Positive definite functions and spherical h-harmonic expansions with negative indices on spheres

Abstract

This thesis consists of the following two parts: In the first part, we investigate the relationship between positive definite functions on the unit sphere and in the Euclidean space with the same dimensions. For the dimension d is odd, a new technique is developed to establish the inheritance of positive (semi-)definite property from d-dimensional Euclidean space to d-dimensional sphere and the converse. For d=2, it is proved that a class of truncated functions is positive definite on the unit sphere with a range restriction. Our results can verify a conjecture proposed by R.K. Beatson, W. Zu Castell, Y. Xu and a sharp Polya type criterion for positive definite functions on spheres. In the second part, we study the Cesàro means of the weighted orthogonal polynomial expansions (WOPEs) with respect to the weight function $\prod_{i=1}^{d}|x_i|^{2\kappa_i}$ on the unit sphere for all parameters $\kappa_i>-1/2$ as $i=1,...,d$. A sharp pointwise estimate is established for the corresponding Cesàro kernels and for the reproducing kernels of the spaces of orthogonal polynomials, which allows us to establish sharp results on Cesàro summability of the WOPEs with less restriction on the parameters. Similar results are also established for the corresponding weight functions on the unit ball and the simplex.