Wave packet frames generated by hyponormal operators on $L^2(\mathbb{R})$

TL;DR

This paper investigates the conditions under which wave packet systems generated by hyponormal operators form frames in L^2(R), providing new characterizations and exploring their linear combinations.

Contribution

It introduces necessary and sufficient conditions for wave packet frames via hyponormal operators and characterizes hyponormal operators using tight wave packet frames.

Findings

Characterization of hyponormal operators using tight wave packet frames

Necessary and sufficient conditions for wave packet frames in L^2(R)

Analysis of linear combinations of wave packet frames

Abstract

In this paper we study frame-like properties of a wave packet system by using hyponormal operators on $L^2(\mathbb{R})$. We present necessary and sufficient conditions in terms of relative hyponormality of operators for a system to be a wave packet frame in $L^2(\mathbb{R})$. A characterization of hyponormal operators by using tight wave packet frames is proved. This is different from a method proved by Djordjevi$\acute{c}$ by using the Moore-Penrose inverse of a bounded linear operator with a closed range. The linear combinations of wave packet frames generated by hyponormal operators are discussed.

Equations

$$\Vert \{c_k(f)\}{\Vert }_{{\ell }^2}^2\le C\Vert f{\Vert }^2\text{for all}f\in {\mathcal{H}}_0$$

$$f=\sum _{k=1}^{\infty }{c}_k(f)f_k\text{for all}f\in {\mathcal{H}}_0.$$

$$A\Vert K^{\ast }f{\Vert }^2\le \sum _{k=1}^{\infty }|⟨f,f_k⟩{|}^2\le B\Vert f{\Vert }^2\text{for all}f\in \mathcal{H}.$$

$$\{D_{a_j}{T}_{bk}{E}_{c_m}\psi {\}}_{j,k,m\in \mathbb{Z}},$$

$$a_o\Vert f{\Vert }^2\le \sum _{k=1}^{\infty }|⟨f,f_k⟩{|}^2\le b_o\Vert f{\Vert }^2\text{for all}f\in \mathcal{H}.$$

$${\gamma }_0=\inf \{b_o>0:b_o\text{ satisfies (2.1)}\}$$

$${\delta }_0=\sup \{a_o>0:a_o\text{ satisfies (2.1)}\}$$

$$\text{synthesis operator}V:{\ell }^2\to \mathcal{H},V(\{c_k\})=\sum _{k=1}^{\infty }{c}_k{f}_k,\{c_k\}\in {\ell }^2,$$

$$\text{analysis operator}{V}^{\ast }:\mathcal{H}\to {\ell }^2,V^{\ast }(f)=\{⟨f,f_k⟩\},f\in \mathcal{H},$$

$$\text{frame operator}S=V{V}^{\ast }:\mathcal{H}\to \mathcal{H},S(f)=\sum _{k=1}^{\infty }⟨f,f_k⟩f_k,f\in \mathcal{H}.$$

$$f=S{S}^{-1}f=\sum _{k=1}^{\infty }⟨S^{-1}f,f_k⟩f_k\left(=\sum _{k=1}^{\infty }⟨f,S^{-1}{f}_k⟩f_k\right).$$

$$\text{Translation by}a\leftrightarrow T_{a}f(t)=f(t-a),$$

$$\text{Modulation by}b\leftrightarrow E_{b}f(t)=e^{2\pi ibt}f(t),$$

$$\text{Dilation by}c\leftrightarrow D_{c}f(t)=|c{|}^{\frac{1}{2}}f(ct).$$

$${\alpha }_0\Vert {\Theta }^{\ast }f{\Vert }^2\le \sum _{j,k,m\in \mathbb{Z}}|⟨f,D_{a_j}{T}_{bk}{E}_{c_m}\psi ⟩{|}^2\le {\beta }_0\Vert \Theta f{\Vert }^2\text{for all}f\in L^2(\mathbb{R}).$$

$${\alpha }_0\Vert {\Theta }^{\ast }f{\Vert }^2\le \sum _{k=1}^{\infty }|⟨f,f_k⟩{|}^2\le {\beta }_0\Vert \Theta f{\Vert }^2\text{for all}f\in \mathcal{H},$$

$$\Theta (\{{\xi }_1,{\xi }_2,{\xi }_3,.......\})=\{{\xi }_2,{\xi }_3,.......\},\{{\xi }_j\}\in \mathcal{H}.$$

$${\Theta }^{\ast }(\{{\xi }_1,{\xi }_2,{\xi }_3,.......\})=\{0,{\xi }_1,{\xi }_2,{\xi }_3,.......\},\{{\xi }_j\}\in \mathcal{H}.$$

$$\Vert {\Theta }^{\ast }f{\Vert }^2=\Vert f{\Vert }^2=\sum _{j=1}^{\infty }|⟨f,f_k⟩{|}^2\text{for all}f=\{{\xi }_j\}\in \mathcal{H}.$$

$$a_o\Vert {\Theta }^{\ast }f{\Vert }^2\le \sum _{k=1}^{\infty }|⟨f,f_k⟩{|}^2\le b_o\Vert \Theta f{\Vert }^2\text{for all}f\in \mathcal{H}.$$

$$\Theta (f=\{{\xi }_1,{\xi }_2,{\xi }_3,.......\})=\{{\xi }_1,{\xi }_1+{\xi }_2,{\xi }_2+{\xi }_3,.......\},f=\{{\xi }_j\}\in \mathcal{H}.$$

$${\Theta }^{\ast }(\{{\xi }_1,{\xi }_2,{\xi }_3,.......\})=\{{\xi }_1+{\xi }_2,{\xi }_2+{\xi }_3,.......\},\{{\xi }_j\}\in \mathcal{H}.$$

$$\gamma \Vert {\Theta }^{\ast }f{\Vert }^2\le \sum _{j=1}^{\infty }|⟨f,f_k⟩{|}^2\le \Vert \Theta f{\Vert }^2\text{for all}f\in \mathcal{H}.$$

