On the best linear method of approximation of some classes analytic functions in the weighted Bergman space

M. S. Saidusaynov

Čebyševskij sbornik, Tome 17 (2016) no. 1, pp. 240-253

Résumé

In this paper the exact values of different widths in the space $B_{q,\gamma}$, $1\leq q\leq\infty$ with the weight $\gamma$ for classes $W_{q,a}^{(r)}(\Phi,\mu)$ were calculated. These classes is consist from functions $f$, which are analytic in a circle $U_{R}:=\{z: |z|\leq R\}$ $(0$ whose $n (n\in\mathbb{N})$-th derivatives by argument $f_{a}^{(r)}$ is belong to the space $B_{q,\gamma} (1\leq q\leq\infty, 0$ and have an averaged modulus of smoothness of second order majorized by function $\Phi$, and everywhere further assumed that the function $\Phi(t), t>0$ is an arbitrary function that $\Phi(0)=0$. The exact inequalities between the best polynomial approximation of analytic functions in a unit disk and integrals consisted from averaged modulus of smoothness of second order functions with $r$-th derivatives order and concrete weight which is flow out from substantial meaning of problem statement. The obtained result is guarantee to calculate the exact values of Bernshtein and Kolmogorov's widths. Method of approximation which is used for obtaining the estimation from above the Kolmogorov n-width is learn on L. V. Taykov work which earlier is proved for modulus of smoothness of complex polynomials. The special interest is offer the problem about constructing the best linear methods of approximation of classes functions $W_{q,a}^{(r)}(\Phi,\mu)$ and connected to it the problem in calculating the exact values of Linear and Gelfand $n$-widths. The founded best linear methods is depend on given number $\mu\geq 1$ and in particular when $\mu=1$ is contain the previous proved results. Also showed the explicit form an optimal subspaces given dimension which are implement the values of widths. Bibliography: 18 titles.

Détail

Keywords: the best linear method, $n$-widths, module of smoothness.