Approximation properties of polynomials $\hat{l}_{n,n}^\alpha(x),$ orthogonal on any sets

Z. M. Magomedova ; A. A. Nurmagomedov

Vladikavkazskij matematičeskij žurnal, Tome 24 (2022) no. 2, pp. 101-116

Résumé

Let $\Omega=\{x_0, x_1, \dots, x_j, \dots\}$ — discrete system of points such that $0=x_0$ $\lim_{j\rightarrow\infty}x_j=+\infty$ 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 $\hat{L}_n^\alpha(x).$