On infinitely differentiable positive definite functions

Remarks on Lp-approximation of infinitely differentiable multivariate functions

We study the Lp-approximation problem (1 ≤ p < ∞) for infinitely differentiable d-variate functions with respect to the worst case error. In particular, we correct a mistake in the argumentation of Novak and Woźniakowski [2], who showed that the problem is intractable. The main ingredients are arguments from convex geometry, as well as a probabilistic calculation.

Construction of Characteristic Functions for Classes of Infinitely Differentiable Functions

Characteristic functions are constructed for classes of infinitely differentiable functions defined on a half-line and for Paley-Wiener classes. A corresponding result is given for normal operators defined in Hilbert space.

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