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

Abstract

As to the Bessel integrals of typewe improve known positivity results by making use of new positivity criteria for 1 F 2 and 2 F 3 generalized hypergeometric functions. As an application, we extend Buhmann's class of compactly supported radial basis functions.

Rational Extension of the Newton Diagram for the Positivity of $${}_1F_2$$ Hypergeometric Functions and Askey–Szegö Problem

“…Theorem 4.1 unifies many of earlier positivity results and we refer to our recent paper [7] in which it is applied to improve the results of Misiewicz and Richards [19], Buhmann [6] at the same time.…”

Triple Integral involving the Bessel-Integral Function Jiv(z): Derivation and Evaluation

“…from equation ( 4) in [2], equation (3.37.4.1) in [1] and equation (3.326.2) in [4] where 0 < Re(w + m) < 2, Re(m) < Re(v) < 2, α ∈ R + and using the reflection formula (8.334.3) in [4] for the Gamma function. We are able to switch the order of integration over x, y and z using Fubini's theorem since the integrand is of bounded measure over the space…”

The Newton Polyhedron and positivity of ${}_2F_3$ hypergeometric functions

“…Appendix. The following is a summary of our work [8], [9] concerning the positivity of 1 F 2 hypergeometric functions of similar type (see also [6], [7] for relevant applications and [15] for a probabilistic approach).…”