Geodesic
Asymptotic properties and weighted estimation of polynomials, orthogonal on the nonuniform grids with Jacobi weight
Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 1, pp. 38-47 
M. S. Sultanakhmedov. Asymptotic properties and weighted estimation of polynomials, orthogonal on the nonuniform grids with Jacobi weight. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 14 (2014) no. 1, pp. 38-47. http://geodesic.mathdoc.fr/item/ISU_2014_14_1_a4/
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Voir la notice de l'article provenant de la source Math-Net.Ru

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 $[-1,1]$.

[1] Sharapudinov I. I., Mixed series of orthogonal polynomials. Theory and applications, Makhachkala, 2004, 276 pp.

[2] Sharapudinov I. I., “Asymptotics of polynomials orthogonal on grids of the unit circle and the number line”, Sovremennye problemy matematiki, mekhaniki, informatiki, Materialy mezhdunar. nauch. konf., Russia, Tula, 2009, 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 funktsii i ikh prilozheniia, Materialy 15-i Saratovskoi zimnei shkoly, posviashchennoi 125-letiiu so dnia rozhdeniia V. V. Golubeva i 100-letiiu SGU, Saratov, 2010, 187

[4] Sharapudinov I. I., “Asymptotic properties of the polynomials orthogonal on the finite nets of the unite circle”, Vestnik Dagestanskogo nauchnogo tsentra, 2011, no. 42, 5–14

[5] Sharapudinov I. I., “Polynomials, orthogonal on grids from unit circle and number axis”, Daghestan electronic mathematical reports, 1 (2013), 1–55

[6] Nurmagomedov A. A., “About approximation polynomials, orthogonal on random grids”, Izv. Sarat. Univ. (N.S.), Ser. Math. Mech. Inform., 8:1 (2008), 25–31

[7] Nurmagomedov A. A., “Asymptotic properties of polynomials $\hat p_n^{\alpha,\beta}(x)$, orthogonal on any sets in the case of integers $\alpha$ and $\beta$”, Izv. Sarat. Univ. (N.S.), Ser. Math. Mech. Inform., 10:2 (2010), 10–19

[8] Baik J., Kriecherbauer T., McLaughlin K. T.-R., Miller P. D., Discrete orthogonal polynomials. Asymptotics and applications, Princeton Univ. Press, Princeton, 2007, 184 pp. | MR | Zbl

[9] Ou C., Wong R., “The Riemann–Hilbert approach to global asymptotics of discrete orthogonal polynomials with infinite nodes”, Analysis and Applications, 8 (2010), 247–286 | DOI | MR | Zbl

[10] Ferreira C., López J. L., Sinusía E. P., “Asymptotic relations between the Hahn-type polynomials and Meixner–Pollaczek, Jacobi, Meixner and Krawtchouk polynomials”, J. of Comp. and Appl. Math., 217 (2008), 88–109 | DOI | MR | Zbl

[11] Szego G., Orthogonal Polynomials, AMS Colloq. Publ., 23, 1939, 154 pp. | MR | Zbl

[12] Bari N. K., “Generalization of inequalities of S. N. Bernshtein and A. A. Markov”, Izv. AS USSR. Ser. matem., 18:2 (1954), 159–176 | MR | Zbl

[13] Konyagin S. V., “V. A. Markov's inequality for polynomials in the metric of $L$”, Trudy Matematicheskogo Instituta im. V. A. Steklova, 145, 1980, 117–125 | MR | Zbl