k)}.$$

$$A\Vert f{\Vert }^2\le \sum _{k\in \mathbb{Z}}|⟨f,f_k⟩{|}^2\le B\Vert f{\Vert }^2,\forall f\in \mathcal{H}.$$

$$A\sum _k|c_k{|}^2\le \Vert \sum _k{c}_k{f}_k{\Vert }^2\le B\sum _k|c_k{|}^2.$$

$$S:\mathcal{H}\to \mathcal{H},Sf=\sum _{k\in \mathbb{Z}}⟨f,f_k⟩f_k$$

$$f=\sum _{k\in \mathbb{Z}}⟨f,g_k⟩f_k,\forall f\in \mathcal{H}$$

$$f=\sum _{k\in \mathbb{Z}}{c}_k{f}_k,f\in \mathcal{H},$$

$$(x,y,t)(u,v,s)=\left(x+u,y+v,t+s+\frac{1}{2}(u\cdot y-v\cdot x)\right).$$

$${\pi }_{\lambda }(x,y,t)\phi (\xi )=e^{2\pi i\lambda t}{e}^{2\pi i\lambda (x\cdot \xi +\frac{1}{2}x\cdot y)}\phi (\xi +y),\phi \in L^2({\mathbb{R}}^n).$$

$$\hat{f}(\lambda )=\underset{{\mathbb{C}}^n\times \mathbb{R}}{\int }f(z,t){\pi }_{\lambda }(z,t)dzdt.$$

$$f^{\lambda }(z)=\underset{\mathbb{R}}{\int }f(z,t)e^{2\pi i\lambda t}dt.$$

$$W_{\lambda }(f)=\underset{{\mathbb{C}}^n}{\int }f(z){\pi }_{\lambda }(z,0)dz.$$

$$\hat{f}(\lambda )=W_{\lambda }(f^{\lambda }).$$

$$K_f^{\lambda }(\xi ,\eta )=\underset{{\mathbb{R}}^n}{\int }f(x,\eta -\xi )e^{\pi i\lambda x\cdot (\xi +\eta )}dx.$$

$$f\ast g(z,t)=\underset{{\mathbb{C}}^n\times \mathbb{R}}{\int }f((z,t)(-w,-s))g(w,s)dwds.$$

$$(f\ast g\widehat{)}(\lambda )=\hat{f}(\lambda )\hat{g}(\lambda ).$$

$$\Vert W_{\lambda }(f){\Vert }_{{\mathcal{B}}_2}=\Vert K_f^{\lambda }{\Vert }_{L^2({\mathbb{C}}^n)}=\frac{1}{|\lambda {|}^{n/2}}\Vert f{\Vert }_{L^2({\mathbb{C}}^n)},$$

$$⟨W_{\lambda }(f),W_{\lambda }(g){⟩}_{{\mathcal{B}}_2}=⟨K_f^{\lambda },K_g^{\lambda }{⟩}_{L^2({\mathbb{C}}^n)}=\frac{1}{|\lambda {|}^n}⟨f,g{⟩}_{L^2({\mathbb{C}}^n)}.$$

$$\Vert \hat{f}{\Vert }_{L^2({\mathbb{R}}^{\ast },{\mathcal{B}}_2;d\mu )}=\Vert f{\Vert }_{L^2({\mathbb{H}}^n)},$$

$$F_{\lambda }({\xi }_1,{\xi }_2)={\pi }_{\lambda }({\xi }_1,{\xi }_2,0)\hat{f}(\lambda ){\pi }_{\lambda }(-{\xi }_1,-{\xi }_2,0),({\xi }_1,{\xi }_2)\in {\mathbb{R}}^{2n}.$$

$$L_{(2k,l,m)}\phi (x,y,t)$$

$$=\phi \left(x-2k,y-l,t-m+\frac{1}{2}(y\cdot 2k-x\cdot l)\right),$$

$$V(\phi )=\overline{span}\{L_{(2k,l,m)}\phi :(k,l,m)\in {\mathbb{Z}}^n\times {\mathbb{Z}}^n\times \mathbb{Z}\}.$$

$$\widehat{L_{(2k,l,m)}\phi }(\lambda )=e^{2\pi im\lambda }\widehat{L_{(2k,l,0)}\phi }(\lambda )$$

