Sharp Jackson–Stechkin type inequalities in Hardy space $H_2$ and widths of functional classes
M. Sh. Shabozov,
Z. Sh. Malakbozov Tajik National University, Rudaki av., 17, 734025, Dushanbe, Tajikistan
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
$H_2$.
Keywords:
Jackson–Stechkin type inequalities, continuity modulus, Steklov function, $n$-width, Hardy space.
UDC:
517.5
MSC: 30E05,
30E10,
42A10 Received: 04.05.2022