On positive definiteness of some radial functions

Abstract

We consider the functions h_μ,ν introduced by Zastavnyi in 2002. The family of these functions is a subfamily of Buhmann’s functions and contains the families of functions introduced by Trigub in 1987 and Wendland in 1995. We investigate the problems of positive definiteness and smoothness at zero for the linear combinations β₂^εh_μ,ν (x/β₂) − β₁^ε_μ,ν (x/β₁).