Gabor frame bound optimizations

TL;DR

This paper investigates the optimization of Gabor frame bounds over rectangular lattices, revealing cases where optimality is achieved by square or rectangular lattices, and situations where no optimal lattice exists.

Contribution

It provides new results on the optimality of Gabor frame bounds for various lattices and windows, including cases with no optimal solutions.

Findings

Optimality results for square lattices in certain cases

Rectangular lattices optimize frame bounds and condition numbers in others

Existence of no optimal lattice in some scenarios

Abstract

We study sharp frame bounds of Gabor systems over rectangular lattices for different windows and integer oversampling rate. In some cases we obtain optimality results for the square lattice, while in other cases the lattices optimizing the frame bounds and the condition number are rectangular lattices which are different for the respective quantities. Also, in some cases optimal lattices do not exist at all and a degenerated system is optimal.