Fractional smoothness of distributions of trigonometric polynomials on a space with a Gaussian measure

G. I. Zelenov

The Bulletin of Irkutsk State University. Series Mathematics, Tome 31 (2020), pp. 78-95

Résumé

In this paper we study properties of images of a gaussian measure under trigonometric polynomials of a fixed degree, defined on finite-dimensional space with fixed number of dimensions. We prove that the images of the $n$-dimensional Gaussian measure under trigonometric polynomials have densities from the Nikolskii–Besov class of fractional parameter. This property of images of a gaussian measure is used for estimating the total variation distance between such images via the Fortet–Mourier distance. We also generalize these results to the case of $k$-dimensional mappings whose components are trigonometric polynomials.

Détail

Mots-clés : Nikolskii–Besov class
Keywords: Gaussian measure, distribution of a trigonometric polynomial.

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