$$\{D_{a_j}{T}_{bk}{E}_{c_m}\psi {\}}_{j,k,m\in \mathbb{Z}}=\{D_{a^j}{T}_{bk}\psi {\}}_{j,k\in \mathbb{Z}}$$

$$\Vert {\Theta }^{\ast }f{\Vert }^2={\alpha }_o\Vert E_{\beta }^{\ast }f{\Vert }^2$$

$$=\sum _{j,k,m\in \mathbb{Z}}|⟨f,D_{a_j}{T}_{bk}{E}_{d_m}\psi ⟩{|}^2$$

$$=\Vert \Theta f{\Vert }^2,\text{for all}f\in L^2(\mathbb{R}).$$

$$\Theta (f)=f.{\chi }_{[0,1]},f\in L^2(\mathbb{R}).$$

$$\{D_{a_j}{T}_{bk}{E}_{c_m}\psi {\}}_{j,k,m\in \mathbb{Z}}=\{T_k\psi {\}}_{k\in \mathbb{Z}}=\{{\chi }_{[k,k+1]}{\}}_{k\in \mathbb{Z}}.$$

$$h(x)$$

$$\sum _{j,k,m\in \mathbb{Z}}|⟨h,D_{a_j}{T}_{bk}{E}_{c_m}\psi ⟩{|}^2$$

$$=|⟨h,{\chi }_{[0,1]}⟩{|}^2+|⟨h,{\chi }_{[2,3]}⟩{|}^2$$

$$=1+B.$$

$$T\{c_{jkm}{\}}_{j,k,m\in \mathbb{Z}}=\sum _{j,k,m\in \mathbb{Z}}{c}_{jkm}{D}_{a_j}{T}_{bk}{E}_{c_m}\psi ,$$

$$T^{\ast }\text{f}=\{⟨f,D_{a_j}{T}_{bk}{E}_{c_m}\psi ⟩{\}}_{j,k,m\in \mathbb{Z}}$$

Full text

Lalit was supported by R $\&$ D Doctoral Research Programme, University of Delhi, Delhi-110007, India. Grant No. : RC/2014/6820.

1. Introduction

Frames in Hilbert spaces are a redundant system of vectors which provides a series representation for each vector in the space. Duffin and Schaffer [11] in 1952, introduced frames for Hilbert spaces, in the context of nonharmonic Fourier series. Frames were revived by Daubechies, Grossmann and Meyer in [8]. For applications of frames in various directions, see [3, 4]

Feichtinger and Werther [12] introduced a family of analysis and synthesis systems with frame-like properties for closed subspaces of a separable Hilbert space $\mathcal{H}$ and call it an atomic system (or local atoms). The motivation for the atomic system is based on examples arising in sampling theory. One of the important properties of the atomic system is that it can generate a proper subspace even though they do not belong to them.

Definition 1.1*.*

[12] Let $\mathcal{H}$ be a Hilbert space and let $\mathcal{H}_{0}$ be a closed subspace of $\mathcal{H}$. A sequence $\{f_{k}\}\subset\mathcal{H}$ is called a family of local atoms (or atomic system) for $\mathcal{H}_{0}$, if

1. $(i)$

there exists a real number $B>0$ such that $\|\{\langle f,f_{k}\rangle\}\|^{2}_{\ell^{2}}\leq B\|f\|^{2}$ for all $f\in\mathcal{H}$, 2. $(ii)$

there exists a sequence of linear functionals $\{c_{k}\}$ and a real number $C>0$ such that

[TABLE]

G$\check{a}$vruta in [14] introduced and studied $K$-frames in Hilbert spaces to study atomic systems with respect to a bounded linear operator $K$ on Hilbert spaces.

Definition 1.2*.*

[14] Let $\mathcal{H}$ be a Hilbert space and let $K$ be a bounded linear operator on $\mathcal{H}$. A sequence $\{f_{k}\}\subset\mathcal{H}$ is called a $K$-frame for $\mathcal{H}$, if there exist constants $A,B>0$ such that

[TABLE]

The lower inequality in $(1.1)$ is controlled by a bounded linear operator on $\mathcal{H}$. It is observed in [14] that $K$-frames are more general than standard frames in the sense that the lower frame bound only holds for the elements in the range of $K$, where $K$ is a bounded linear operator on the underlying Hilbert space. G$\check{a}$vruta in [14] characterize $K$-frames in Hilbert spaces by using bounded linear operators.

It would be interesting to control both lower and upper frame condition in $(1.1)$ by bounded linear operators on $\mathcal{H}$. In this direction, we study frame-like properties of an irregular wave packet system in $L^{2}(\mathbb{R})$, where both lower and upper frame conditions are controlled by bounded linear operators on $L^{2}(\mathbb{R})$ (see Definition 3.1). The wave packet system is a family of functions generated by combined action of dilation, translation and modulation operators on $L^{2}(\mathbb{R})$. More precisely, we consider a system of the form

[TABLE]

where $\psi\in L^{2}(\mathbb{R})$, $\{a_{j}\}_{j\in\mathbb{Z}}\subset\mathbb{R}^{+}$, $b\neq 0$ and $\{c_{m}\}_{m\in\mathbb{Z}}\subset\mathbb{R}$ and call it irregular Weyl-Heisenberg wave packet system (or simply* wave packet system*) in $L^{2}(\mathbb{R})$. A frame for $L^{2}(\mathbb{R})$ of the form $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is called an irregular wave packet frame (or* wave packet frame*). The wave packet system was introduced by Cordoba and Fefferman [6] by applying certain collections of dilations, modulations and translations to the Gaussian function in the study of some classes of singular integral operators. Later, Labate et al. [20] adopted the same expression to describe, more generally, any collection of functions which are obtained by applying the same operations to a finite family of functions in $L^{2}(\mathbb{R}^{d})$. More precisely, Gabor systems, wavelet systems and the Fourier transform of wavelet systems are special cases of wave packet systems. Lacey and Thiele [21, 22] gave applications of wave packet systems in boundedness of the Hilbert transforms. The wave packet systems have been studied by several authors, see [7, 15, 17, 18, SLV, SV].

