Positive Definiteness on Spheres and Hyperbolic Spaces

Spheres and hyperbolic spaces

The group-invariant geometry on real and complex n-balls is hyperbolic geometry, in the sense that there are infinitely many straight lines (geodesics) through a given point not on a given straight line, thus contravening the parallel postulate for Euclidean geometry. We will not directly consider geometric notions, since the transitive group action determines structure in a more useful form. S...

Strict Positive Definiteness on Spheres via Disk Polynomials

compactifications and function spaces on weighted semigruops

chapter one is devoted to a moderate discussion on preliminaries, according to our requirements. chapter two which is based on our work in (24) is devoted introducting weighted semigroups (s, w), and studying some famous function spaces on them, especially the relations between go (s, w) and other function speces are invesigated. in fact this chapter is a complement to (32). one of the main fea...

Positive Definiteness, Reproducing Kernel Hilbert Spaces and Beyond

Positive definiteness, reproducing kernel Hilbert spaces, integral operators and Mercer’s theorem in its various formats are common topics in many branches of mathematics. In this paper we review and upgrade upon some recent results that always involve at least two of them and indicate a few directions in which additional research could be carried out.

Strictly positive definite functions on spheres in Euclidean spaces

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...