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Izv. Saratov Univ. Math. Mech. Inform., 2014, Volume 14, Issue 1, Pages 38–47 (Mi isu484)  

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

Mathematics

Asymptotic properties and weighted estimation of polynomials, orthogonal on the nonuniform grids with Jacobi weight

M. S. Sultanakhmedov

Department of Mathematics and Computer Science, Daghestan Scientific Center, 45, M. Gadzhieva str., 367000, Makhachkala, Daghestan, Russia

Abstract: Let $-1=\eta_0<\eta_1<\eta_2<…<\eta_{N-1}<\eta_N=1$, $\lambda_N=\max_{0\leq j\leq N-1}(\eta_{j+1}-\eta_j)$. Current work is devoted to investigation of properties of polynomials, orthogonal with Jacobi weight $\kappa^{\alpha,\beta}(t)=(1-t)^\alpha (1+t)^\beta$ on nonuniform grid $\Omega_N=\{t_j\}_{j=0}^{N-1}$, where $\eta_j\leq t_j\leq\eta_{j+1}$. In case of integer $\alpha,\beta\geq0$ for such discrete orthonormal polynomials $\hat P_{n,N}^{\alpha,\beta}(t)$ ($n=0,\ldots,N-1$) asymptotic formula $\hat P_{n,N}^{\alpha,\beta}(t)=\hat P_n^{\alpha,\beta}(t)+\upsilon_{n,N}^{\alpha,\beta}(t)$ with $n=O(\lambda_N^{-1/3})$ ($\lambda_N\to0$) was obtained, where $\hat P_n^{\alpha,\beta}(t)$ – classical Jacobi polynomial, $\upsilon_{n,N}^{\alpha,\beta}(t)$ – remainder term. As corollary of asymptotic formula it was deduced weighted estimation of $\hat P_{n,N}^{\alpha,\beta}(t)$ polynomials on segment $[-1,1]$.

Key words: orthogonal polynomials, nonuniform grid, asymptotic formula, weighted estimation.

DOI: https://doi.org/10.18500/1816-9791-2014-14-1-38-47

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UDC: 517.518.82

Citation: M. S. Sultanakhmedov, “Asymptotic properties and weighted estimation of polynomials, orthogonal on the nonuniform grids with Jacobi weight”, Izv. Saratov Univ. Math. Mech. Inform., 14:1 (2014), 38–47

Citation in format AMSBIB
\Bibitem{Sul14}
\by M.~S.~Sultanakhmedov
\paper Asymptotic properties and weighted estimation of polynomials, orthogonal on the nonuniform grids with Jacobi weight
\jour Izv. Saratov Univ. Math. Mech. Inform.
\yr 2014
\vol 14
\issue 1
\pages 38--47
\mathnet{http://mi.mathnet.ru/isu484}
\crossref{https://doi.org/10.18500/1816-9791-2014-14-1-38-47}
\elib{https://elibrary.ru/item.asp?id=21510771}


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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. M. S. Sultanakhmedov, “On the convergence of the least square method in case of non-uniform grids”, Probl. anal. Issues Anal., 8(26):3 (2019), 166–186  mathnet  crossref  elib
    2. M. S. Sultanakhmedov, “Approximation of Functions by Discrete Fourier Sums in Polynomials Orthogonal on a Nonuniform Grid with Jacobi Weight”, Math. Notes, 110:3 (2021), 418–431  mathnet  crossref  crossref  isi
  • Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
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