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This article is cited in 3 scientific papers (total in 3 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 $-1=\eta_0<\eta_1<\eta_2<\dots<\eta_{N-1}<\eta_N=1$, $\lambda_N=\max_{0\leq j\leq N-1}(\eta_{j+1}-\eta_j)$. 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.
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
Linking options:
https://www.mathnet.ru/eng/isu484 https://www.mathnet.ru/eng/isu/v14/i1/p38
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