Positive-definiteness and integral representations for special functions

Abstract

It is known that a holomorphic positive definite function f defined on a horizontal strip of the complex plane may be characterized as the Fourier-Laplace transform of a unique exponentially finite measure on R. In this paper we apply this complex integral representation to specific families of special functions, including the , ζ 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 ζ function this process leads to a new proof of an integral representation on the critical strip.