Abstract:
Systems of functions are considered which are associated with a given orthogonal system and are orthogonal with respect to an inner product of Sobolev type involving terms with masses concentrated at a point. Special attention is paid to such systems generated by classical orthogonal systems such as the cosine system, the Haar system, and the systems of Legendre, Jacobi, and Laguerre polynomials. The approximation properties of Fourier series in Sobolev-orthogonal systems are investigated in several cases. For (generally speaking, non-linear) systems of differential equations deep connections between Sobolev-orthogonal systems and the Cauchy problem are considered.
Bibliography: 54 titles.
Keywords:
Sobolev-orthogonal systems; Cauchy problem for a system of ordinary differential equations; systems generated by the Haar polynomials, the cosines, the Legendre, Jacobi, Laguerre polynomials.
\Bibitem{Sha19}
\by I.~I.~Sharapudinov
\paper Sobolev-orthogonal systems of functions and some of their applications
\jour Russian Math. Surveys
\yr 2019
\vol 74
\issue 4
\pages 659--733
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\crossref{https://doi.org/10.1070/RM9846}
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Linking options:
https://www.mathnet.ru/eng/rm9846
https://doi.org/10.1070/RM9846
https://www.mathnet.ru/eng/rm/v74/i4/p87
This publication is cited in the following 23 articles:
R. M. Gadzhimirzaev, “Convergence of the Fourier Series in Meixner–Sobolev
Polynomials and Approximation Properties of Its Partial Sums”, Math. Notes, 115:3 (2024), 301–316
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R. M. Gadzhimirzaev, “Approximation properties of de la Vallée Poussin means of partial Fourier series in Meixner–Sobolev polynomials”, Sb. Math., 215:9 (2024), 1202–1223
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M. G. Magomed-Kasumov, “Weighted Sobolev Orthogonal Systems with Two Discrete Points and Fourier Series with Respect to Them”, Russ Math., 68:11 (2024), 29
M. G. Magomed-Kasumov, “The uniform convergence of Fourier series in a system of polynomials orthogonal in the sense of Sobolev and associated to Jacobi polynomials”, Siberian Math. J., 64:2 (2023), 338–346
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M. G. Magomed-Kasumov, “Otsenki skorosti skhodimosti ryadov Fure po ortogonalnoi v smysle Soboleva sisteme funktsii, porozhdennoi sistemoi Uolsha”, Materialy 20 Mezhdunarodnoi Saratovskoi zimnei shkoly «Sovremennye problemy teorii funktsii i ikh prilozheniya», Saratov, 28 yanvarya — 1 fevralya 2020 g. Chast 2, Itogi nauki i tekhn. Sovrem. mat. i ee pril. Temat. obz., 200, VINITI RAN, M., 2021, 73–80
Ó. Ciaurri, J. Mínguez Ceniceros, “Fourier series for coherent pairs of Jacobi measures”, Integral Transforms Spec. Funct., 32:5-8 (2021), 437–457
M. G. Magomed-Kasumov, S. R. Magomedov, “Bystroe preobrazovanie Fure po sisteme funktsii, ortogonalnykh po Sobolevu i porozhdennykh sistemoi Uolsha”, Dagestanskie elektronnye matematicheskie izvestiya, 2021, no. 15, 55–66
M. G. Magomed-Kasumov, “Sobolev orthogonal systems with two discrete points and Fourier series”, Russian Math. (Iz. VUZ), 65:12 (2021), 47–55
J. F. Mañas-Manas, J. J. Moreno-Balcázar, R. Wellman, “Eigenvalue problem for discrete Jacobi-Sobolev orthogonal polynomials”, Mathematics, 8:2 (2020), 182
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