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Chebyshevskii Sbornik, 2022, Volume 23, Issue 1, Pages 167–182
DOI: https://doi.org/10.22405/2226-8383-2022-23-1-167-182
(Mi cheb1162)
 

This article is cited in 2 scientific papers (total in 2 papers)

Mean-squared approximation of some classes of complex variable functions by Fourier series in the weighted Bergman space $B_{2,\gamma}$

M. Sh. Shabozov, M. S. Saidusainov

Tajik National University (Dushanbe)
Full-text PDF (636 kB) Citations (2)
References:
Abstract: The article considers extremal problems of mean-square approximation of functions of a complex variable, regular in the domain $\mathscr{D}\subset\mathbb{C}$, by Fourier series orthogonal in the system of functions $\{\varphi_{k}(z)\}_{k=0}^{\infty}$ in $\mathscr{D}$ belonging to the weighted Bergman space $B_{2,\gamma}$ with finite norm
\begin{equation*} \|f\|_{2,\gamma}:=\|f\|_{B_{2,\gamma}}=\left(\frac{1}{2\pi}\iint\limits_{(\mathscr{D})}\gamma(|z|)|f(z)|^{2}d\sigma\right)^{1/2},\end{equation*}
where $\gamma:=\gamma(|z|)\geq 0$ is a real integrable function in the domain $\mathscr{D}$, and the integral is understood in the Lebesgue sense, $d\sigma:=dxdy$ is an element of area.
The formulated problem is investigated in more detail in the case when $\mathscr{D}$ is the unit disc in the space $B_{2,\gamma_{\alpha,\beta}}, \gamma_{\alpha,\beta}=|z|^{\alpha}(1-|z|)^{\beta}, \alpha,\beta>-1$ – Jacobi weight. Sharp Jackson-Stechkin-type inequalities that relate the value of the best mean-squared polynomial approximation of $f\in \mathcal{B}_{2,\gamma_{\alpha,\beta}}^{(r)}$ and the Peetre $\mathscr{K}$-functional were proved. In case when $\gamma_{\alpha,\beta}\equiv 1$ we will obtain the earlier known results.
Keywords: Fourier's sum, mean-squared approximation, upper bound best approximation, Peetre $\mathscr{K}$-functional.
Received: 16.12.2021
Accepted: 27.02.2022
Document Type: Article
UDC: 517.5
Language: Russian
Citation: M. Sh. Shabozov, M. S. Saidusainov, “Mean-squared approximation of some classes of complex variable functions by Fourier series in the weighted Bergman space $B_{2,\gamma}$”, Chebyshevskii Sb., 23:1 (2022), 167–182
Citation in format AMSBIB
\Bibitem{ShaSai22}
\by M.~Sh.~Shabozov, M.~S.~Saidusainov
\paper Mean-squared approximation of some classes of complex variable functions by Fourier series in the weighted Bergman space $B_{2,\gamma}$
\jour Chebyshevskii Sb.
\yr 2022
\vol 23
\issue 1
\pages 167--182
\mathnet{http://mi.mathnet.ru/cheb1162}
\crossref{https://doi.org/10.22405/2226-8383-2022-23-1-167-182}
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  • This publication is cited in the following 2 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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