Optimal Gabor frame bounds for separable lattices and estimates for Jacobi theta functions

TL;DR

This paper proves that the square lattice optimizes Gabor frame bounds with Gaussian windows among rectangular lattices, confirming a conjecture through advanced analysis of Jacobi theta functions.

Contribution

It establishes the optimality of the square lattice for Gabor frame bounds with Gaussian windows, using new estimates for Jacobi theta functions.

Findings

Square lattice maximizes and minimizes Gabor frame bounds among rectangular lattices.

The proof relies on refined log-convexity and concavity estimates of Jacobi theta functions.

Confirms a conjecture by Floch, Alard & Berrou.

Abstract

We study sharp frame bounds of Gabor frames with the standard Gaussian window and prove that the square lattice optimizes both the lower and the upper frame bound among all rectangular lattices. This proves a conjecture of Floch, Alard & Berrou (as reformulated by Strohmer & Beaver). The proof is based on refined log-convexity/concavity estimates for the Jacobi theta functions $\theta_3$ and $\theta_4$.