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DOI: 10.23671/VNC.2018.1.11400 MeanSquare Approximation of Complex Variable Functions by Fourier Series in the Weighted Bergman Space
Abstract:
In this paper we consider the problem of meansquare 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 JacksonStechkin 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
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For citation: Shabozov M. S., Saidusaynov M. S. MeanSquare Approximation of Complex Variable Functions by Fourier Series in the Weighted Bergman Space. Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 1, pp.8697. DOI 10.23671/VNC.2018.1.11400 ← Contents of issue 
 

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