On Zeros of Fourier Transforms

TL;DR

This paper verifies a Jensen-based criterion for entire functions of genus 0 or 1, showing they have only real zeros, and applies this to special functions like Bessel and Xi functions.

Contribution

It establishes the sufficiency of Jensen's condition for certain Fourier transforms to have only real zeros, extending the understanding of zero distributions in special functions.

Findings

Proves Fourier transforms of specific functions have only real zeros.

Validates Jensen's condition as sufficient for zero location.

Applies results to Bessel and Xi functions confirming their real zeros.

Abstract

In this work we verify the sufficiency of a Jensen's necessary and sufficient condition for a class of genus 0 or 1 entire functions to have only real zeros. They are Fourier transforms of even, positive, indefinitely differentiable, and very fast decreasing functions. We also apply our result to several important special functions in mathematics, such as modified Bessel function $K_{iz}(a),\ a>0$ as a function of variable $z$, Riemann Xi function $\Xi(z)$, and character Xi function $\Xi(z;\chi)$ when $\chi$ is a real primitive non-principal character satisfying $\varphi(u;\chi)\ge0$ on the real line, we prove these entire functions have only real zeros.