1.1. Outline:

This paper is organized as follows: In Section 2, we give basic definitions and results which will be used throughout the paper. Section 3 is devoted to the study of frame-like properties of irregular Weyl-Heisenberg wave packet systems. We introduce $\Theta$-irregular Weyl-Heisenberg wave packet frame (in short, $\Theta$-$IWH$ wave packet frame) for $L^{2}(\mathbb{R})$, where $\Theta$ is a bounded linear operator on $L^{2}(\mathbb{R})$ (see Definition 3.1). This type of wave packet frame can control both lower and upper frame conditions by bounded linear operators on $L^{2}(\mathbb{R})$ . The $\Theta$-$IWH$ wave packet frame (in the context of standard Hilbert frame) for a Hilbert space is a $K$-frame, but converse is not true (see Example 3.2). Furthermore, the $\Theta$-$IWH$ wave packet frame control both lower and upper frame conditions by bounded linear operators. Necessary and sufficient conditions for a certain system to be a $\Theta$-$IWH$ wave packet frames for $L^{2}(\mathbb{R})$ by using hyponormality of operators on $L^{2}(\mathbb{R})$ have been obtained. A characterization of hyponormal operator in terms of a special type of tight wave packet frames for $L^{2}(\mathbb{R})$ is given. This is different from a method proved by Djordjevi$\acute{c}$ in [9] by using the Moore-Penrose inverse of a bounded linear operator with a closed range (see Theorem 3.7). The linear combinations of frames or redundant building blocks are important in applied mathematics, we discuss linear combinations of $\Theta$-$IWH$ wave packet frames for $L^{2}(\mathbb{R})$ in Section 4.

2. Preliminaries

In this section, we recall basic notations and definitions to make the paper self-contained. Let $\mathcal{H}$ be a separable real (or complex) Hilbert space with inner product $\langle.,.\rangle$ linear in the first entry. A countable sequence $\{f_{k}\}\subset\mathcal{H}$ is called a frame (or Hilbert frame) for $\mathcal{H}$, if there exist numbers $0<a_{o}\leq b_{o}<\infty$ such that

[TABLE]

The numbers $a_{o}$ and $b_{o}$ are called lower and upper frame bounds, respectively. They are not unique. If it is possible to choose $a_{o}=b_{o}$, then the frame $\{f_{k}\}$ is called *Parseval frame *(or tight frame).

The scalars

[TABLE]

are called the optimal bounds or best bounds of the frame.

Associated with a frame $\{f_{k}\}$ for $\mathcal{H}$, there are three bounded linear operators:

[TABLE]

The frame operator $S$ is a positive, self-adjoint and invertible operator on $\mathcal{H}$. This gives the reconstruction formula for all $f\in\mathcal{H}$,

[TABLE]

The scalars $\{\langle S^{-1}f,f_{k}\rangle\}$ are called frame coefficients of the vector $f\in\mathcal{H}$. The representation of $f$ in the reconstruction formula need not be unique. This reflects one of the important properties of frames in applied mathematics.

Let $a,b\in\mathbb{R}$ and $c\in\mathbb{R}\setminus\{0\}$. We consider operators $T_{a},\ E_{b},\ D_{c}:L^{2}(\mathbb{R})\rightarrow~{}L^{2}(\mathbb{R})$ given by

[TABLE]

A bounded linear operator $T$ defined on $\mathbb{H}$ is said to be positive, if $\langle Tf,f\rangle\geq 0$ for all $f\in\mathbb{H}$. In symbol we write $T\geq 0$. If $T_{1},T_{2}$ are bounded linear operator on $\mathbb{H}$ such that $T_{1}-T_{2}\geq 0$, then we write $T_{1}\geq T_{2}$. A bounded linear operator $T:\mathbb{H}\rightarrow\mathbb{H}$ is said to be hyponormal, if $T^{*}T-TT^{*}\geq 0$, or equivalently if $\|T^{*}f\|\leq\|Tf\|$ for all $f\in\mathbb{H}$. The characteristic function of any set $E$ is denoted by $\chi_{E}$. By $\mathcal{R}(T)$ we denote the range of a bounded linear operator $T$ from a normed space $X$ into a normed space $Y$.

Theorem 2.1**.**

[10]** Let $\mathbb{H},\mathbb{H}_{1},\mathbb{H}_{2}$ be Hilbert spaces. Assume that $T_{1}:\mathbb{H}_{1}\rightarrow\mathbb{H}$ and $T_{2}:\mathbb{H}_{2}\rightarrow\mathbb{H}$ be bounded linear operators. The following statement are equivalent:

1. $(i)$

$\mathcal{R}(T_{1})\subset\mathcal{R}(T_{2})$. 2. $(ii)$

$T_{1}T_{1}^{*}\leq\lambda^{2}T_{2}T_{2}^{*}$* for some $\lambda\geq 0$.* 3. $(iii)$

There exists a bounded linear operator $S:\mathbb{H}_{1}\rightarrow\mathbb{H}_{2}$ such that $T_{1}=T_{2}S.$

3. Wave Packet Frames in $L^{2}(\mathbb{R})$

Definition 3.1*.*

Let $\psi\in L^{2}(\mathbb{R})$, $\{a_{j}\}_{j\in\mathbb{Z}}\subset\mathbb{R}^{+}$, $\{c_{m}\}_{m\in\mathbb{Z}}\subset\mathbb{R}$ and $b\neq 0$ and let $\Theta$ be a bounded linear operator on $L^{2}(\mathbb{R})$. A system $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is called a $\Theta$-irregular Weyl-Heisenberg wave packet frame (in short, $\Theta$-$IWH$ wave packet frame) for $L^{2}(\mathbb{R})$, if there exist constants $0~{}<~{}\mathrm{\alpha_{0}}\leq\mathrm{\beta_{0}}<\infty$ such that

[TABLE]

The scalars $\alpha_{0}$ and $\beta_{0}$ are called lower and upper bounds of the $\Theta$-$IWH$ wave packet frame $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$, respectively. If upper inequality in $(3.1)$ is satisfied, then $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is called a

• Bessel sequence* in $L^{2}(\mathbb{R})$ with Bessel bound $\mathrm{\beta_{0}}$. If $\Theta$ is the identity operator on $L^{2}(\mathbb{R})$, then $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ is the standard $IWH$ wave packet frame for $L^{2}(\mathbb{R})$.

