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

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 $Z_b$.