Cite this article as:
Sultanakhmedov M. S. Asymptotic properties and weighted estimation of polynomials, orthogonal on the nonuniform grids with Jacobi weight . Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 2014, vol. 14, iss. 1, pp. 38-47. DOI: https://doi.org/10.18500/1816-9791-2014-14-1-38-47
Language:
Russian
Heading:
UDC:
517.518.82
Asymptotic properties and weighted estimation of polynomials, orthogonal on the nonuniform grids with Jacobi weight
Abstract:
Current work is devoted to investigation of properties of polynomials, orthogonal with Jacobi weight on nonuniform grid where. In case of integer for such discrete orthonormal polynomials asymptotic formula with was obtained, where classical Jacobi polynomial, remainder term. As corollary of asymptotic formula it was deduced weighted estimation polynomials on segment
References
1. Sharapudinov I. I. Smeshannye riady po ortogonal’nym
polinomam. Teoriia i prilozheniia [Mixed series of
orthogonal polynomials. Theory and applications].
Makhachkala, 2004, 276 p. (in Russian).
2. Sharapudinov I. I. Asimptotika polinomov,
ortogonal’nykh na setkakh iz edinichnoi okruzhnosti
i chislovoi priamoi [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, pp. 100-106 (in Russian) .
3. Sharapudinov I. I. Nekotorye svoistva polinomov, ortogonal’nykh na neravnomernykh setkakh iz edinichnoi
okruzhnosti i otrezka [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, pp. 187
(in Russian).
4. Sharapudinov I. I. Asymptotic properties of the polynomials orthogonal on the finite nets of the unite circle
[Asimptoticheskie svoistva polinomov, ortogonal’nykh nakonechnykh setkakh edinichnoi okruzhnosti]. Vestnik
Dagestanskogo nauchnogo tsentra, 2011, no. 42, pp. 5–14
(in Russian).
46 Научный отдел
А. Б. Шишкин. Проективное и инъективное описания в комплексной области. Двойственность
5. Sharapudinov I. I. Polynomials, orthogonal on grids
from unit circle and number axis. Dagestanskie elektronnye matematicheskie izvestiia [Daghestan electronic
mathematical reports], 2013, vol. 1, pp. 1–55 (in Russian).
6. Nurmagomedov A. A. About approximation polynomials, orthogonal on random grids. Izv. Sarat. Univ.
(N. S.), Ser. Math. Mech. Inform., 2008, vol. 8, iss. 1,
pp. 25–31 (in Russian).
7. Nurmagomedov A. A. Asymptotic properties of polynomials ˆ p α,β n
(x), orthogonal on any sets in the case
of integers α and β. Izv. Sarat. Univ. (N. S.), Ser.
Math. Mech. Inform., 2010, vol. 10, iss. 2, pp. 10–19
(in Russian).
8. Baik J., Kriecherbauer T., McLaughlin K. T.-R., Miller P. D. Discrete orthogonal polynomials. Asymptotics
and applications. Princeton, Princeton Univ. Press, 2007,
184 p.
9. Ou C., Wong R. The Riemann–Hilbert approach to
global asymptotics of discrete orthogonal polynomials
with infinite nodes. Analysis and Applications, 2010,
vol. 8, pp. 247–286.
10. Ferreira C., L ´ opez J. L., Sinus´
ia E. P. Asymptotic relations between the Hahn-type polynomials and Meixner–
Pollaczek, Jacobi, Meixner and Krawtchouk polynomials.
Journal of Computational and Applied Mathematics,
2008, vol. 217, pp. 88–109.
11. Szego G. Orthogonal Polynomials. AMS Colloq. Publ,
1939, vol. 23, 154 p.
12. Bari N. K. Generalization of inequalities of
S. N. Bernshtein and A. A. Markov. Izv. AS USSR. Ser.
matem., 1954, vol. 18, no. 2, pp. 159–176 (in Russian).
13. Konyagin S. V. V. A. Markov’s inequality for
polynomials in the metric of L [O neravenstve
V. A. Markova dlia mnogochlenov v metrike L]. Trudy
Matematicheskogo Instituta im. V. A. Steklova, 1980,
no. 145, pp. 117–125 (in Russian).
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