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

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Home Editorial board Publication ethics Peer review guidelines Current Archive Rules for authors Send an article Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	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 + References 1. Shabozov M. Sh., Shabozov O. Sh. Best approximation and the value of the widths of some classes of functions in the Bergman space \(B_p, 1\leq p\leq\infty\), Doklady Akademii Nauk [Dokl. Akad. Nauk], 2006, vol. 410, no. 4, pp. 461-464 (in Russian). 2. Shabozov M. Sh., Shabozov O. Sh. On the best approximation of some classes of analytic functions in the weighted Bergman spaces, Doklady Akademii Nauk [Dokl. Akad. Nauk], 2007, vol. 412, no. 4, pp. 466-469 (in Russian). DOI 10.1134/S1064562407010279. 3. Vakarchuk S. B., Shabozov M. Sh. The widths of classes of analytic functions in a disc, Matematicheskij sbornik [Sbornik: Mathematics], 2010, vol. 201, no. 8, pp. 3-22 (in Russian). DOI 10.4213/sm7505. 4. Shabozov M. Sh., Saidusaynov M. S. The values of \(n\) widths and best linear methods of approximation for some analytic classes functions in the weighted Bergman space, Izvestija Tul'skogo Gosudarstvennogo Universiteta. Estestvennye nauki [News of the Tula state university. Natural sciences], 2014, no. 3, pp. 40-57 (in Russian). 5. Saidusaynov M. S. On the Values of widths and the best linear methods of approximation in the weighted Bergman space, Izvestija Tul'skogo Gosudarstvennogo Universiteta. Estestvennye nauki [News of the Tula state university. Natural sciences], 2015, no. 3, pp. 91-104 (in Russian). 6. Saidusaynov M. S. On the best linear method of approximation of some classes analytic functions in the weighted Bergman space, Chebyshevskij sbornik [Chebyshevskii Sb.], 2016, vol. 17, no. 1, pp. 240-253 (in Russian). 7. Langarshoev M. R. On the Best Linear Methods of Approximation and the Exact Values of Widths for Some Classes of Analytic Functions in the Weighted Bergman Space, Ukrainian Mathematical Journal, 2016, vol. 67, no. 10, pp. 1537-1551. DOI 10.1007/s11253-016-1171-z. 8. Smirnov V. I., Lebedev N. A. Konstruktivnaja teorija funkcij kompleksnogo peremennogo [Constructive Theory of Functions of Complex Variables]. Nauka, Moscow, 1964, 440 p. (in Russian). 9. Abilov V. A., Abilova F. V., Kerimov M. K. Sharp estimates for the convergence rate of Fourier series of complex variable functions in \(L_{2(D,p(z))}\), Computational Mathematics and Mathematical Physics, 2010, vol. 50, no. 6, pp. 946-950. DOI 10.1134/S0965542510060023. 10. Pinkus A. \(n\)-Widths in Approximation Theory. Berlin, etc., Springer-Verlag, 1985, 287 p. 11. Tikhomirov V. M. Nekotorye voprosy teorii priblizhenij [Some Questions of Approximation Theory]. Moscow, Izd. Moscow. Univ., 1976, 325 p. ← Contents of issue


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