Mean-square approximation of complex variable functions by Fourier series in the weighted Bergman space

M. Sh. Shabozov ; M. S. Saidusaynov

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 1, pp. 86-97

Résumé

In this paper we consider the problem of mean-square approximation of functions of a complex variable by Fourier series in orthogonal system. The functions $f$ under consideration are assumed to be regular in some simply connected domain $\mathcal{D}\subset\mathbb{C}$ and square integrable with a nonnegative weight function $\gamma:=\gamma(|z|)$ which is integrable in $\mathcal{D}$, that is, when $f\in L_{2,\gamma}:=L_{2}(\gamma(|z|),D)$. Earlier, V. A. Abilov, F. V. Abilova and M. K. Kerimov investigated the problems of finding exact estimates of the rate of convergence of Fourier series for functions $f\in L_{2,\gamma}$ [9]. They proved some exact Jackson type inequalities and found the values of the Kolmogorov's $n$-width for certain classes of functions. In doing so, a special form of the shift operator was widely used to determine the generalized modulus of continuity of $m$th order and classes of functions defined by a given increasing in $\mathbb{R}_{+}:=[0,+\infty)$ majorant. The article continues the research of these authors, namely, the exact Jackson–Stechkin type inequality between the best approximation of a functions $f\in L_{2,\gamma}$ by algebraic complex polynomials and $L_{p}$ norm of generalized module of continuity is proved; аpproximative properties of classes of functions are studied for which the $L_{p}$ norm of the generalized modulus of continuity has a given majorant. Under certain assumptions on the majorant,the values of Bernstein, Kolmogorov, linear, Gelfand, and projection $n$-widths for classes of functions in $L_{2,\gamma}$ were calculated. It was proved that all widths are coincide and an optimal subspace is the subspace of complex algebraic polynomials.