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DOI: 10.46698/g5860-8517-3109-i

Approximation Properties of Polynomials \(\hat{l}_{n,N}^\alpha(x)\), Orthogonal on Any Sets

Magomedova, Z. M.  , Nurmagomedov, A. A.
Vladikavkaz Mathematical Journal 2022. Vol. 24. Issue 2.
Abstract:
Let \(\Omega=\{x_0, x_1, \dots, x_j, \dots\}\) - discrete system of points such that \(0=x_0 < x_1 < {x_2 < \dots < x_j < \dots,}\) \(\lim_{j\rightarrow\infty}x_j=+\infty\) and \(\Delta{x_j}=x_{j+1}-x_j\), \(\delta=\sup_{0\leq j < \infty}\Delta x_j < \infty,N=1/\delta\). Asymptotic properties of polynomials \(\hat{l}_{n,N}^\alpha(x)\) orthogonal with weight \(\rho_1^\alpha(x_j)=e^{-x_j}(x_{j+1}^{\alpha+1}-x_j^{\alpha+1})/(\alpha+1)\) in the case \(-1 < \alpha\leq 0\) and \(\rho_2^\alpha(x_j)=e^{-x_{j+1}}(x_{j+1}^{\alpha+1}-x_j^{\alpha+1}/(\alpha+1)\) in the case \(\alpha > 0\) on arbitrary grids consisting of an infinite many points on the semi-axis \([0, +\infty)\) are investigated. Namely an asymptotic formula is proved in which asymptotic behavior of these polynomials as \(n\) tends to infinity together with \(N\) is closely related to asymptotic behavior of the orthonormal Laguerre polynomials \(\hat{L}_n^\alpha(x)\).
Keywords: polynomial, orthogonal system, set, weight, asymptotic formula.
Language: Russian Download the full text  
For citation: Magomedova, Z. M. and Nurmagomedov, A. A. Approximation Properties of Polynomials \(\hat{l}_{n,N}^\alpha(x)\), Orthogonal on Any Sets, Vladikavkaz Math. J., 2022, vol. 24, no. 2, pp.101-116 (in Russian). DOI 10.46698/g5860-8517-3109-i
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