Sharp Jackson–Stechkin type inequalities in Hardy space $H_2$ and widths of functional classes

Abstract

In this work we obtain sharp Jackson–Stechkin type inequalities relating the best joint polynomial approximation of functions analytic in the unit disk and a special generalization of the continuity modulus, which is defined by means of the Steklov function. While solving a series of problems in the theory on approximation of periodic functions by trigonometric polynomials in the space $L_2$, a modification of the classical definition of the continuity modulus of $m$th order generated by the Steklov function was employed by S.B. Vakarchuk, M.Sh. Shabozov and A.A. Shabozova. Here the proposed construction is employed for defining a modification of the continuity modulus of $m$th order for functions analytic in the unit disk generated by the Steklov function in the Hardy space $H_2$. By using this smoothness characteristic we solve a problem on finding a sharp constant in the Jackson–Stechkin type inequalities for joint approximation of the functions and their intermediate derivatives. For the classes of function, averaged with a weight, the generalized continuity moduli of which are bounded by a given majorant, we find exact values of various $n$-widths. We also solve the problem on finding sharp upper bounds for best joint approximations of the mentioned classes of functions in the Hardy space $H_2$.