Mathematical Developments on Isotropic Positive Definite Functions on Spheres

Abstract

In this thesis, three open problems related to the isotropic positive definite functions on the sphere have been solved. Firstly, we provide necessary and sufficient conditions for the equivalence of two Gaussian measures with two different covariance models with associated d-Schoenberg sequences. Secondly, we find the d-Schoenberg coefficients associated with an isotropic positive definite function on the sphere in terms of the Fourier coefficients on the circle. Thirdly, we drive the infimum of the second derivative at zero for the class of the locally supported isotropic positive definite functions on the sphere. Finally, we propose a new family of Markov processes in maximal compact subgroups of a semisimple Lie group.