$$\Vert \widehat{L_{(2k,l,0)}\phi }(\lambda ){\Vert }_{{\mathcal{B}}_2}=\Vert \hat{\phi }(\lambda ){\Vert }_{{\mathcal{B}}_2},$$

$$G_{k,l}^{\phi }(\lambda )=\sum _{r\in \mathbb{Z}}⟨\hat{\phi }(\lambda +r),\widehat{L_{(2k,l,0)}\phi }(\lambda +r){⟩}_{{\mathcal{B}}_2}|\lambda +r{|}^n,\lambda \in (0,1].$$

$$G_{k,l}^{\phi }(\lambda )=\sum _{r\in \mathbb{Z}}\underset{{\mathbb{R}}^n}{\int }\underset{{\mathbb{R}}^n}{\int }$$

$$A\le G_{0,0}^{\phi }(\lambda )\le B\text{a.e. }\lambda \in {\Omega }_{\phi },$$

$$f=\sum _{(k,l,m)\in {\mathbb{Z}}^{2n+1}}f(2k,l,m)L_{(2k,l,m)}\psi ,\forall f\in V_0,$$

$$\underset{(x,y,t)\in {\mathbb{T}}^{2n+1}}{\sup }\sum _{m\in \mathbb{Z}}|L_{(2k,l,m)}\phi (x,y,t){|}^2<\infty ,$$

$$A{\chi }_{{\Omega }_{\phi }}(\lambda )\le \Vert P(\lambda ){\Vert }_{l^2({\mathbb{Z}}^{2n})}^2\le B{\chi }_{{\Omega }_{\phi }}(\lambda )\text{a.e. }\lambda \in \mathbb{R}$$

$$P(\lambda )=\{P_{k,l}(\lambda ){\}}_{(k,l)\in {\mathbb{Z}}^{2n}}\text{ with }{P}_{k,l}(\lambda )=\sum _{m\in \mathbb{Z}}\phi (2k,l,m)e^{2\pi im\lambda },$$

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1. Introduction

The classical Shannon sampling theorem [25] states that if $f\in L^{2}(\mathbb{R})$ whose Fourier transform has support in $[-\frac{1}{2},\frac{1}{2}]$, then $f$ can be recovered from its uniform samples at integers by the formula

[TABLE]

The reconstruction of such band limited functions $f$ from nonuniform samples dates back to the works of Paley and Wiener [19] and Duffin and Eachus [8] in Mathematics literature. Later, Kadec [14] proved that in order to obtain a stable set of sampling for such functions from the nonuniform samples the perturbation should be at the most $\tfrac{1}{4}$ from the integers. The more general sampling condition was given in terms of Beurling density in [15]. We refer to the work of Butzer and Stens [6] for the classical historical review of sampling theory.

In [31], Walter extended the Shannon sampling theorem to wavelet subspaces. In 1999, Zhou and Sun [34] provided a necessary and sufficient condition under which every function in a closed subspace $V_{0}$ of $L^{2}(\mathbb{R})$ has a sampling expansion. In other words, they proved necessary and sufficient conditions to obtain the sampling formula $f(x)=\sum_{k}f(k)g(x-k)$, $\forall\,f\in V_{0}$ which holds for some $\{g(\cdot-n):n\in\mathbb{Z}\}$ in $V_{0}$ with the convergence being both in $L^{2}(\mathbb{R})$ and pointwise on $\mathbb{R}$.

The problem of sampling and reconstruction has been studied in a shift invariant space with a single generator by several authors. (See for example [16, 2, 28, 12, 24].) For the sampling problem in a shift invariant space with multiple generators, we refer to [4, 3, 26, 32, 11, 1, 17, 27, 18, 33, 10, 21].

In [22], a Shannon type sampling theorem was proved for the Heisenberg group. Recently, Radha and Saswata obtained a sampling theorem on a subspace of a twisted shift-invariant space in [23]. In fact, they gave a necessary and sufficient condition for obtaining a reconstruction formula for functions belonging to a subspace $V^{0,t}(\varphi)=\overline{span}\{T^{t}_{(k,0)}\varphi:k\in\mathbb{Z}^{n}\}$ of the principal twisted shift invariant space $V^{t}(\varphi)=\overline{span}\{T^{t}_{(k,l)}\varphi:(k,l)\in\mathbb{Z}^{2n}\}$ in $L^{2}(\mathbb{R}^{2n})$ from their samples $\{f(k,j):k\in\mathbb{Z}^{n}\}$ for each fixed $j\in\mathbb{Z}^{n}$.

