Izvestiya of Saratov University.

Mathematics. Mechanics. Informatics

ISSN 1816-9791 (Print)
ISSN 2541-9005 (Online)


For citation:

Sharapudinov I. I., Gadzhieva Z. D. Sobolev Orthogonal Polynomials Generated by Meixner Polynomials. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2016, vol. 16, iss. 3, pp. 310-321. DOI: 10.18500/1816-9791-2016-16-3-310-321, EDN: WMIIIB

This is an open access article distributed under the terms of Creative Commons Attribution 4.0 International License (CC-BY 4.0).
Published online: 
14.09.2016
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Russian
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517.587
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WMIIIB

Sobolev Orthogonal Polynomials Generated by Meixner Polynomials

Autors: 
Sharapudinov Idris Idrisovich, Daghestan Scientific Centre of Russian Academy of Sciences
Gadzhieva Zul?ya Dzhamaldinovna, Daghestan Scientific Centre of Russian Academy of Sciences
Abstract: 

The problem of constructing Sobolev orthogonal polynomials mα r,n(x, q) (n = 0, 1, . . .), generated by classical Meixner’s polynomials is considered.

References: 
  1. Iserles A., Koch P. E., Norsett S. P., SanzSerna J. M. On polynomials orthogonal with respect to certain Sobolev inner products // J. Approx. Theory. 1991. Vol. 65, iss. 2. P. 151–175. DOI: https://doi.org/10.1016/0021-9045(91)90100-O.
  2. Marcellan F., Alfaro M., Rezola M. L. Orthogonal polynomials on Sobolev spaces: old and new directions // J. Comput. Appl. Math. 1993. Vol. 48, iss. 1–2. P. 113–131. DOI: https://doi.org/10.1016/0377-0427(93)90318-6.
  3. Meijer H. G. Laguerre polynimials generalized to a certain discrete Sobolev inner product space // J. Approx. Theory. 1993. Vol. 73, iss. 1. P. 1–16. DOI: https://doi.org/10.1006/jath.1993.1029.
  4. Kwon K. H., Littlejohn L. L. The orthogonality of the Laguerre polynomials {L (−k) n (x)} for positive integers k // Ann. Numer. Anal. 1995. Vol. 2. P. 289– 303.
  5. Kwon K. H., Littlejohn L. L. Sobolev orthogonal polynomials and second-order differential equations // Ann. Numer. Anal. 1998. Vol. 28. P. 547– 594.
  6. Marcellan F., Yuan Xu On sobolev orthogonal polynomials. arXiv: 6249v1 [math.C.A]. 25 Mar 2014. P. 1–40.
  7. Sharapudinov I. I. Approximation of discrete functions and Chebyshev polynomials orthogonal on the uniform grid. Math. Notes, 2000, 67, iss. 3, pp. 389–397. DOI: https://doi.org/10.1007/BF02676675.
  8. Sharapudinov I. I. Approximation of functions of variable smoothness by Fourier – Legendre sums. Sb. Math., 2000, vol. 191, no. 5, pp. 759–777. DOI: https://doi.org/10.1070/SM2000v191n05ABEH000480.
  9. Sharapudinov I. I. Approximation properties of the operators Yn+2r(f) and of their discrete analogs. Math. Notes, 2002, vol. 72, iss. 5, pp. 705–732. DOI: https://doi.org/10.1023/A:1021421425474.
  10. Sharapudinov I. I. Mixed series in ultraspherical polynomials and their approximation properties. Sb. Math., 2003, vol. 194, no. 3, pp. 423–456. DOI: https://doi.org/10.1070/SM2003v194n03ABEH000723.
  11. Sharapudinov I. I. Smeshannye rjady po ortogonal’nym polinomam [Mixed series of orthogonal polynomials]. Makhachkala, Dagestan Scientific Center RAS, 2004, 176 p. (in Russian).
  12. Sharapudinov I. I. Mixed series of Chebyshev polynomials orthogonal on a uniform grid. Math. Notes, 2005, vol. 78, iss. 3, pp. 403–423. DOI: https://doi.org/10.1007/s11006-005-0139-3.
  13. Sharapudinov I. I. Approximation properties of mixed series in terms of Legendre polynomials on the classes Wr . Sb. Math, 2006, vol. 197, no. 3, pp. 433–452. DOI: https://doi.org/10.1070/SM2006v197n03ABEH003765.
  14. Sharapudinov T. I. Approximative properties of mixed series by Chebyshev polynomials, orthogonal on an uniform net. Vestnik Dagestan. nauch. centra RAN, 2007, vol. 29, pp. 12–23 (in Russian).
  15. Sharapudinov I. I. Approximation properties of the Vallee – Poussin means of partial sums of a mixed series of Legendre polynomials. Math. Notes, 2008, vol. 84, iss 3–4, pp. 417–434. DOI: https://doi.org/10.1134/S0001434608090125.
  16. Sharapudinov I. I., Muratova G. N. Same Properties r-fold Integration Series on Fourier – Haar System. Izv. Saratov Univ. (N.S), Ser. Math. Mech. Inform., 2009, vol. 9, iss. 1, pp. 68–76 (in Russian).
  17. Sharapudinov I. I., Sharapudinov T. I. Mixed series of Jacobi and Chebyshev polynomials and their discretization. Math. Notes, vol. 88, iss. 1–2, pp. 112– 139. DOI: https://doi.org/10.1134/S0001434610070114.
  18. Sharapudinov I. I. Sistemy funktsii, ortogonal’nykh po Sobolevu, porozhdennye ortogonal’nymi funktsiiami [System functions orthogonal on Sobolev generated orthogonal functions]. Sovremennye problemy teorii funktsii i ikh prilozheniia : materialy 18-i mezhdunar. Sarat. zimnei shkoly [Modern problems of function theory and their applications : Proc. 18th Intern. Sarat. Winter School]. Saratov, OOO Izd-vo "Nauchnaia kniga", 2016, pp. 329–332.
  19. Trefethen L. N. Spectral methods in Matlab. Fhiladelphia : SIAM, 2000. 160 p.
  20. Trefethen L. N. Finite difference and spectral methods for ordinary and partial differential equation. Cornell University, 1996. 325 p.
  21. Magomed-Gasimov M. G. Priblizhennoe reshenie obyknovennykh differentsial’nykh uravnenii s ispol’zovaniem smeshannykh riadov po sisteme Khaara [Approximate solution of ordinary differential equations with the use of mixed series on the Haar system]. Sovremennye problemy teorii funktsii i ikh prilozheniia : materialy 18-i mezhdunar. Sarat. zimnei shkoly [Modern problems of function theory and their applications : Proc. 18th Intern. Sarat. Winter School]. Saratov, OOO Izd-vo "Nauchnaia kniga", 2016, pp. 176–178.
  22. Sharapudinov I. I. Mnogochleny, ortogonal’nye na dickretnyh setkah [Polynomials orthogonal on discrete grids]. Mahachkala, Izd-vo Dag. Gos. Ped. Un-ta, 1997, 252 p. (in Russian).
  23. Gasper G. Positivity and special function // Theory and Application of Special Functions : Proceedings of an Advanced Seminar Sponsored by the Mathematics Research Center, the University of WisconsinMadison, March 31-April 2, 1975 / ed. R. Askey. N. Y. ; San Francisco ; L. : Academic Press, Inc., 1975. P. 375–433.
Received: 
23.04.2016
Accepted: 
28.08.2016
Published: 
30.09.2016