Geodesic
Fractional smoothness of distributions of trigonometric polynomials on a space with a Gaussian measure
The Bulletin of Irkutsk State University. Series Mathematics, Tome 31 (2020), pp. 78-95

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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.
Mots-clés : Nikolskii–Besov class
Keywords: Gaussian measure, distribution of a trigonometric polynomial. 
@article{IIGUM_2020_31_a5,
     author = {G. I. Zelenov},
     title = {Fractional smoothness of distributions of trigonometric polynomials on a space with a {Gaussian} measure},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {78--95},
     publisher = {mathdoc},
     volume = {31},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2020_31_a5/}
}
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G. I. Zelenov. Fractional smoothness of distributions of trigonometric polynomials on a space with a Gaussian measure. The Bulletin of Irkutsk State University. Series Mathematics, Tome 31 (2020), pp. 78-95. http://geodesic.mathdoc.fr/item/IIGUM_2020_31_a5/