Multivariate inequalities of Chernoff type for classical orthogonal polynomials

Abstract

In this paper we present two-sided Chernoff-type inequalities for the error of the best approximation of a smooth d-variate function by polynomials of total degree less than k in the L w 2 -norm. It is supposed that the d-variate weight function w has one-dimensional classical component weights, which satisfy Pearson differential equations. Similarly as in the univariate case, the leading coefficients of the multivariate classical polynomials, orthonormal with respect to the weight function w, play an important role in the presented estimates.