On the Realization of Exact Upper Bounds of the Best Approximations on the Classes H1,1 by Favard Sums

Abstract

In this paper, we find the sets of all extremal functions for approximations of the Hölder classes of H1 2π-periodic functions of one variable by the Favard sums, which coincide with the set of all extremal functions realizing the exact upper bounds of the best approximations of this class by trigonometric polynomials. In addition, we obtain the sets of all of extremal functions for approximations of the class H1 by linear methods of summation of Fourier series. Furthermore, we receive the set of all extremal functions for the class H1 in the Korneichuk–Stechkin lemma and its analogue, the Stepanets lemma, for the Hölder class H1,1 functions of two variables being 2π-periodic in each variable.