Geodesic
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
Matematičeskie zametki, Tome 103 (2018) no. 4, pp. 617-631

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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 $\{\varphi_k(z)\}$, $k=0,1,2,\dots$ . For the case in which $\mathscr D=\{z\in\mathbb C:|z|1\}$, we obtain sharp estimates for the rate of convergence of the Fourier series in the orthogonal system $\{z^k\}$, $k=0,1,2,\dots$, 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.
Mots-clés : Fourier sum
Keywords: mean-square approximation, generalized modulus of continuity, Jackson–Stechkin inequality, upper bounds for best approximations, $n$-widths. 
@article{MZM_2018_103_4_a13,
     author = {M. Sh. Shabozov and M. S. Saidusajnov},
     title = {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}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {617--631},
     publisher = {mathdoc},
     volume = {103},
     number = {4},
     year = {2018},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2018_103_4_a13/}
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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. Matematičeskie zametki, Tome 103 (2018) no. 4, pp. 617-631. http://geodesic.mathdoc.fr/item/MZM_2018_103_4_a13/