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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.
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