Zeros of the Zak transform of totally positive functions

TL;DR

This paper investigates the zeros of the Zak transform for totally positive functions, proving they have only one zero in their fundamental domain under certain conditions using complex analysis and properties of exponential B-splines.

Contribution

It establishes a new zero distribution property for the Zak transform of TP functions without Gaussian factors, connecting finite type TP functions and exponential B-splines.

Findings

Zak transforms of TP functions without Gaussian factors have exactly one zero in the fundamental domain.

The proof utilizes convergence of Zak transforms of finite type TP functions and complex analysis techniques.

The results deepen understanding of the structure of TP functions in harmonic analysis.

Abstract

We study the Zak transform of totally positive (TP) functions. We use the convergence of the Zak transform of TP functions of finite type to prove that the Zak transforms of all TP functions without Gaussian factor in the Fourier transform have only one zero in their fundamental domain of quasi-periodicity. Our proof is based on complex analysis, especially the Theorem of Hurwitz and some real analytic arguments, where we use the connection of TP functions of finite type and exponential B-splines.