If a countable sequence $\{f_{k}\}$ in a Hilbert space $\mathcal{H}$ satisfies the inequality (3.1), i.e., if

[TABLE]

then we say that $\{f_{k}\}$ is a $\Theta$-Hilbert frame for $\mathcal{H}$.

3.1. Examples and comments:

Every $\Theta$-Hilbert frame for $\mathcal{H}$ is a $K$-frame for $\mathcal{H}$, but not conversely. More precisely, if $\{f_{k}\}$ is a $\Theta$-Hilbert frame for $\mathcal{H}$ with frame bounds $\alpha_{0}$ and $\beta_{0}$. Then, $\{f_{k}\}$ is a $K$-frame for $\mathcal{H}$ with frame bounds $\alpha_{0}$ and $\beta_{0}\ \|\Theta\|^{2}$. The following example shows that a $K$-frame for $\mathcal{H}$ need not be a $\Theta$-Hilbert frame for $\mathcal{H}$.

Example 3.2*.*

Let $\{\chi_{k}\}$ be the canonical orthonormal basis for the discrete signal space $\mathcal{H}=L^{2}(\Omega,\mu)$ (where $\Omega=\mathbb{N}$ and $\mu$ is the counting measure) and let $\Theta$ be the backward shift operator on $\mathcal{H}$ given by

[TABLE]

Then, its conjugate $\Theta^{*}$ is the forward shift operator on $\mathcal{H}$ which is given by

[TABLE]

Choose $f_{k}=\chi_{k}$ for all $k\in\mathbb{N}$.

We compute

[TABLE]

Hence $\{f_{k}\}$ is a $K$-frame (with a choice $K=\Theta)$ for $\mathcal{H}$ with frame bounds $A=B=1$. But $\{f_{k}\}$ is not a $\Theta$-Hilbert frame for $\mathcal{H}$. Indeed, let $a_{o}$ and $b_{o}$ be positive numbers such that

[TABLE]

Then, for $f_{o}=\chi_{1}\in\mathcal{H}$, we obtain $\Theta f_{o}=0$. Therefore, by using upper inequality in $(3.2)$, we have $f_{o}=0$, a contradiction.

Remark 3.3*.*

A $\Theta$-Hilbert frame for $\mathcal{H}$ ($\Theta\neq I$, the identity operator on $\mathcal{H}$) need not be a standard Hilbert frame for $\mathcal{H}$ and vice-versa. Indeed, let $\mathcal{H}$ be the discrete signal space given in Example 3.2 with canonical orthonormal basis $\{\chi_{k}\}$.

Choose $f_{k}=\chi_{k}+\chi_{k+1},k\in\mathbb{N}$.

Define $\Theta:\mathcal{H}\rightarrow\mathcal{H}$ by

[TABLE]

Then, $\Theta$ is a bounded linear operator on $\mathcal{H}$ and its conjugate operator $\Theta^{*}$ is given by

[TABLE]

One can verify that there exists a $\gamma\in(0,1)$ such that

[TABLE]

Hence $\mathcal{F}\equiv\{f_{k}\}$ is a $\Theta$-Hilbert frame for $\mathcal{H}$. But $\mathcal{F}$ is not a standard Hilbert frame for $\mathcal{H}$ (see Example $5.4.6$ in [4],  p.   98).

To show that a standard Hilbert frame for $\mathcal{H}$ need not be $\Theta$-Hilbert frame for $\mathcal{H}$. Choose $g_{k}=\chi_{k},k\in\mathbb{N}$ and let $\Theta$ be the backward shift operator on $\mathcal{H}$. Then, $\mathcal{G}=\{g_{k}\}$ is a Hilbert frame for $\mathcal{H}$, but not a $\Theta$-Hilbert frame for $\mathcal{H}$.

Regarding the existence of $\Theta$-$IWH$ wave packet frames for $L^{2}(\mathbb{R})$, we have following examples.

Example 3.4*.*

Let $a>1$ and $b>0$ and $c_{m}=0$ for all $m\in\mathbb{Z}$. Choose $a_{j}=a^{j}$ for all $j\in\mathbb{Z}$. Then, there exist $\psi\in L^{2}(\mathbb{R})$ such that $\hat{\psi}=\chi_{E}$, where $E$ is a compact subset of $\mathbb{R}$. Therefore,

[TABLE]

is an orthonormal basis for $L^{2}({\mathbb{R})}$ (see Theorem 12.3 in [16]   p. 357), hence a tight $IWH$ wave packet frame for $L^{2}(\mathbb{R})$ .

Let $\beta\in\mathbb{R}$ be arbitrary, but fixed.

Choose $\Theta=E_{\beta}$ (the modulation operator on $L^{2}(\mathbb{R})$) and $d_{m}=c_{m}+\beta\ (m\in\mathbb{Z})$.

We compute

[TABLE]

Hence $\{D_{a_{j}}T_{bk}E_{d_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$.

Example 3.5*.*

Let $\Theta:L^{2}(\mathbb{R})\rightarrow L^{2}(\mathbb{R})$ be the multiplication operator given by

[TABLE]

Then, $\Theta$ is a bounded linear operator on $L^{2}(\mathbb{R})$.

Choose $b=1,\ a_{j}=1,\ c_{m}=0$  $(j,m\in\mathbb{Z})$ and $\psi=\chi_{[0,1]}$.

Then

[TABLE]

The system $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is not a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$. Indeed, let $B$ be an upper $\Theta$-$IWH$ wave packet frame bound for $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$. Let $h\in L^{2}(\mathbb{R})$ be a function given by

[TABLE]

We compute

[TABLE]

On the other hand, $\|\Theta h\|^{2}=\|h.\chi_{[0,1]}\|^{2}=1$.

