Wronskians of Fourier and Laplace Transforms

TL;DR

This paper explores the properties of Wronskians of Fourier and Laplace transforms, establishing their positivity and monotonicity, and characterizes entire functions related to the Riemann hypothesis through correlation functions.

Contribution

It introduces a correlation function framework linking Wronskians of transforms to positivity and monotonicity, and characterizes functions in the Laguerre-Pólya class with implications for the Riemann hypothesis.

Findings

Wronskians of Fourier transforms of nonnegative functions are positive definite.

Wronskians of Laplace transforms of nonnegative functions are completely monotone.

A new characterization of functions in the Laguerre-Pólya class related to the Riemann hypothesis.

Abstract

Associated with a given suitable function, or a measure, on $\mathbb{R}$, we introduce a correlation function, so that the Wronskian of the Fourier transform of the function is the Fourier transform of the corresponding correlation function, and the same holds for the Laplace transform. We obtain two types of results. First, we show that Wronskians of the Fourier transform of a nonnegative function on $\mathbb{R}$ are positive definite functions and the Wronskians of the Laplace transform of a nonnegative function on $\mathbb{R}_+$ are completely monotone functions. Then we establish necessary and sufficient conditions in order that a real entire function, defined as a Fourier transform of a positive kernel $K$, belongs to the Laguerre-P\'olya class, which answers an old question of P\'olya. The characterization is given in terns of a density property of the correlation kernel related to $K$, via classical results of Laguerre and Jensen and employing Wiener's $L^1$ Tauberian theorem. As a consequence we provide a necessary and sufficient condition for the Riemann hypothesis in terms of a density of the translations of the correlation function related to the Riemann $\xi$-function.