Shchytov O. Extremal problems of polynomial approximation of functions of one and several real variables

Exact upper bounds of approximation errors of functions by Fourier-Haar partial sums, exact upper bounds of the best and one-sided best approximation by means of Haar polynomials for classes of functions of one and several variables in spaces with integral and uniform metrics are calculated. There are obtained the exact upper bounds of approximation errors of functions by Faber-Schauder partial sums in integral metric on classes of differentiable functions of one variable. Exact upper bounds of Fourier-Haar coefficient modules on certain classes of functions of one and several variables are calculated. Exact constants are established in Jackson-type inequalities under the best approximation of functions by trigonometric polynomials in the spaces S^p , introduced by O.I. Stepanets. Exact values of various widths - Kolmogorov's, Bernstein's and so on - of functional classes in the spaces S^p are found.