Therefore, $\sum_{j,k,m\in\mathbb{Z}}|\langle h,D_{a_{j}}T_{bk}E_{c_{m}}\psi\rangle|^{2}=1+B>B\|\Theta h\|^{2}$. Hence $B$ is not an upper $\Theta$-$IWH$ wave packet frame bound for $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$, a contradiction.

3.2. Operators associated with $\Theta$-$IWH$ wave packet frames

Suppose that $\mathcal{F}\equiv\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$. The operator $T:\ell^{2}(\mathbb{Z}^{3})\rightarrow L^{2}(\mathbb{R})$ given by

[TABLE]

is called the* pre-frame operator* or synthesis operator associated with $\mathcal{F}$ and the adjoint operator $T^{*}:L^{2}(\mathbb{R})\rightarrow\ell^{2}(\mathbb{Z}^{3})$ is given by

[TABLE]

is called the analysis operator associated with $\mathcal{F}$. Composing $T$ and $T^{*}$, we obtain the frame operator $\mathcal{S}:L^{2}(\mathbb{R})\rightarrow L^{2}(\mathbb{R})$ given by

[TABLE]

Since $\mathcal{F}$ is a $\Theta$-$IWH$ wave packet Bessel sequence in $L^{2}(\mathbb{R})$, the series defining $\mathcal{S}$ converges unconditionally for all $f\in L^{2}(\mathbb{R}).$ Notice that, in general, frame operator of the $\Theta$- $IWH$ wave packet frame $\mathcal{F}$ is not invertible on $L^{2}(\mathbb{R})$, but it is invertible on a subspace $\mathcal{R}(\Theta)\subset L^{2}(\mathbb{R})$. In fact, if $\mathcal{R}(\Theta)$ is closed , then there exist a pseudoinverse $\Theta^{\dagger}$ of $\Theta$ such that $\Theta\Theta^{\dagger}f=f$ for all $f\in\mathcal{R}(\Theta)$, i.e., $\Theta\Theta^{\dagger}|_{\mathcal{R}(\Theta)}=I_{\mathcal{R}(\Theta)}$, so we have $\left(\Theta^{\dagger}|_{\mathcal{R}(\Theta)}\right)^{*}\Theta^{*}=I_{\mathcal{R}(\Theta)}^{*}$. Hence for any $f\in\mathcal{R}(\Theta)$, we obtain

[TABLE]

Therefore, by using (3.3), we can write

[TABLE]

That is

[TABLE]

Thus, the operator $\mathcal{S}:\mathcal{R}(\Theta)\rightarrow\mathcal{S}(\mathcal{R}(\Theta))$ is a homeomorphism. Furthermore, we have

[TABLE]

Next, we characterizes a system $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}\subset L^{2}(\mathbb{R})$ as $\Theta$-$IWH$ wave packet frame. Let $T_{1}:\mathbb{H}\rightarrow\mathbb{H}$ and $T_{2}:\mathbb{H}_{1}\rightarrow\mathbb{H}$ be bounded linear operators, where $\mathbb{H},\mathbb{H}_{1}$ are Hilbert spaces. We say that the pair $(T_{1},T_{2})$ is relatively hyponormal, if

[TABLE]

In this case we say that $T_{1}$ and $T_{2}$ are relatively hyponormal. Aldroubi in [1] characterized operators on a Hilbert space $\mathcal{H}$, which can generate Hilbert frames (as images of given frames) for $\mathcal{H}$. Actually, Aldroubi considered operators which are relative hyponormal with the identity operator on $\mathcal{H}$. The following theorem characterizes a certain system as a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ in terms of the relative hyponormality of operators.

Theorem 3.6**.**

Let $\psi\in L^{2}(\mathbb{R})$, $\{a_{j}\}_{j\in\mathbb{Z}}\subset\mathbb{R}^{+}$, $\{c_{m}\}_{m\in\mathbb{Z}}\subset\mathbb{R}$ and $b\neq 0$ and let $\Theta$ be a bounded linear operator on $L^{2}(\mathbb{R})$. Then, $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ if and only if there exist a bounded linear operator $\Xi:\ell^{2}(\mathbb{Z}^{3})\rightarrow L^{2}(\mathbb{R})$ such that

1. $(i)$

the pair $(\Theta,\Xi)$ is relative hyponormal, i.e., $\lambda\Theta^{*}\Theta\geq\Xi\ \Xi^{*}\ \ \text{for some}\ \lambda~{}>~{}0$, 2. $(ii)$

$\Xi(e_{j,k,m})=D_{a_{j}}T_{bk}E_{c_{m}}\psi$*  $(j,k,m\in\mathbb{Z})$  and $\mathcal{R}(\Theta)\subset\mathcal{R}(\Xi)$,*

where $\{e_{j,k,m}\}_{j,k,m\in\mathbb{Z}}$ is an orthonormal basis for $\ell^{2}(\mathbb{Z}^{3})$.

Proof.

Suppose first that $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$. Then, we can find positive constants $\mathrm{a}_{0},\mathrm{b}_{0}$ such that

[TABLE]

Define $\mathcal{W}:L^{2}(\mathbb{R})\rightarrow\ell^{2}(\mathbb{Z}^{3})$ by

[TABLE]

Clearly, $\mathcal{W}$ is a well defined bounded linear operator on $L^{2}(\mathbb{R})$.

We compute

[TABLE]

This gives

[TABLE]

By using (3.5) and lower frame inequality in (3.4), we obtain

[TABLE]

This gives $\mathrm{a}_{0}\Theta\Theta^{*}\leq\mathcal{W}^{*}\mathcal{W}$.

Choose $\Xi=\mathcal{W}^{*}$. Then, by Theorem 2.1, we have $\mathcal{R}(\Theta)\subset$ $\mathcal{R}(\Xi).$ The condition $(ii)$ in the result is proved.