The aim of our paper is to look for necessary and sufficient conditions for obtaining a sampling theorem and a reconstruction formula for every function belonging to the principal shift invariant space of $L^{2}(\mathbb{H}^{n})$. We also consider a transform $F_{\lambda}$ of functions $f$ in $L^{2}(\mathbb{H}^{n})$ which is an operator valued function on $\mathbb{R}^{2n}$ and derive reconstruction formula in the transform domain of $f$. We provide an inversion theorem which recovers $f$ from $F_{\lambda}$, thereby paving way for another reconstruction formula for $f$ in the time domain. We may also view the reconstruction formulae as matrix equations which aids in getting corresponding sampling theorems.

2. Necessary background

Let $\mathcal{H}$ be a separable Hilbert space.

Definition 2.1**.**

A sequence $\{f_{k}:k\in\mathbb{Z}\}$ in $\mathcal{H}$ is a frame for $\mathcal{H}$ if there exist constants $A,B>0$ such that

[TABLE]

The numbers $A$ and $B$ are called frame bounds. If the right hand side inequality holds, then $\{f_{k}:k\in\mathbb{Z}\}$ is said to be a Bessel sequence with bound $B$. A sequence $\{f_{k}:k\in\mathbb{Z}\}$ in $\mathcal{H}$ is said to be a frame sequence if it is a frame for $\overline{\text{span}}\{f_{k}:k\in\mathbb{Z}\}$.

Definition 2.2**.**

A Riesz basis for $\mathcal{H}$ is a family of the form $\{Ue_{k}:k\in\mathbb{Z}\}$, where $\{e_{k}:k\in\mathbb{Z}\}$ is an orthonormal basis for $\mathcal{H}$ and $U:\mathcal{H}\rightarrow\mathcal{H}$ is a bounded invertible operator. Alternatively, a sequence $\{f_{k}:k\in\mathbb{Z}\}$ is a Riesz basis for $\mathcal{H}$ if $\{f_{k}:k\in\mathbb{Z}\}$ is complete in $\mathcal{H}$, and there exist constants $A,B>0$ such that for every finite scalar sequence $\{c_{k}\}$, one has

[TABLE]

A sequence $\{f_{k}:k\in\mathbb{Z}\}$ in $\mathcal{H}$ is a Riesz sequence if it is a Riesz basis for $\overline{\text{span}}\{f_{k}:k\in\mathbb{Z}\}$.

Let $\{f_{k}:k\in\mathbb{Z}\}$ be a frame for $\mathcal{H}$. The operator

[TABLE]

is called the frame operator. It is bounded, invertible, self-adjoint and positive.

Definition 2.3**.**

Suppose $\{f_{k}:k\in\mathbb{Z}\}$ is a frame for $\mathcal{H}$. The canonical dual frame of $\{f_{k}:k\in\mathbb{Z}\}$ is the frame $\{S^{-1}f_{k}:k\in\mathbb{Z}\}$ for $\mathcal{H}$, where $S$ is the frame operator.

Any element $f\in\mathcal{H}$ can be expressed in terms of $S^{-1}f_{k}$ as follows.

1. (i)

$f=\sum_{k\in\mathbb{Z}}\langle f,S^{-1}f_{k}\rangle f_{k}$, 2. (ii)

$f=\sum_{k\in\mathbb{Z}}\langle f,f_{k}\rangle S^{-1}f_{k}$.

Definition 2.4**.**

Let $\{f_{k}:k\in\mathbb{Z}\}$ be a frame for $\mathcal{H}$. A frame $\{g_{k}:k\in\mathbb{Z}\}$ in $\mathcal{H}$ such that

[TABLE]

is called a dual frame of $\{f_{k}:k\in\mathbb{Z}\}$.

