An extension of positivity for integrals of Bessel functions and Buhmann's radial basis functions

TL;DR

This paper improves positivity results for Bessel integrals using hypergeometric function criteria and extends Buhmann's class of radial basis functions, enhancing their mathematical properties and potential applications.

Contribution

It introduces new positivity criteria for hypergeometric functions and extends Buhmann's radial basis functions, advancing the theoretical understanding and applicability of these functions.

Findings

Enhanced positivity conditions for Bessel integrals.

Extended Buhmann's radial basis functions class.

Improved mathematical tools for function analysis.

Abstract

As to the Bessel integrals of type \begin{equation*} \int_0^x \left(x^\mu-t^\mu\right)^\lambda t^\alpha J_\beta(t)dt\qquad(x>0), \end{equation*} we improve known positivity results by making use of new positivity criteria for ${}_1F_2$ and ${}_2F_3$ generalized hypergeometric functions. As an application, we extend Buhmann's class of compactly supported radial basis functions.