Isotropic positive definite functions on spheres generated from those in Euclidean spaces
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In this thesis, author have reviewed the classical characterizations of isotropic positive definite functions on Euclidean spaces and spheres, and had applied them to develop a construction of isotropic positive de nite functions on spheres. However, this construction only works when the dimension of the sphere is odd. The next work is to develop isotropic positive de nite functions on all spheres.
Table of Contents
1. Introduction -- 2. Background information: definitions, the Gamma function; Orthogonal polunomial; Bessel functions; Abel-summability -- 3/ Isotropic positive definite functions in Euclidean space: Integral transforms; Characterizations of positive definite functions; Schoenberg's characterization; Properties of positive definite functions -- 4. Izotropic positive definite functions of spheres: Some classical results concerning ultraspherical polynomials; Schoenber's characterization; Recent results -- 5. Isotropic positive definite functions on spheres generated from those in Euclidean spaces: Main results; Some lemmas and their proofs; Proof of theorems 5.1.1 and 5.1.2 -- 6. Plans for future work. -- References
Thesis (Ph.D.)-- Wichita State University, College of Liberal Arts and Sciences, Dept. of Mathematics, Statistics and Physics