Let $\{f_{k}:k\in\mathbb{Z}\}$ be a frame for $\mathcal{H}$ which is not a Riesz basis. Then there exist dual frames other than the canonical dual frame. So, every element in $\mathcal{H}$ has a representation in terms of the frame elements, namely,

[TABLE]

where $\{c_{k}\}\in l^{2}(\mathbb{Z})$. The above representation is called a frame expansion of $f$.

For more on frames and bases, we refer to the books by Christensen[7] and Heil[13].

The Heisenberg group $\mathbb{H}^{n}$ is a nilpotent Lie group whose underlying manifold is $\mathbb{R}^{n}\times\mathbb{R}^{n}\times\mathbb{R}$ which satisfies the group law

[TABLE]

It is a nonabelian noncompact locally compact group. The Haar measure on $\mathbb{H}^{n}$ is the Lebesgue measure $dxdydt$. From the well known Stone-von Neumann theorem it follows that every infinite dimensional irreducible unitary representation on the Heisenberg group is unitarily equivalent to the representation $\pi_{\lambda},\,\lambda\in\mathbb{R}^{\ast}$, where $\pi_{\lambda}$ is defined by

[TABLE]

For $f\in L^{1}(\mathbb{H}^{n})$, the group Fourier transform $\hat{f}$ is defined as follows. For $\lambda\in\mathbb{R}^{\ast},\,\hat{f}(\lambda)$ is given by

[TABLE]

More explicitly, $\hat{f}(\lambda)$ is the bounded operator acting on $L^{2}(\mathbb{R}^{n})$ (i.e., $\hat{f}(\lambda)\in\mathscr{B}(L^{2}(\mathbb{R}^{n}))$) given by $\hat{f}(\lambda)\varphi=\int\limits_{\mathbb{C}^{n}\times\mathbb{R}}f(z,t)\pi_{\lambda}(z,t)\varphi dzdt,\,\varphi\in L^{2}(\mathbb{R}^{n})$, where the integral is a Bochner integral taking values in the Hilbert space $L^{2}(\mathbb{R}^{n})$. Further, $\|\hat{f}(\lambda)\|_{\mathscr{B}}\leq\|f\|_{L^{1}(\mathbb{H}^{n})}$. The inverse Fourier transform of $f\in L^{1}(\mathbb{H}^{n})$ in the $t$ variable, denoted by $f^{\lambda}$, is defined as

[TABLE]

It can be seen that $f^{\lambda}\in L^{1}(\mathbb{C}^{n})$. For $f\in L^{1}(\mathbb{C}^{n})$, the operator $W_{\lambda}(f)$ on $L^{2}(\mathbb{R}^{n})$ is defined as

[TABLE]

Clearly, there is a relation between group Fourier transform and $W_{\lambda}$ given by

[TABLE]

Moreover, $W_{\lambda}(f)$ is an integral operator on $L^{2}(\mathbb{R}^{n})$ with kernel $K^{\lambda}_{f}$ given by

[TABLE]

In particular when $\lambda=1,\,W_{\lambda}(f)$ is denoted by $W(f)$ which is called the Weyl transform of $f$ and the associated kernel is denoted by $K_{f}$.

If $f$ and $g$ are in $L^{1}(\mathbb{H}^{n})$, then their convolution is defined by

[TABLE]

Under this convolution, $L^{1}(\mathbb{H}^{n})$ turns out to be a noncommutative Banach algebra. The group Fourier transform takes convolution into products like in the classical case, i.e.,

[TABLE]

Analogous to the case of the Euclidean Fourier transform, the definitions of $W_{\lambda}$ and the group Fourier transform $\hat{f}$ can be extended to functions in $L^{2}(\mathbb{C}^{n})$ and $L^{2}(\mathbb{H}^{n})$ respectively through the density argument. In fact, for $f\in L^{2}(\mathbb{C}^{n}),\,W_{\lambda}(f)$ is a Hilbert-Schmidt operator on $L^{2}(\mathbb{R}^{n})$ which satisfies

[TABLE]

where $\mathcal{B}_{2}=\mathcal{B}_{2}(L^{2}(\mathbb{R}^{n}))$ denotes the class of Hilbert-Schmidt operators on $L^{2}(\mathbb{R}^{n})$. In other words, for $f,g\in L^{2}(\mathbb{C}^{n})$,

[TABLE]

Furthermore, the group Fourier transform satisfies the Plancherel formula

[TABLE]

where $L^{2}(\mathbb{R}^{\ast},\mathcal{B}_{2};d\mu)$ stands for the space of functions on $\mathbb{R}^{\ast}$ taking values in $\mathcal{B}_{2}$ and square integrable with respect to the measure $d\mu(\lambda)=|\lambda|^{n}d\lambda.$

For further study on the Heisenberg group, we refer to [9] and [30].

