A Question About Total Positivity and Newman's Fourier Transforms with Real Zeroes

TL;DR

This paper explores the relationship between measures on Hermitian matrices, their characteristic functions, and the real zeroes of associated Fourier transforms, connecting random matrix spectra with classical Fourier analysis.

Contribution

It investigates the link between random Hermitian matrix measures and the real zeroes of Fourier transforms, extending Newman’s classification.

Findings

Fourier transforms of certain measures have entire functions with real zeroes.

The measure's characteristic function uniquely determines the measure.

Connections between matrix spectra and Fourier zeroes are proposed but not fully established.

Abstract

Given a unitarily invariant ergodic measure on $\infty\times \infty$ Hermitian matrices, it is known that the characteristic function determines (and is determined by) a Polya frequency function $p(t)$. In turn the (finite) measure $d\rho(u):=\frac{1}{p(-iu^2)}du$ has the property that the Fourier transform $Z_b$ of $exp(-bu^2)d\rho(u)$ is an entire function and has real zeroes, for all $b\ge 0$; this is very close (but not identical) to a classification of such measures due to Newman. This raises the question of whether there is a direct connection between (e.g. the spectrum of) random Hermitian matrices and the reality of the zeroes of $Z_b$.