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DOI: 10.23671/VNC.2018.1.11400

Mean-Square Approximation of Complex Variable Functions by Fourier Series in the Weighted Bergman Space

Shabozov M. S. , Saidusaynov M. S.
Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 1.
Abstract:
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; àpproximative 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.
Keywords: weighted Bergman space, generalized module of continuity, \(n\)-width, generalized shift operator
Language: Russian Download the full text  
For citation: Shabozov M. S., Saidusaynov M. S. Mean-Square Approximation of Complex Variable Functions by Fourier Series in the Weighted Bergman Space. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 1, pp.86-97. DOI 10.23671/VNC.2018.1.11400
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