On norm-dependent positive definite functions

A Note on Positive Definite Norm Dependent Functions

Let K be an origin symmetric star body in R. We prove, under very mild conditions on the function f : [0,∞) → R, that if the function f(‖x‖K) is positive definite on R , then the space (R, ‖ · ‖K) embeds isometrically in L0. This generalizes the solution to Schoenberg’s problem and leads to progress in characterization of n-dimensional versions, i.e. random vectors X = (X1, ...,Xn) in R n such ...

GENERALIZED POSITIVE DEFINITE FUNCTIONS AND COMPLETELY MONOTONE FUNCTIONS ON FOUNDATION SEMIGROUPS

A general notion of completely monotone functionals on an ordered Banach algebra B into a proper H*-algebra A with an integral representation for such functionals is given. As an application of this result we have obtained a characterization for the generalized completely continuous monotone functions on weighted foundation semigroups. A generalized version of Bochner’s theorem on foundation se...

Multivariate positive definite functions on spheres

In 1942 I.J. Schoenberg proved that a function is positive definite in the unit sphere if and only if this function is a positive linear combination of the Gegenbauer polynomials. In this paper we extend Schoenberg’s theorem for multivariate Gegenbauer polynomials. This extension derives new positive semidefinite constraints for the distance distribution which can be applied for spherical codes.

Schoenberg’s Problem on Positive Definite Functions

Strictly Positive Definite Functions on Spheres

In this paper we study strictly positive definite functions on the unit sphere of the m-dimensional Euclidean space. Such functions can be used for solving a scattered data interpolation problem on spheres. Since positive definite functions on the sphere were already characterized by Schoenberg some fifty years ago, the issue here is to determine what kind of positive definite functions are act...