Local properties of gaussian random fields on compact symmetric spaces and theorems of the Jackson-Bernstein type
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We consider local properties of smaple functions of Gaussian isotropic random fields on compact Riemannian symmetric spacesM of rank 1. 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 a theorem of the Bernstein type for optimal approximations of functions of this sort by harmonic polynomials in the metric of the spaceL2(M). We use theorems of the Jackson-Bernstein-type to obtain sufficient conditions for the sample functions of a field to almost surely belong to the classes of functions associated with the Riesz and Cesàro means.
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