For $f\in L^{2}(\mathbb{H}^{n})$ and $\lambda\in\mathbb{R}^{\ast}$, an operator valued function, $F_{\lambda}$, on $\mathbb{R}^{2n}$ is defined as

[TABLE]

The above operator was used by Thangavelu [29] in studying Paley-Wiener theorem for the Heisenberg group.

Let $\mathscr{L}$ denote a lattice in $\mathbb{H}^{n}$. In other words, $\mathscr{L}$ is a discrete subgroup of the Heisenberg group $\mathbb{H}^{n}$ such that $\mathbb{H}^{n}/\mathscr{L}$ is compact. For $\varphi\in L^{2}(\mathbb{H}^{n})$, the shift invariant space, $V(\varphi)$, is defined to be $\overline{span}\{{L_{l}\varphi:l\in\mathscr{L}}\}$, where $L_{l}\varphi(X)=\varphi(l^{-1}\cdot X),~{}X\in\mathbb{H}^{n}$. However, from the computational point of view, one can work with the standard lattice $\{(2k,l,m):k,l\in\mathbb{Z}^{n},m\in\mathbb{Z}\}$ in place of $\mathscr{L}$. Explicitly, the action of the left translation $L_{(2k,l,m)}$ on $L^{2}(\mathbb{H}^{n})$ for $(2k,l,m)\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}$ is given by

[TABLE]

for $\varphi\in L^{2}(\mathbb{H}^{n})$ and the principal shift invariant space generated by $\varphi$ is given by

[TABLE]

The left translates satisfy the following equations.

[TABLE]

for $\lambda\in\mathbb{R}^{\ast}$. For a study of frames and Riesz bases in connection with shift invariant spaces on $\mathbb{H}^{n}$, we refer to [5, 20]. In [20], the system of left translates on $\mathbb{H}^{n}$ has been characterized to be a frame sequence and a Riesz sequence in terms of the weight function $G^{\varphi}_{k,l}$ defined below.

Definition 2.5**.**

For $\varphi\in L^{2}(\mathbb{H}^{n})$ and $k,l\in\mathbb{Z}^{n}$, the function $G^{\varphi}_{k,l}$ is defined by

[TABLE]

It can also be written in terms of the kernel of $W_{\lambda}$ as

[TABLE]

We refer to [20] in this connection.

Definition 2.6** ([20]).**

A function $\varphi\in L^{2}(\mathbb{H}^{n})$ is said to satisfy Condition C if $G_{k,l}^{\varphi}(\lambda)=0$ a.e. $\lambda\in(0,1]$, for all $(k,l)\in\mathbb{Z}^{2n}\backslash\{(0,0)\}$.

The characterizations for the system of left translates on $\mathbb{H}^{n}$ to be a Bessel sequence and a frame sequence are as follows.

Theorem 2.7** ([20]).**

Let $\varphi\in L^{2}(\mathbb{H}^{n})$. Suppose $\{L_{(2k,l,m)}\varphi:(k,l,m)\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}\}$ is a Bessel sequence in $L^{2}(\mathbb{H}^{n})$ with bound $B$. Then $G^{\varphi}_{0,0}(\lambda)\leq B\quad\text{a.e. }\lambda\in(0,1]$. Conversely, suppose there exists $B>0$ such that $G^{\varphi}_{0,0}(\lambda)\leq B\quad\text{a.e. }\lambda\in(0,1]$ and $\varphi$ satisfies Condition $C$. Then $\{L_{(2k,l,m)}\varphi:(k,l,m)\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}\}$ is a Bessel sequence in $L^{2}(\mathbb{H}^{n})$ with bound $B$.

