Fourier transforms of positive definite kernels and the Riemann $\xi$-Function

TL;DR

This paper explores the zeros of entire functions represented as Fourier transforms of positive definite kernels, revealing connections with classical inequalities and special functions, and discusses open problems in the field.

Contribution

It establishes a novel link between positive definite kernels, Fourier transforms, and the distribution of zeros of entire functions, utilizing properties of special functions.

Findings

Connection between positive definite kernels and zeros of entire functions

Role of Jacobi theta function's convexity in zero distribution

Open problems related to Fourier transforms and kernel properties

Abstract

The purpose of this paper is to investigate the distribution of zeros of entire functions which can be represented as the Fourier transforms of certain admissible kernels. The principal results bring to light the intimate connection between the Bochner-Khinchin-Mathias theory of positive definite kernels and the generalized real Laguerre inequalities. The concavity and convexity properties of the Jacobi theta function play a prominent role throughout this work. The paper concludes with several questions and open problems.