Positive-definiteness and integral representations for special functions

TL;DR

This paper characterizes holomorphic positive definite functions on a strip as Fourier-Laplace transforms of unique measures, unifying classical theorems and applying the framework to special functions like Gamma, zeta, and Bessel functions.

Contribution

It introduces a new integral representation for positive definite functions, generalizing classical theorems and explicitly constructing measures for special functions.

Findings

Unified integral representation for positive definite functions.

Explicit measures constructed for Gamma, zeta, and Bessel functions.

New proof of the zeta function's integral representation on the critical strip.

Abstract

We characterize a holomorphic positive definite function $f$ defined on a horizontal strip of the complex plane as the Fourier-Laplace transform of a unique exponentially finite measure on $\mathbb{R}$. The classical theorems of Bochner on positive definite functions and of Widder on exponentially convex functions become special cases of this characterization: they are respectively the real and pure imaginary sections of the complex integral representation. We apply this representation to special cases, including the $\Gamma$, $\zeta$ and Bessel functions, and construct explicitly the corresponding measures, thus providing new insight into the nature of complex positive and co-positive definite functions: in the case of the zeta function this process leads to a new proof of an integral representation on the critical strip.