Minimal Operator Norms via Minimal Theta Functions

TL;DR

This paper studies optimal configurations of Gabor frames with Gaussian windows, showing that for certain redundancies, the hexagonal lattice minimizes the upper frame bound, using theta function minimization results.

Contribution

It establishes the minimality of the hexagonal lattice for the upper frame bound in Gabor frames with Gaussian windows and specific redundancies, leveraging theta function theory.

Findings

Hexagonal lattice minimizes the upper frame bound for even redundancy with Gaussian windows.

The result applies to Gabor frames with chirped Gaussians and rectangular lattices.

Utilizes Montgomery's minimal theta function results to prove optimality.

Abstract

We investigate sharp frame bounds of Gabor frames with chirped Gaussians and rectangular lattices or, equivalently, the case of the standard Gaussian and general lattices. We prove that for even redundancy and standard Gaussian window the hexagonal lattice minimizes the upper frame bound using a result by Montgomery on minimal theta functions.