Frames of multi-windowed exponentials on subsets of ${\mathbb R}^d$

TL;DR

This paper characterizes when collections of windowed exponential functions form frames in L^2 spaces over subsets of R^d, providing necessary and sufficient conditions and exploring cases with unbounded or finite measure domains.

Contribution

It establishes a precise criterion for the existence of frames of windowed exponentials on subsets of R^d, including unbounded and finite measure cases, and constructs counterexamples.

Findings

Necessary and sufficient condition for frames involving window bounds.

No frames exist for unbounded sets with infinite measure.

Sufficient conditions and counterexamples for finite measure unbounded sets.

Abstract

Given discrete subsets $\Lambda_j\subset {\Bbb R}^d$, $j=1,...,q$, consider the set of windowed exponentials $\bigcup_{j=1}^{q}\{g_j(x)e^{2\pi i <\lambda,x>}: \lambda\in\Lambda_j\}$ on $L^2(\Omega)$. We show that a necessary and sufficient condition for the windows $g_j$ to form a frame of windowed exponentials for $L^2(\Omega)$ with some $\Lambda_j$ is that $m\leq \max_{j\in J}|g_j|\leq M$ almost everywhere on $\Omega$ for some subset $J$ of $\{1,..., q\}$. If $\Omega$ is unbounded, we show that there is no frame of windowed exponentials if the Lebesgue measure of $\Omega$ is infinite. If $\Omega$ is unbounded but of finite measure, we give a sufficient condition for the existence of Fourier frames on $L^2(\Omega)$. At the same time, we also construct examples of unbounded sets with finite measure that have no tight exponential frame.