Related papers: Weyl-Heisenberg Frame Wavelets with Basic Supports
Let $a$, $b$ be two fixed positive constants. A function $g\in L^2({\mathbb R})$ is called a \textit{mother Weyl-Heisenberg frame wavelet} for $(a,b)$ if $g$ generates a frame for $L^2({\mathbb R})$ under modulates by $b$ and translates by…
In 1990, Daubechies proved a fundamental identity for Weyl-Heisenberg systems which is now called the Weyl-Heisenberg Frame Identity. WH-Frame Identity: If $g\in W(L^{\infty},L^{1})$, then for all continuous, compactly supported functions f…
A Weyl-Heisenberg frame for L^2(R) is a frame consisting of translates and modulates of a fixed function. In this paper we give necessary and sufficient conditions for this family to form a tight WH-frame. This allows us to write down…
In this paper we have generalized and studied the $K$-Weyl-Heisenberg frames, where $K$ is a bounded linear operator on $L^2(\mathbb{R}^d)$. We have obtained necessary and sufficient conditions for acertain system to be a…
In a previous Note we established a necessary and sufficient condition for a multivariate Weyl--Heisenberg system G({\phi},{\Lambda}) to be a frame when the window is a generalized Gaussian (squeezed coherent state) and {\Lambda} a…
We prove that for any two lattices $L, M \subseteq \mathbb{R}^d$ of the same volume there exists a measurable, bounded, common fundamental domain of them. In other words, there exists a bounded measurable set $E \subseteq \mathbb{R}^d$ such…
The relationship between the frame bounds of frames (Gabor) for the space $L^2(\mathbb{R})$ with several generators from the Weyl-Heisenberg group and the scalars linked to the sum of frames is examined in this paper. We give sufficient…
Let {\phi} be an arbitrary generalized Gaussian (squeezed coherent state), {\Lambda}_{{\alpha}{\beta}}=({\alpha}_1 Z \times\cdot\cdot\cdot\times \alpha_{n}\mathbb{Z)\times}(\beta_{1}\mathbb{Z}\times\cdot\cdot\cdot…
We give sufficient conditions for translates and modulates of a function g in L^2(R) to be a frame for its closed linear span. Even in the case where this family spans all of L^2(R), wou conditions are significantly weaker than the previous…
We develop a usable perturbation theory for Weyl-Heisenberg frames. In particular, we prove that if $(E_{mb}T_{na}g)_{m,n\inmathbb Z}$ is a WH-frame and $h$ is a function which is close to $g$ in the Wiener Amalgam space norm, then…
We provide a detailed development of a function valued inner product known as the bracket product and used effectively by de Boor, Devore, Ron and Shen to study translation invariant systems. We develop a version of the bracket product…
We present a comprehensive analysis of the convergence properties of the frame operators of Weyl-Heisenberg systems and shift-invariant systems, and relate these to the convergence of the Walnut representation. We give a deep analysis of…
In this work, we analyze Gabor frames for the Weyl--Heisenberg group and wavelet frames for the extended affine group. Firstly, we give necessary and sufficient conditions for the existence of nonstationary frames of translates. Using these…
We study an intriguing question in frame theory we call "Weaving Frames" that is partially motivated by preprocessing of Gabor frames. Two frames $\{\varphi_i\}_{i\in I}$ and $\{\psi_i \}_{i\in I}$ for a Hilbert space ${\mathbb H}$ are…
Let $g \in L^2(\mathbb{R})$ be a rational function of degree $M$, i.e. there exist polynomials $P, Q$ such that $g = {{P} \over {Q}}$ and $deg(P) < deg(Q) \leq M$. We prove that for any $\varepsilon>0$ and any $M \in \mathbb{N}$ there…
The frame set of a function $g\in L^2(\mathbb{R})$ is the subset of all parameters $(a, b)\in \mathbb{R}^2_+$ for which the time-frequency shifts of $g$ along $a\mathbb{Z}\times b\mathbb{Z}$ form a Gabor frame for $L^2(\mathbb{R}).$ In this…
Weyl-Heisenberg ensembles are translation-invariant determinantal point processes on $\mathbb{R}^{2d}$ associated with the Schr\"odinger representation of the Heisenberg group, and include as examples the Ginibre ensemble and the…
A subset $M$ of a separable Hilbert space $H$ is $\ell^1$-bounded if there exists a Riesz basis $\mathcal{F} = \{e_n\}_{n \in \mathbb{N}}$ for $H$ such that $\sup_{x \in M} \sum_{n \in \mathbb{N}} |\langle x, e_n\rangle| < \infty.$ A…
An explicit expression for the Kohn-Nirenberg symbol of a Weyl- Heisenberg frame operator on $L^2(\mathbb{R})$ is obtained directly from the Gabor atom coming from new classes of window functions. This new approach, using only elementary…
We analyze the notion of reproducing pair of weakly measurable functions, which generalizes that of continuous frame. We show, in particular, that each reproducing pair generates two Hilbert spaces, conjugate dual to each other. Several…