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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
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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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http://mi.mathnet.ru/eng/mz11864https://doi.org/10.4213/mzm11864 http://mi.mathnet.ru/eng/mz/v103/i4/p617
Citing articles on Google Scholar:
Russian citations,
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This publication is cited in the following articles:
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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
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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
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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
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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
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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
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