To show $\lambda\Theta^{*}\Theta\geq\Xi\ \Xi^{*}$ $(\lambda>0)$, we consider upper frame inequality in (3.4):

[TABLE]

This gives $\mathrm{b}_{0}\Theta^{*}\Theta\geq\mathcal{W}^{*}\mathcal{W}$. That is, $\lambda\Theta^{*}\Theta\geq\Xi\ \Xi^{*}$ $(\lambda=\mathrm{b}_{0}>0)$. This proves the condition $(i)$ in the result.

Conversely, assume that both conditions $(i)$ and $(ii)$ given in the theorem hold.

We compute

[TABLE]

for all $f\in L^{2}(\mathbb{R})$ and for all $h\in\ell^{2}(\mathbb{Z}^{3})$.

This gives

[TABLE]

Therefore, by using $(3.6)$ and the condition $(i)$, we have

[TABLE]

By hypothesis $\mathcal{R}(\Theta)\subset\mathcal{R}(\Xi)$ (see condition $(ii)$). So, by Theorem 2.1, we can find a positive constant $\beta$ such that $\Theta\Theta^{*}\leq\beta\ \Xi\ \Xi^{*}$ (note that $\beta$ is positive, since otherwise $\Theta=O$). Again by using the condition $(ii)$, we have

[TABLE]

By using (3.7) and (3.2), we conclude that $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R}).$ ∎

Djordjevi$\acute{c}$ in [9] characterized hyponormal operators by using the Moore-Penrose inverse of a bounded linear operator with a closed range. There may be other conditions for a bounded linear operator on a Hilbert space to be hyponormal. Let $H$ and $K$ be Hilbert spaces and $A:H\rightarrow K$ be a bounded linear operator. The Moore-Penrose inverse of $A$ is denoted by $A^{{\dagger}}$, see [2]. Djordjevi$\acute{c}$ proved the following result by using the Moore-Penrose inverse of a bounded linear operator with a closed range.

Theorem 3.7**.**

[9]** Let $A$ and $AA^{*}+A^{*}A$ have closed ranges. Then the following statements are equivalent:

1. $(i)$

$A$* is hyponormal* 2. $(ii)$

$2AA^{*}(AA^{*}+A^{*}A)^{{\dagger}}AA^{*}\leq AA^{*}$.

Thus, a bounded linear operator $A$ defined on a Hilbert space is hyponormal if a certain operator inequality (consisting of adjoint and Moore-Penrose inverse of $A$) is satisfied. Frame can be used to characterizes a hyponormal operator on $L^{2}(\mathbb{R})$. First we define a type of tight frame (or Parseval frame) in $L^{2}(\mathbb{R})$. In Definition 3.1, if $\alpha_{0}=\beta_{0}$, then $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is not a standard tight frame, in general. This is the motivation for new type of tight frames in $L^{2}(\mathbb{R})$.

Definition 3.8*.*

Let $\Theta\neq I$ (where $I$ the identity operator on $L^{2}(\mathbb{R})$). A $\Theta$-Hilbert frame $\{f_{n}\}\subset\mathcal{H}$ for $\mathcal{H}$ with frame bounds $\alpha_{0}=\beta_{0}$ is called a $(\Theta,\alpha_{0})$-Hilbert tight frame.

The following theorem characterizes hyponormal operators on $L^{2}(\mathbb{R})$ in terms of $(\Theta,\alpha_{0})$- Hilbert tight frames for $L^{2}(\mathbb{R})$.

Theorem 3.9**.**

A bounded linear operator $\Theta$ on $L^{2}(\mathbb{R})$ is hyponormal if and only if there exists a $(\Theta,1)$-Hilbert tight frame for $L^{2}(\mathbb{R})$.

Proof.

Assume first that $\Theta$ is a hyponormal operator on $L^{2}(\mathbb{R})$. Let $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ be a tight $IWH$ wave packet frame for $L^{2}(\mathbb{R})$.

Then

[TABLE]

Choose $f_{n}(n\in\mathbb{N})\leftrightarrow\varphi_{j,k,m}=\Theta(D_{a_{j}}T_{bk}E_{c_{m}}\psi),j,k,m\in\mathbb{Z}$.

Then, by using (3.9) and hyponormality of $\Theta$, we compute

[TABLE]

For the lower frame inequality, we compute

[TABLE]

By using (3.2) and (3.2) we have

[TABLE]

Hence $\{\varphi_{j,k,m}\}_{j,k,m\in\mathbb{Z}}$ is a $(\Theta,1)$-Hilbert tight frame for $L^{2}(\mathbb{R})$.

For the reverse part, suppose that $\{f_{n}\}$ is a $(\Theta,1)$-Hilbert tight frame for $L^{2}(\mathbb{R})$.

Then

[TABLE]

This gives $\|\Theta^{*}f\|\leq\|\Theta f\|$ for all $f\in\mathcal{H}$. Hence $\Theta$ is a hyponormal operator on $L^{2}(\mathbb{R})$. ∎

Favier and Zalik proved in [13] that the image of a Hilbert frame for $\mathcal{H}$ under a linear homeomorphism is a Hilbert frame for $\mathcal{H}$. They established relation between optimal bounds of a given Hilbert frame and its image (as frame). This is not true for $\Theta$-$IWH$ wave packet frame (see Example 3.12), in general. The problem (regarding invariance behaviour as a frame under linear homeomorphism) for $\Theta$-$IWH$ wave packet frames can be solved, provided the given linear homeomorphism commutes with $\Theta^{*}$. This is proved in the following theorem.

Theorem 3.10**.**

Let $\mathcal{F}\equiv\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ be a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ and $U$ be a linear homeomorphism on $L^{2}(\mathbb{R})$ such that $U$ commutes with $\Theta^{*}$. Then, $\mathcal{F}_{U}\equiv\{U(D_{a_{j}}T_{bk}E_{c_{m}}\psi)\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$. Furthermore, if $A_{1}$ and $B_{1}$ are optimal bounds of the frame $\mathcal{F}$ and the pair $(\Theta,U^{*})$ is relatively hyponormal, then the optimal bounds $A_{2}$ and $B_{2}$ of the frame $\mathcal{F}_{U}$ satisfy the inequalities

[TABLE]

Proof.

