Gabor frame bound optimizations (Q6051157)

The authors find extremal lattices for the spectral bounds of Gabor systems with specific windows. The quantities that they optimize are the lower and upper frame bounds as well as their ratio, which is the condition number of the associated frame operator. They study the cases provided by \textit{A. J. E. M. Janssen} [Indag. Math., New Ser. 7, No. 2, 165--183 (1996; Zbl 1056.42512)], where sharp spectral bounds for Gabor frames over rectangular lattices of the form \(a Z \times b Z\) were computed for several different window functions and \((ab)^{-1} \in N\). Their results hold for rectangular lattices of integer density and for different windows. This paper is organized as follows. Section 1 is introductory. The results are presented in Section 2. Section 3 settles the notation and provides some background information and motivation. Section 4 contains explanations of auxiliary techniques used in the manuscript. The proofs of the results follow in Section 5 for the hyperbolic secant, Section 6 for cut-off exponentials, Section 7 for one-sided exponentials, and Section 8 for two-sided exponentials. The rigorous analytic study of optimal lattices for Gabor systems is relatively new. The first work on this topic is apparently due to \textit{M. Faulhuber} and \textit{S. Steinerberger} [J. Math. Anal. Appl. 445, No. 1, 407--422 (2017; Zbl 1351.42039)]. A number of the computations involved in this work are performed using software and are available for download as a supplement.