Izvestiya of Saratov University. Mathematics. Mechanics. Informatics
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. Saratov Univ. Math. Mech. Inform.: Year: Volume: Issue: Page: Find

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

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

Full text: PDF file (183 kB)
References: PDF file   HTML file

Bibliographic databases:

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} 

• http://mi.mathnet.ru/eng/isu484
• http://mi.mathnet.ru/eng/isu/v14/i1/p38

 SHARE:

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. 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
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
•  Number of views: This page: 213 Full text: 70 References: 39