On the Derivatives of Radial Positive Definite Functions

Convolution Roots of Radial Positive Definite Functions with Compact Support

A classical theorem of Boas, Kac, and Krein states that a characteristic function φ with φ(x) = 0 for |x| ≥ τ admits a representation of the form φ(x) = ∫ u(y)u(y + x) dy, x ∈ R, where the convolution root u ∈ L2(R) is complex-valued with u(x) = 0 for |x| ≥ τ/2. The result can be expressed equivalently as a factorization theorem for entire functions of finite exponential type. This paper examin...

Smoothing splines using compactly supported , positive definite , radial basis functions

In this paper, we develop a fast algorithm for a smoothing spline estimator in multivariate regression. To accomplish this, we employ general concepts associated with roughness penalty methods in conjunction with the theory of radial basis functions and reproducing kernel Hilbert spaces. It is shown that through the use of compactly supported radial basis functions it becomes possible to recove...

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