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
Full text:
(downloads: 204)
Language:
Russian
Heading:
UDC:
517.587
EDN:
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.
Key words:
References:
- 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.
- 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.
- 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.
- 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.
- Kwon K. H., Littlejohn L. L. Sobolev orthogonal polynomials and second-order differential equations // Ann. Numer. Anal. 1998. Vol. 28. P. 547– 594.
- Marcellan F., Yuan Xu On sobolev orthogonal polynomials. arXiv: 6249v1 [math.C.A]. 25 Mar 2014. P. 1–40.
- 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.
- 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.
- 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.
- 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.
- Sharapudinov I. I. Smeshannye rjady po ortogonal’nym polinomam [Mixed series of orthogonal polynomials]. Makhachkala, Dagestan Scientific Center RAS, 2004, 176 p. (in Russian).
- 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.
- 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.
- 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).
- 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.
- 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).
- 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.
- 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.
- Trefethen L. N. Spectral methods in Matlab. Fhiladelphia : SIAM, 2000. 160 p.
- Trefethen L. N. Finite difference and spectral methods for ordinary and partial differential equation. Cornell University, 1996. 325 p.
- 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.
- 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).
- 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
- 1190 reads