Article
Ufa Mathematical Journal
Volume 15, Number 2, pp. 74-84
Sharp Jackson--Stechkin type inequalities in the Hardy space $H_2$ and widths of functional classes
Shabozov M.Sh., Malakbozov Z.Sh.
DOI:10.13108/2023-15-2-74
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In the 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 $H_2$.