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. Shabozov ; M. S. Saidusajnov

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

Résumé

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.

Détail

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

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