The Strohmer and Beaver Conjecture for Gaussian Gabor Systems - A Deep Mathematical Problem (?)

Abstract

In this article we are going to discuss the conjecture of Strohmer and Beaver for Gaussian Gabor systems. It asks for an optimal sampling pattern in the time-frequency plane, where optimality is measured in terms of the condition number of the frame operator. From a heuristic point of view, it seems obvious that a hexagonal (sometimes called triangular) lattice should yield the solution. The conjecture is now open for 16 years and only recently partial progress has been made. One point this article aims to make, is to show up parallels to a long standing, open problem from geometric function theory, Landau's problem posed in 1929, suggesting that the conjecture of Strohmer and Beaver is a very deep mathematical problem.