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

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.