2018
Том 70
№ 6

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

Malyarenko A. A.

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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.

English version (Springer): Ukrainian Mathematical Journal 51 (1999), no. 1, pp 66-75.

Citation Example: Malyarenko A. A. Local properties of gaussian random fields on compact symmetric spaces and theorems of the Jackson-Bernstein type // Ukr. Mat. Zh. - 1999. - 51, № 1. - pp. 60–68.

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