Chebyshev–Hermite polynomials and distributions of polynomials in Gaussian random variables

This paper gives a survey of several directions of research connected with Chebyshev–Hermite polynomials on finite-dimensional and infinite-dimensional spaces, in particular, of approaches using the Malliavin calculus and other methods of investigation of distributions of polynomials in Gaussian random variables. We give estimates for measures of sets of large and small values, estimates of distances in total variation norm between distributions of polynomials, and results on membership of such distributions in Nikolskii–Besov classes of fractional differentiability. New results are obtained on weak convergence of measures given by polynomial densities with respect to Gaussian measures.