Sharp Jackson–Stechkin type inequalities in Hardy space H_2 and widths of functional classes
DOI:
https://doi.org/10.13108/2023-15-2-74Keywords:
Jackson–Stechkin type inequalities, continuity modulus, Steklov function, n-width, Hardy space.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 mth 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 mth 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.Downloads
Published
20.06.2023
Issue
Section
Article