Discretized Gabor Frames of Totally Positive Functions

TL;DR

This paper constructs a broad class of universal window functions for Gabor frames, ensuring that all overcritical lattices form frames and all undercritical lattices form Riesz sequences, advancing the understanding of Gabor frame universality.

Contribution

It introduces a new class of totally positive functions serving as universal windows for Gabor frames across all lattice configurations.

Findings

Universal windows generate Gabor frames for all overcritical lattices.

These windows form Riesz sequences for all undercritical lattices.

The approach broadens the applicability of Gabor analysis.

Abstract

In this paper a large class of universal windows for Gabor frames (Weyl-Heisenberg frames) is constructed. These windows have the fundamental property that every overcritical rectangular lattice generates a Gabor frame. Likewise, every undercritical rectangular lattice generates a Riesz sequence.