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

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

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

I. I. Sharapudinovab, Z. D. Gadzhievaab, R. M. Gadzhimirzaeva

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, hebyshev polynomials of the first kind, Laguerre polynomials, Hermite polynomials

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

Full text: PDF file (574 kB)
Full text: http://mathreports.ru/.../sistemy-funktsiy-ortogonalnykh-otnositelno-skalyarnykh-proizvedeniy-tipa-soboleva-s-diskretnymi-mass
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UDC: 517.538
Received: 29.07.2016
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
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    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  mathnet  crossref  elib
    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  mathnet  crossref
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