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 \(\hat{L}_n^\alpha(x)\).
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
1. Sharapudinov, I. I.Smeshannye ryady po ortogonal'nym polinomam. Teoriya i prilozheniya
[Mixed Series of Orthogonal Polynomials. Theory and Applications],
Makhachkala, DSC RAS, 2004, 276 p. (in Russian).
2. Sharapudinov, I. I. Asymptotics of Polynomials Orthogonal on
Grids of the Unit Circle and the Number Line, Sovremennye Problemy Matematiki,
Mekhaniki, Informatiki: Materialy Mezhdunarodnoy Nauchnoy Konferentsii,
Tula, Izd-vo TulGU, 2009, pp. 100-106 (in Russian).
3. Sharapudinov, I. I. Some Properties of Polynomials
Orthogonal on Nonuniform Grids of the Unit Circle and the Segment,
Sovremennye Problemy Teorii Funktsiy
i ikh Prilozheniya: Materialy 15-i Saratovskoi Zimney Shkoly,
Posviashchennoi 125-letiyu so Dnya Rozhdeniya V. V. Golubeva i
100-Letiyu SGU, Saratov, Nauchnaya Kniga, 2010, p. 187 (in Russian).
4. Sharapudinov, I. I. Asymptotic Properties of the Polynomials Orthogonal
on the Finite Nets of the Unite Circle,
Vestnik Dagestanskogo Nauchnogo Centra, 2011, no. 42, pp. 5-14 (in Russian).
5. Sharapudinov, I. I. Polynomials, Orthogonal on Grids
From Unit Circle and Number Axis, Daghestan Electronic Mathematical Reports, 2013, vol. 1,
pp. 1-55 (in Russian). DOI: 10.31029/demr.1.1.
6. Nurmagomedov, A. A.
Asymptotic Properties of Polynomials \(hatp_n^\alpha,\beta(x),\) Orthogonal
on Any Sets in the Case of Integers \(\alpha\) and \(\beta\),
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2010,
vol. 10, no. 2, pp. 10-19 (in Russian). DOI: 10.18500/1816-9791-2010-10-2-10-19.
7. Nurmagomedov, A. A. Polynomials, Orthogonal on Non-Uniform Grids,
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics,
2011, vol. 11, no. 3(2), pp. 29-42 (in Russian). DOI: 10.18500/ 1816-9791-2011-11-3-2-29-42.
8. Sultanakhmedov, M. S. Asymptotic Properties and Weighted Estimation of Polynomials,
Orthogonal on the Nonuniform Grids with Jacobi Weight,
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2014, vol. 14, no. 1,
pp. 38-47 (in Russian). DOI: 10.18500/1816-9791-2014-14-1-38-47.
9. Dorfler, P. Asymptotics of the Best Constant in a Certain Markov-Type Inequality,
Journal of Approximation Theory, 2002, vol. 114, no. 1, pp. 84-97.
DOI: 10.1006/jath.2001.3638.
10. Magomedova, Z. M. About Asymptotic Polynomials, Orthogonal on Any Grids,
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2011,
vol. 11, no. 3(1), pp. 32-41 (in Russian).
DOI: 10.18500/1816-9791-2011-11-3-1-32-41.
11. Szego, G. Ortogonal'nye mnogochleny [Orthogonal Polynomials],
Moscow, Fizmatgiz, 1962, 500 p. (in Russian).
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