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 Mat. Zametki, 2017, Volume 101, Issue 4, Pages 611–629 (Mi mz10987)

Approximation Properties of Fourier Series of Sobolev Orthogonal Polynomials with Jacobi Weight and Discrete Masses

I. I. Sharapudinovab

a Daghestan Scientific Centre of the Russian Academy of Sciences, Makhachkala
b Daghestan State Pedagogical University

Abstract: We study Fourier series of Jacobi polynomials $P_k^{\alpha-r,-r}(x)$, $k=r,r+1,…$, orthogonal with respect to the Sobolev-type inner product of the following form:
$$\langle f,g\rangle=\sum_{\nu=0}^{r-1} f^{(\nu)}(-1)g^{(\nu)}(-1) +\int_{-1}^1f^{(r)}(t)g^{(r)}(t)(1-t)^\alpha dt.$$
It is shown that such series are a particular case of mixed series of Jacobi polynomials $P_k^{\alpha,\beta}(x)$, $k=0,1,…$, considered earlier by the author. We study the convergence of mixed series of general Jacobi polynomials and their approximation properties. The results obtained are applied to the study of the approximation properties of Fourier series of Sobolev orthogonal Jacobi polynomials $P_k^{\alpha-r,-r}(x)$.

Keywords: mixed series of Sobolev orthogonal Jacobi polynomials Jacobi polynomials, Fourier–Sobolev series of Jacobi polynomials and their approximation properties.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00486 This work was supported by the Russian Foundation for Basic Research under grant 16-01-00486.

DOI: https://doi.org/10.4213/mzm10987

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English version:
Mathematical Notes, 2017, 101:4, 718–734

Bibliographic databases:

UDC: 517.538
Revised: 30.04.2016

Citation: I. I. Sharapudinov, “Approximation Properties of Fourier Series of Sobolev Orthogonal Polynomials with Jacobi Weight and Discrete Masses”, Mat. Zametki, 101:4 (2017), 611–629; Math. Notes, 101:4 (2017), 718–734

Citation in format AMSBIB
\Bibitem{Sha17} \by I.~I.~Sharapudinov \paper Approximation Properties of Fourier Series of Sobolev Orthogonal Polynomials with Jacobi Weight and Discrete Masses \jour Mat. Zametki \yr 2017 \vol 101 \issue 4 \pages 611--629 \mathnet{http://mi.mathnet.ru/mz10987} \crossref{https://doi.org/10.4213/mzm10987} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3629050} \elib{http://elibrary.ru/item.asp?id=28931422} \transl \jour Math. Notes \yr 2017 \vol 101 \issue 4 \pages 718--734 \crossref{https://doi.org/10.1134/S0001434617030300} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000401454600030} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85018828482} 

• http://mi.mathnet.ru/eng/mz10987
• https://doi.org/10.4213/mzm10987
• http://mi.mathnet.ru/eng/mz/v101/i4/p611

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This publication is cited in the following articles:
1. I. I. Sharapudinov, M. G. Magomed-Kasumov, “Chislennyi metod resheniya zadachi Koshi dlya sistem obyknovennykh differentsialnykh uravnenii s pomoschyu ortogonalnoi v smysle Soboleva sistemy, porozhdennoi sistemoi kosinusov”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 8, 53–60
2. M. G. Magomed-Kasumov, “Sistema funktsii, ortogonalnaya v smysle Soboleva i porozhdennaya sistemoi Uolsha”, Matem. zametki, 105:4 (2019), 545–552
3. R. M. Gadzhimirzaev, “Sobolev-orthonormal system of functions generated by the system of Laguerre functions”, Probl. anal. Issues Anal., 8(26):1 (2019), 32–46
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