Frames for compactly supported functions with irrational density

TL;DR

This paper establishes sufficient conditions for compactly supported functions to generate Gabor frames with irrational density, providing explicit examples and broad applicability for such functions.

Contribution

It introduces explicit, verifiable conditions on functions to ensure Gabor systems form frames with irrational density, expanding the class of functions known to generate frames.

Findings

Conditions are satisfied by almost all such functions

Explicit examples of functions generating Gabor frames

Applicable for all lpha < b-a, lphaeta < 1, lphaeta Q

Abstract

We find sufficient conditions on a compactly supported function $g$, $\supp g = [a,b]$ which guarantee that the Gabor system $$\mathcal{G}(g;\alpha,\beta)=\{e^{2\pi i \beta m x}g(x-\alpha n)\}_{m,n\in\mathbb{Z}}$$ is a frame for all $\alpha < b-a, \alpha\beta < 1, \alpha\beta \notin\Q$. These conditions are on one hand satisfied by almost all such functions, and on the other hand are explicit enough that we can give many concrete examples of the functions $g$ which give us a frame e.g. $g(x) = \exp(\frac{1}{x^4-1})\chi_{(-1,1)}(x)$.