Theorem 2.8** ([20]).**

Let $\varphi\in L^{2}(\mathbb{H}^{n})$ satisfy Condition C. Then the collection $\{L_{(2k,l,m)}\varphi:(k,l,m)\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}\}$ is a frame for $V(\varphi)$ with frame bounds $A,B>0$ if and only if

[TABLE]

where $\Omega_{\varphi}=\{\eta\in(0,1]:G^{\varphi}_{0,0}(\eta)>0\}.$

3. The main results

We shall now state some necessary conditions when a sampling formula holds for the principal shift invariant space $V(\varphi)$, $\varphi\in L^{2}(\mathbb{H}^{n})$.

Theorem 3.1**.**

Let $V_{0}$ be a closed subspace of $L^{2}(\mathbb{H}^{n})$. Let $\varphi\in L^{2}(\mathbb{H}^{n})$ satisfy Condition C and the collection $\{L_{(2k,l,m)}\varphi:(k,l,m)\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}\}$ be a frame for $V_{0}$. Suppose, for each $(k,l)\in\mathbb{Z}^{2n}$, $\sum_{m\in\mathbb{Z}}c_{m}L_{(2k,l,m)}\varphi$ converges pointwise to a continuous function, for any $\{c_{m}\}\in l^{2}(\mathbb{Z})$. Further, suppose there exists a function $\psi\in L^{2}(\mathbb{H}^{n})$ satisfying Condition C such that $\{L_{(2k,l,m)}\psi:(k,l,m)\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}\}$ is a frame for $V_{0}$ and

[TABLE]

with the convergence in $L^{2}(\mathbb{H}^{n})$. Then $\varphi\in C(\mathbb{H}^{n})$,

[TABLE]

for each $(k,l)\in\mathbb{Z}^{2n}$ and

[TABLE]

for some constants $A,B>0$, where

[TABLE]

$\Omega_{\varphi}=\{\lambda\in\mathbb{R}:G^{\varphi}_{0,0}(\lambda)>0\}$* and $G^{\varphi}_{0,0}$ is as in Definition 2.5.*

The following theorem gives sufficient conditions for obtaining a reconstruction formula for the principal shift invariant subspace of $L^{2}(\mathbb{H}^{n})$.

Theorem 3.2**.**

Let $V_{0}$ be a closed subspace of $L^{2}(\mathbb{H}^{n})$. Let $\varphi\in L^{2}(\mathbb{H}^{n})$ satisfy Condition C and the collection $\{L_{(2k,l,m)}\varphi:(k,l,m)\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}\}$ be a frame for $V_{0}$. For $f\in V_{0}$, let

[TABLE]

where $g_{2k,l}=\sum_{m\in\mathbb{Z}}c_{2k,l,m}L_{(2k,l,m)}\varphi$ in $L^{2}(\mathbb{H}^{n})$. Suppose $\varphi\in C(\mathbb{H}^{n})$ and for each $(k,l)\in\mathbb{Z}^{2n}$,

[TABLE]

Further, suppose there exist constants $A,B>0$ such that

[TABLE]

where $P(\lambda)$ and $\Omega_{\varphi}$ are as in Theorem 3.1. Then, for each $(k,l)\in\mathbb{Z}^{2n}$, $\sum_{m\in\mathbb{Z}}c_{m}L_{(2k,l,m)}\varphi$ converges pointwise to a continuous function, for any $\{c_{m}\}\in l^{2}(\mathbb{Z})$ and there exists a function $\psi\in L^{2}(\mathbb{H}^{n})$ satisfying Condition C such that $\{L_{(2k,l,m)}\psi:(k,l,m)\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}\}$ is a frame for $V_{0}$ with the following reconstruction formula. For every $f\in V_{0}$,

[TABLE]

with the convergence in $L^{2}(\mathbb{H}^{n})$, where $\alpha_{2k,l,m}=g_{2k,l}\ast_{d}\tilde{\varphi}(2k,l,m)$, $\tilde{\varphi}(X)=\overline{\varphi(X^{-1})}$, $X\in\mathbb{H}^{n}$ and $g_{2k,l}\ast_{d}\tilde{\varphi}(2k^{\prime},l^{\prime},m^{\prime})$ denotes

[TABLE]

for $(2k^{\prime},l^{\prime},m^{\prime})\in\mathbb{Z}^{2n+1}.$

Next, we provide a sufficient condition for obtaining an exact sampling formula for a smaller class of functions, namely, $\{L_{(0,0,m)}\varphi:m\in\mathbb{Z}\}$. Let the closed linear span of this collection be denoted by $V^{0}(\varphi)$.

