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This article is cited in 1 scientific paper (total in 1 paper)
Scientific articles
The best approximation and the values of the widths of some classes of analytical functions in the weighted Bergman space $\mathscr{B}_{2,\gamma}$
M. R. Langarshoev College near Moscow “Energia”
Abstract:
In the paper, exact inequalities are found for the best approximation of an arbitrary analytic function $f$ in the unit circle by algebraic complex polynomials in terms of the modulus of continuity of the $m$th order of the $r$th order derivative $f^{(r)}$ in the weighted Bergman space $ \mathscr{B}_{2,\gamma}.$ Also using the modulus of continuity of the $m$-th order of the derivative $f^{(r)}$, we introduce a class of functions $W_{m}^{(r)}(h,\Phi)$ analytic in the unit circle and defined by a given majorant $\Phi,$ $h\in (0,\pi/n],$ $n>r,$ monotonically increasing on the positive semiaxis. Under certain conditions on the majorant $\Phi,$ for the introduced class of functions, the exact values of some known $n$-widths are calculated. We use methods for solving extremal problems in normed spaces of functions analytic in a circle, as well as the method for estimating from below the $n$-widths of functional classes in various Banach spaces developed by V. M. Tikhomirov. The results presented in this paper are a continuation and generalization of some earlier results on the best approximations and values of widths in the weighted Bergman space $\mathscr{B}_{2,\gamma}.$
Keywords:
analytic function, best approximation, modulus of higher-order continuity, weighted Bergman space, widths.
Received: 03.05.2023 Accepted: 09.06.2023
Citation:
M. R. Langarshoev, “The best approximation and the values of the widths of some classes of analytical functions in the weighted Bergman space $\mathscr{B}_{2,\gamma}$”, Russian Universities Reports. Mathematics, 28:142 (2023), 182–192
Linking options:
https://www.mathnet.ru/eng/vtamu288 https://www.mathnet.ru/eng/vtamu/v28/i142/p182
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