Local properties of gaussian random fields on compact symmetric spaces and theorems of the Jackson-Bernstein type

Abstract

We consider local properties of sample functions of Gaussian isotropic random fields on the compact Riemann symmetric spaces $\mathcal{M}$ of rank one. We give conditions under which the sample functions of a field almost surely possess logarithmic and power modulus of continuity. As a corollary, we prove the Bernshtein-type theorem for optimal approximations of functions of this sort by harmonic polynomials in the metric of space $L_2(\mathcal{M})$. We use the Jackson-Bernshtein-type theorems to obtain sufficient conditions of almost surely belonging of the sample functions of a field to classes of functions associated with Riesz and Cesaro means.