Theorem 3.3**.**

Let $\varphi\in L^{2}(\mathbb{H}^{n})$ be such that $\{L_{(0,0,m)}\varphi:m\in\mathbb{Z}\}$ is a frame for $V^{0}(\varphi)$. Suppose $\varphi\in C(\mathbb{H}^{n})$ and

[TABLE]

Further, suppose there exist constants $A,B>0$ such that

[TABLE]

where $P_{0}(\lambda)=\sum_{m\in\mathbb{Z}}\varphi(0,0,m)e^{2\pi im\lambda}$ and $\Omega_{\varphi}$ is as in Theorem 3.1. Then, $\sum_{m\in\mathbb{Z}}c_{m}L_{(0,0,m)}\varphi$ converges pointwise to a continuous function, for any $\{c_{m}\}\in l^{2}(\mathbb{Z})$ and there exists a function $\psi\in L^{2}(\mathbb{H}^{n})$ such that $\{L_{(0,0,m)}\psi:m\in\mathbb{Z}\}$ is a frame for $V^{0}(\varphi)$ with the following sampling formula. For every $f\in V^{0}(\varphi)$,

[TABLE]

where the convergence is both in $L^{2}(\mathbb{H}^{n})$ and uniform on $\mathbb{H}^{n}$.

Now, we shall obtain a sampling theorem for $f$, belonging to the principal shift invariant space of $L^{2}(\mathbb{H}^{n})$, in the transform domain (Theorem 3.4) using the operator valued function $F_{\lambda}$ on $\mathbb{R}^{2n}$ given in (2.1). Using the inversion formula for $f$ from $F_{\lambda}$, we can also obtain a reconstruction formula for $f$ in the time domain (Corollary 3.6).

We note that for $(\xi_{1},\xi_{2})\in\mathbb{R}^{2n}$, $F_{\lambda}(\xi_{1},\xi_{2})$ can also be written as

[TABLE]

Theorem 3.4**.**

Let $\varphi\in L^{2}(\mathbb{H}^{n})$ and $\{L_{(2k,l,m)}\varphi:(k,l,m)\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}\}$ be a frame sequence. For $f\in V(\varphi)$ and $\lambda\in\mathbb{R}^{\ast}$, the transform $F_{\lambda}$ of $f$, as defined in (2.1), satisfies

[TABLE]

where $\{c_{2k,l,m}\}$ is the sequence of frame coefficients in a frame expansion of $f$ and $(\Phi^{M_{\lambda,\xi_{1},\xi_{2}}})_{\lambda}$ is the corresponding transform of the modulated $\varphi$, namely $M_{(\lambda\xi_{2},-\lambda\xi_{1},0)}\varphi$, in $L^{2}(\mathbb{H}^{n})$.

The following is an inversion formula that gives $f\in L^{2}(\mathbb{H}^{n})$ from $F_{\lambda}(\xi_{1},\xi_{2})$.

Theorem 3.5**.**

For $f\in L^{2}(\mathbb{H}^{n})$ and any $(\xi_{1},\xi_{2})\in\mathbb{R}^{2n}$, one has

[TABLE]

where $(u,v,w)\in\mathbb{H}^{n}$. In particular,

[TABLE]

Using Theorems 3.5 and 3.4, we obtain another reconstruction formula for $f$ in the time domain which is as follows.

Corollary 3.6**.**

Let $\varphi\in L^{2}(\mathbb{H}^{n})$ and $\{L_{(2k,l,m)}\varphi:(k,l,m)\in\mathbb{Z}^{n}\times\mathbb{Z}^{n}\times\mathbb{Z}\}$ be a frame sequence. For $f\in V(\varphi)$ and $(u,v,w)\in\mathbb{H}^{n}$,

[TABLE]

where $\{c_{2k,l,m}\}$ is the sequence of frame coefficients in a frame expansion of $f$ and $\Phi_{\lambda}$ is the transform of $\varphi$, as defined in (2.1).