We compute

[TABLE]

By using the fact that $A_{1}$ is one of the choice for lower $\Theta$-$IWH$ wave packet frame bound for $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ and $U$ commutes with $\Theta^{*}$, we compute

[TABLE]

By using (3.2) and (3.2), we obtain

[TABLE]

Hence $\{U(D_{a_{j}}T_{bk}E_{c_{m}}\psi)\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ with one of the choice of frame bounds $A_{1}\|U\|^{-2},\ B_{1}\|U\|^{2}$.

Since $A_{2}$ and $B_{2}$ are best frame bounds for $\{U(D_{a_{j}}T_{bk}E_{c_{m}}\psi)\}_{j,k,m\in\mathbb{Z}}$, we have

[TABLE]

Again $\{U(D_{a_{j}}T_{bk}E_{c_{m}}\psi)\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ with $A_{2},B_{2}$ as one of the choice of frame bounds. So, for all $f\in L^{2}(\mathbb{R}),$ we have

[TABLE]

For all $f\in L^{2}(\mathbb{R})$, we have

[TABLE]

By using (3.16), (3.17) and relative hyponormality of the pair $(\Theta,U^{*})$ , we have

[TABLE]

where $\lambda$ is a positive constant which appears in the relative hyponormality of the pair $(\Theta,U^{*})$.

Since $A_{1}$ and $B_{1}$ are the best $\Theta$-$IWH$ wave packet frame bounds for $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$, by using (3.2), we have

[TABLE]

The inequalities in (3.12) are obtained from (3.15) and (3.19). The result is proved. ∎

Remark 3.11*.*

The condition that the linear homeomorphism $U$ commutes with $\Theta^{*}$ in Theorem 3.10 cannot be relaxed. This is justified in the following example.

Example 3.12*.*

Consider the multiplication operator $\Theta:L^{2}(\mathbb{R})\longrightarrow L^{2}(\mathbb{R})$ given by

[TABLE]

Then, $\Theta$ is a bounded linear self-adjoint operator on $L^{2}(\mathbb{R})$.

Choose $a_{j}=1,c_{m}=m$ for all $j,m\in\mathbb{Z}$, $b=0$ and $\psi=\chi_{[0,1)}$. Then, $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$. Indeed, for all $f\in L^{2}(\mathbb{R})$, we have

[TABLE]

Hence $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$.

Choose $U_{\circ}=T_{1}$, the translation operator on $L^{2}(\mathbb{R})$, i.e., $U_{\circ}f(\bullet)=f(\bullet-1)$. Then, $U_{\circ}$ is a linear homeomorphism on $L^{2}(\mathbb{R})$. First we show that the operator $U_{\circ}$ and $\Theta^{*}(=\Theta)$ does not commutes. For this, we compute

[TABLE]

By using $(3.20)$ and $(3.21)$, we conclude that the operators $U_{\circ}$ and $\Theta^{*}$ does not commutes.

Next, we show that the system $\mathcal{F}_{U_{\circ}}\equiv\{U_{\circ}(D_{a_{j}}T_{bk}E_{c_{m}}\psi)\}_{j,k,m\in\mathbb{Z}}$ is not a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$. Let $a_{\circ}$ and $b_{\circ}$ be a choice of frame bounds for $\mathcal{F}_{U_{\circ}}$.

Then

[TABLE]

Choose $f_{\circ}=\chi_{[0,1[}\in L^{2}(\mathbb{R})$. Then, $\|\Theta^{*}f_{\circ}\|=1$.

Then, by using lower inequality in $(3.22)$, we compute

[TABLE]

a contradiction. Hence $\mathcal{F}_{U_{\circ}}$ is not a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$.

4. Linear Combinations of $\Theta$-$IWH$ Wave Packet Frames

Linear combination of frames (or redundant building blocks) is important in applied mathematics. Aldroubi in [1] considered the following problem: given a Hilbert frame $\{f_{k}\}$ for $\mathcal{H}$, define a set of functions $\Phi_{j}$ by taking linear combinations of the frame elements $f_{k}$. What are the conditions on the coefficients in the linear combinations, so that the new system $\{\Phi_{j}\}$ constitutes a frame for $\mathcal{H}~{}?$ More precisely, Aldroubi considered a linear combination of the form

[TABLE]

where $\alpha_{j,k}$ are scalars. Aldroubi proved sufficient conditions on $\{\alpha_{j,k}\}$ such that $\{\Phi_{j}\}$ constitutes a frame for $\mathcal{H}$. Christensen in [5] gave sufficient conditions which are different from those proved by Aldroubi. In this section, we extend some results by Kaushik et al. in [19] to $\Theta$-$IWH$ wave packet frames for $L^{2}(\mathbb{R})$.

Let $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ be a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$. First we consider a linear combination of the form:

[TABLE]

where $\bigcup\limits_{r,s,t\in\mathbb{Z}}\mathbb{I}_{r,s,t}=\mathbb{Z}\times\mathbb{Z}\times\mathbb{Z}$,  $\mathbb{I}_{r,s,t}\bigcap\mathbb{I}_{r^{\prime},s^{\prime},t^{\prime}}=\emptyset$, $(r,s,t)\neq(r^{\prime},s^{\prime},t^{\prime})$ for all $r,s,t,r^{\prime},s^{\prime},t^{\prime}\in~{}\mathbb{Z}$ and $\alpha_{j,k,m}$ are scalars. The system $\{\Phi_{r,s,t}\}_{r,s,t\in\mathbb{Z}}$ is not a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$, in general. This type of combinations under the WH-packet for Gabor system were studied by Kaushik et al. [19]. The following theorem gives necessary and sufficient conditions for the system $\{\Phi_{r,s,t}\}_{r,s,t\in\mathbb{Z}}$ to be a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$. This is an adaption of [19, Theorem 3.5].

