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 Daghestan Electronic Mathematical Reports, 2016, Issue 6, Pages 31–60 (Mi demr28)

Systems of functions orthogonal with respect to scalar products of Sobolev type with discrete masses generated by classical orthogonal systems

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

Abstract: For some natural number $r$ and a given system of functions $\{\varphi_k(x)\}_{k=0}^\infty$, orthonormal on $(a, b)$ with weight $\rho(x)$, we construct the new system of functions $\{\varphi_{r,k}(x)\}_{k=0}^\infty$, orthonormal with respect to the Sobolev type inner product of the following form
\begin{equation*} \langle f,g\rangle=\sum_{\nu=0}^{r-1}f^{(\nu)}(a)g^{(\nu)}(a)+\int_{a}^{b} f^{(r)}(t)g^{(r)}(t)\rho(t) dt. \end{equation*}
The convergence of the Fourier series by the system $\{\varphi_{r,k}(x)\}_{k=0}^\infty$ is investigated. Moreover, we consider some important special cases of systems of such type and obtain explicit representations for them, which can be used in the study of asymptotic properties of functions $\varphi_{r,k}(x)$ when $k\to\infty$ and the approximative properties of Fourier sums by the system $\{\varphi_{r,k}(x)\}_{k = 0}^\infty$.

Keywords: orthogonal polynomials, Sobolev orthogonal polynomials, Haar system, Jacobi polynomials, Ñhebyshev polynomials of the first kind, Laguerre polynomials, Hermite polynomials

DOI: https://doi.org/10.31029/demr.6.3

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UDC: 517.538
Revised: 07.09.2016
Accepted:08.09.2016

Citation: I. I. Sharapudinov, Z. D. Gadzhieva, R. M. Gadzhimirzaev, “Systems of functions orthogonal with respect to scalar products of Sobolev type with discrete masses generated by classical orthogonal systems”, Daghestan Electronic Mathematical Reports, 2016, no. 6, 31–60

Citation in format AMSBIB
\Bibitem{ShaGadGad16} \by I.~I.~Sharapudinov, Z.~D.~Gadzhieva, R.~M.~Gadzhimirzaev \paper Systems of functions orthogonal with respect to scalar products of Sobolev type with discrete masses generated by classical orthogonal systems \jour Daghestan Electronic Mathematical Reports \yr 2016 \issue 6 \pages 31--60 \mathnet{http://mi.mathnet.ru/demr28} \crossref{https://doi.org/10.31029/demr.6.3} \elib{http://elibrary.ru/item.asp?id=https://elibrary.ru/item.asp?id=29409286} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. R. M. Gadzhimirzaev, “Rekurrentnye sootnosheniya dlya polinomov, ortonormirovannykh po Sobolevu, porozhdennykh polinomami Lagerra”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 18:1 (2018), 17–24
2. 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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