Positive definite functions on spheres

Abstract

Positive de nite functions play a central role in approximation theory as in many other areas of mathematical research. The methods for data interpolation on spheres can be e ectively used for analysis of large data sets arising from geosciences. In this endeavor, studying positive de nite functions on spheres is essential. The characterization of positive de nite functions on spheres in Rm using ultraspherical polynomials is given by I. J. Schoenberg in his celebrated 1942 paper \Positive De nite Functions on Spheres" where he also characterizes positive de nite functions in the unit sphere of a real Hilbert space utilizing the cosine function. In this thesis, our aim is to expand the underlying ideas in Schoenberg's characterization of positive de nite functions and review the results on some of its extensions. For this purpose, we rstly present the fundamental results in the theory of positive de nite functions. We also review basic concepts in ultraspherical polynomials in which we present a proof of the addition formula for ultraspherical polynomials by simplifying the one in Nielsen's book as much as possible. Then, we analyze the proofs of Schoenberg's characterization of positive de nite functions on nite and in nite dimensional unit spheres. Finally, we introduce strictly and conditionally positive de nite functions and review some partial results on their characterizations.