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Mat. Zametki, 2018, Volume 103, Issue 4, Pages 617–631 (Mi mz11864)  

This article is cited in 5 scientific papers (total in 5 papers)

Upper Bounds for the Approximation of Certain Classes of Functions of a Complex Variable by Fourier Series in the Space $L_2$ and $n$-Widths

M. Sh. Shabozova, M. S. Saidusajnovb

a Dzhuraev Institute of Mathematics, Academy of Sciences of Republic of Tajikistan, Dushanbe
b Tajik National University, Dushanbe

Abstract: We consider the problem of the mean-square approximation of complex functions regular in a domain $\mathscr D\subset\mathbb C$ by Fourier series with respect to an orthogonal (in $\mathscr D$) system of functions $\{\varphi_k(z)\}$, $k=0,1,2,…$ . For the case in which $\mathscr D=ż\in\mathbb C:|z|<1\}$, we obtain sharp estimates for the rate of convergence of the Fourier series in the orthogonal system $ż^k\}$, $k=0,1,2,…$, for classes of functions defined by a special $m$th-order modulus of continuity. Exact values of the series of $n$-widths for these classes of functions are calculated.

Keywords: Fourier sum, mean-square approximation, generalized modulus of continuity, Jackson–Stechkin inequality, upper bounds for best approximations, $n$-widths.

DOI: https://doi.org/10.4213/mzm11864

Full text: PDF file (536 kB)
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English version:
Mathematical Notes, 2018, 103:4, 656–668

Bibliographic databases:

UDC: 517.5
Received: 23.05.2017

Citation: M. Sh. Shabozov, M. S. Saidusajnov, “Upper Bounds for the Approximation of Certain Classes of Functions of a Complex Variable by Fourier Series in the Space $L_2$ and $n$-Widths”, Mat. Zametki, 103:4 (2018), 617–631; Math. Notes, 103:4 (2018), 656–668

Citation in format AMSBIB
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\by M.~Sh.~Shabozov, M.~S.~Saidusajnov
\paper Upper Bounds for the Approximation of Certain Classes of Functions of a Complex Variable by Fourier Series in the Space~$L_2$ and $n$-Widths
\jour Mat. Zametki
\yr 2018
\vol 103
\issue 4
\pages 617--631
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\crossref{https://doi.org/10.4213/mzm11864}
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\vol 103
\issue 4
\pages 656--668
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. Sh. Shabozov, V. D. Sainakov, “O neravenstvakh tipa Kolmogorova v prostranstve Bergmana dlya funktsii dvukh peremennykh”, Tr. IMM UrO RAN, 24, no. 4, 2018, 270–282  mathnet  crossref  elib
    2. M. Sh. Shabozov, M. S. Saidusainov, “Srednekvadraticheskoe priblizhenie funktsii kompleksnogo peremennogo summami Fure po ortogonalnym sistemam”, Tr. IMM UrO RAN, 25, no. 2, 2019, 258–272  mathnet  crossref  elib
    3. M. Sh. Shabozov, M. S. Saidusaynov, “Approximation of functions of a complex variable by Fourier sums in orthogonal systems in $L_2$”, Russian Math. (Iz. VUZ), 64:6 (2020), 56–62  mathnet  crossref  crossref  isi
    4. S. B. Vakarchuk, “Estimates of the Values of $n$-Widths of Classes of Analytic Functions in the Weight Spaces $H_{2,\gamma}(D)$”, Math. Notes, 108:6 (2020), 775–790  mathnet  crossref  crossref  mathscinet  isi  elib
    5. M. Sh. Shabozov, E. U. Kadamshoev, “Sharp Inequalities between the Best Root-Mean-Square Approximations of Analytic Functions in the Disk and Some Smoothness Characteristics in the Bergman Space”, Math. Notes, 110:2 (2021), 248–260  mathnet  crossref  crossref  isi  elib
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