Theorem 4.1**.**

Let $\Theta$ be a bounded linear operator on $L^{2}(\mathbb{R})$ such that $\Theta^{*}$ is hyponormal. Assume that $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ and $\{\Phi_{r,s,t}\}_{r,s,t\in\mathbb{Z}}\subset L^{2}(\mathbb{R})$ be the sequence defined in (4). Let $T:\ell^{2}(\mathbb{Z}^{3})\rightarrow\ell^{2}(\mathbb{Z}^{3})$ be a bounded linear operator such that

[TABLE]

Then, $\{\Phi_{r,s,t}\}_{r,s,t\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ if and only if there exists a constant $\lambda>0$ such that

[TABLE]

Proof.

Assume first that $\{\Phi_{r,s,t}\}_{r,s,t\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ with frame bounds $A^{\prime},B^{\prime}$. Then, for any $f\in L^{2}(\mathbb{R})$, we have

[TABLE]

If $B$ is an upper $\Theta$-$IWH$ wave packet frame bound for $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$, then

[TABLE]

i.e.

[TABLE]

Choose $\lambda=\frac{A^{\prime}}{B}>0$. Then, by using hyponormality of $\Theta^{*}$, (4.3) and (4.4), we have

[TABLE]

The inequality given in (4.2) is proved.

For the reverse part, since $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$. There exist positive constants $A,B$ such that

[TABLE]

By using (4.2) and (4.5), we have

[TABLE]

We compute

[TABLE]

By using (4) and (4), we conclude that $\{\Phi_{r,s,t}\}_{r,s,t\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$. ∎

4.1. The case of finite sum:

We now consider a linear combination of the form $\mathcal{F}_{p}\equiv\left\{\sum\limits_{s=1}^{p}\alpha_{s}D_{a_{j}}T_{bk}E_{c_{m}}\psi_{s}\right\}_{j,k,m\in\mathbb{Z}}$, where $\alpha_{1},\alpha_{2},...,\alpha_{p}$ are nonzero scalars, $\psi_{s}\in L^{2}(\mathbb{R})$ and $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi_{s}\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ for each s\in\Lambda_{p}=\{1,2,3,..,p\}\. The finite sum $\mathcal{F}_{p}$ is not a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$, in general. Kaushik, Singh, and Virender [19] showed that if some scalar multiple of a series associated with a Gabor frame is dominated by the series associated with the finite sum of Gabor frames, then the finite sum constitutes a Gabor frame for the underlying space and vice-versa, see Theorem 4.2 of [19]. The following theorem extend this result in the context of $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$.

Theorem 4.2**.**

Assume that $\Theta:L^{2}(\mathbb{R})\rightarrow L^{2}(\mathbb{R})$ is a bounded linear operator such that $\Theta^{*}$ is hyponormal. Let $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi_{s}\}_{{j,k,m\in\mathbb{Z}}\atop{s\in\Lambda_{p}}}$ be a finite family of $\Theta$-$IWH$ frames for $L^{2}(\mathbb{R})$. Then, $\mathcal{F}_{p}\equiv\left\{\sum\limits_{s=1}^{p}\alpha_{s}D_{a_{j}}T_{bk}E_{c_{m}}\psi_{s}\right\}_{j,k,m\in\mathbb{Z}}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ if and only if there exists $\mu>0$ and some $\xi\in\Lambda_{p}$ such that

[TABLE]

for any finite sequence of scalars $\{\alpha_{s}\}$.

Proof.

Let $A_{\xi},B_{\xi}$ be frame bounds for $\Theta$-$IWH$ wave packet frame $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi_{\xi}\}_{j,k,m\in\mathbb{Z}}$ for $L^{2}(\mathbb{R})$ $(1\leq\xi\leq p)$.

Then

[TABLE]

Thus, the lower frame condition is satisfied for the finite system $\mathcal{F}_{p}$.

For the upper frame condition, we compute

[TABLE]

By $(4.8)$ and $(4.9)$, we conclude that the finite sum $\mathcal{F}_{p}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$.

Conversely, assume that the finite sum $\mathcal{F}_{p}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$ with frame bounds $A,B$. Then, for all $f\in L^{2}(\mathbb{R})$, we have

[TABLE]

If $B_{\xi}$ is an upper frame bound for $\{D_{a_{j}}T_{bk}E_{c_{m}}\psi_{\xi}\}_{j,k,m\in\mathbb{Z}}$, then

[TABLE]

Choose $\mu=\frac{A}{B_{\xi}}>0$. Then, using hyponormality of $\Theta^{*}$, $(4.10)$ and $(4.11)$ we have

[TABLE]

The theorem is proved. ∎

Application: The following example gives an application of Theorem 4.2.

Example 4.3*.*

Let $\Theta:L^{2}(\mathbb{R})\rightarrow L^{2}(\mathbb{R})$ be the modulation operator. That is, $\Theta f(t)=e^{2\pi ibt}f(t)$, where $b\in\mathbb{R}$ is fixed. Then, $\Theta^{*}$ is hyponormal on $L^{2}(\mathbb{R})$.

Choose $\psi=\chi_{[0,1[}$, $a_{j}=1,\ b=1,c_{m}=m$ for all $j,m\in\mathbb{Z}$ and $\psi_{s}=\psi$ for all $s\in\Lambda_{p}$. Then, for any nonzero scalars $\alpha_{1},\alpha_{2},...,\alpha_{p}$ with $\sum_{s=1}^{p}\alpha_{s}\neq 0$, we have

[TABLE]

where $\psi_{\xi}=\chi_{[0,1)}$.

Choose $\mu=|\sum_{s=1}^{p}\alpha_{s}|^{2}>0$.

Then

[TABLE]

By Theorem 4.2, the finite system $\mathcal{F}_{p}$ is a $\Theta$-$IWH$ wave packet frame for $L^{2}(\mathbb{R})$.

This work is jointly with A. K. Sah and Deepshikha

[TABLE]

Lalit is supported by R $\&$ D Doctoral Research Programme, University of Delhi, Delhi-110007, India. Grant No. : RC/2014/6820.