Approximation by trigonometric polynomials of functions of several variables on the torus

The paper is devoted to the approximation of classes of periodic functions of several variables whose derivative is given with the aid of the absolute value of mixed moduli of continuity. The author studies best approximations by Fourier sums and by spaces of trigonometric polynomials, the Kolmogorov widths of these classes and other related questions. In the study of these questions, the problem arises in a natural way of estimating integrals and sums over convex sets depending on a parameter or over their complements. Asymptotic orders are computed for such integrals and sums connected with the corresponding questions of approximation.
Bibliography: 46 titles.