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Daghestan Electronic Mathematical Reports, 2015, Issue 4, Pages 1–14 (Mi demr15)  

This article is cited in 2 scientific papers (total in 2 papers)

Sobolev orthogonal polynomials, associated with the Chebyshev polynomials of the first kind

I. I. Sharapudinovab, M. G. Magomed-Kasumovab, S. R. Magomedova

a Daghestan Scientific Centre of RAS
b Vladikavkaz Scientific Centre of the RAS

Abstract: Using Chebyshev polynomials $T_n(x)=\cos(n\arccos x) (n=0,1,\ldots)$, for any natural $r$ we build a new system of polynomials $\{T_{r,k}(x)\}_{k=0}^\infty$, orthonormal with respect to the Sobolev type inner product of the following form
$$ <f,g>=\sum_{\nu=0}^{r-1}f^{(\nu)}(-1)g^{(\nu)}(-1)+\int_{-1}^{1} f^{(r)}(t)g^{(r)}(t)\kappa(t) dt, $$
where $\kappa(t)=\frac2\pi(1-t^2)^{-\frac12}$. The convergence of the Fourier series by the system $\{T_{r,k}(x)\}_{k=0}^\infty$ is investigated. We consider the important special cases of systems of this type. For these instances we obtain explicit representations, that can be used in the study of asymptotic properties of functions $T_{r,k}(x)$ when $k\to\infty$ and study of the approximative properties of Fourier sums by the system $\{T_{r,k}(x)\}_{k = 0}^\infty$.

Keywords: orthogonal polynomials, Sobolev orthogonal polynomials, Chebyshev polynomials of the first kind

DOI: https://doi.org/10.31029/demr.4.1

Full text: PDF file (413 kB)
Full text: http://mathreports.ru/.../polinomy-ortogonalnye-po-sobolevu-assotsiirovannye-s-polinomami-chebysheva-pervogo-roda
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UDC: 517.538
Received: 07.10.2015
Revised: 18.11.2015
Accepted:19.11.2015

Citation: I. I. Sharapudinov, M. G. Magomed-Kasumov, S. R. Magomedov, “Sobolev orthogonal polynomials, associated with the Chebyshev polynomials of the first kind”, Daghestan Electronic Mathematical Reports, 2015, no. 4, 1–14

Citation in format AMSBIB
\Bibitem{ShaMagMag15}
\by I.~I.~Sharapudinov, M.~G.~Magomed-Kasumov, S.~R.~Magomedov
\paper Sobolev orthogonal polynomials, associated with the Chebyshev polynomials of the first kind
\jour Daghestan Electronic Mathematical Reports
\yr 2015
\issue 4
\pages 1--14
\mathnet{http://mi.mathnet.ru/demr15}
\crossref{https://doi.org/10.31029/demr.4.1}
\elib{http://elibrary.ru/item.asp?id=https://elibrary.ru/item.asp?id=27311207}


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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  mathnet  crossref
    2. M. G. Magomed-Kasumov, “Sistema funktsii, ortogonalnaya v smysle Soboleva i porozhdennaya sistemoi Uolsha”, Matem. zametki, 105:4 (2019), 545–552  mathnet  crossref  elib
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