Frames for compactly supported functions with irrational density

TL;DR

The paper investigates when Gabor systems with compactly supported windows form frames for irrational densities, proving sufficiency of conditions on $g$ that ensure the frame property for all $\alpha< b-a$, $\alpha\beta<1$, with $\alpha\beta\notin\mathbb{Q}$. The authors develop a strategy based on the Ron–Shen criterion, reducing the problem to finite blocks and establishing a uniform lower bound via nonvanishing determinants of those blocks, aided by the irrationality of $\alpha\beta$. They prove the nonvanishing of these determinants for broad classes of windows, including analytic and generic random constructions, and provide explicit examples such as $g(x)=\exp\big(1/(x^4-1)\big)\chi_{(-1,1)}(x)$. The rational-density case is shown to be more delicate, with partial results indicating frame behavior away from finitely many problematic densities. These results broaden the class of windows known to yield frames and illuminate the structure of the frame set for Gabor systems with irrational densities.

Abstract

We find sufficient conditions on a compactly supported function $g$, $\supp g = [a,b]$ which guarantee that the Gabor system $$\mathcal{G}(g;α,β)=\{e^{2πi βm x}g(x-αn)\}_{m,n\in\mathbb{Z}}$$ is a frame for all $α< b-a, αβ< 1, αβ\notin\Q$. 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. $g(x) = \exp(\frac{1}{x^4-1})χ_{(-1,1)}(x)$.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Remark 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Theorem 2.2
• proof
• Theorem 3.1
• proof