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

Abstract

As to the Bessel integrals of type ∫ 0 x ( x μ − t μ ) λ t α J β ( t ) d t ( x > 0 ) , \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 1 F 2 {}_1F_2 and 2 F 3 {}_2F_3 generalized hypergeometric functions. As an application, we extend Buhmann’s class of compactly supported radial basis functions.