Geodesic
Polynomials orthogonal in the Sobolev sense, generated by Chebyshev polynomials orthogornal on a mesh
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2017), pp. 67-79.

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We consider the problem of constructing polynomials, orthogonal in Sobolev sense on the finite uniform mesh and associated with classical Chebyshev polynomials of discrete variable. We have found an explicit expression of these polynomials by classical Chebyshev polynomials. Also we have obtained an expansion of new polynomials by generalized powers of Newton type. We obtain expressions for the deviation of a discrete function and its finite differences from respectively partial sums of its Fourier series on the new system of polynomials and their finite differences.
Mots-clés : polynomials orthogonal in Sobolev sence
Keywords: Chebyshev polynomials orthogonal on the mesh, approximation of discrete functions, mixed series of Chebyshev polynomials orthogonal on a uniform mesh. 
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     author = {I. I. Sharapudinov and T. I. Sharapudinov},
     title = {Polynomials orthogonal in the {Sobolev} sense, generated by {Chebyshev} polynomials orthogornal on a mesh},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
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I. I. Sharapudinov; T. I. Sharapudinov. Polynomials orthogonal in the Sobolev sense, generated by Chebyshev polynomials orthogornal on a mesh. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2017), pp. 67-79. http://geodesic.mathdoc.fr/item/IVM_2017_8_a6/

[1] Iserles A., Koch P. E., Norsett S. P., Sanz-Serna J. M., “On polynomials orthogonal with respect to certain Sobolev inner products”, J. Approx. Theory, 65 (1991), 151–175 | DOI | MR | Zbl

[2] Marcellan F., Alfaro M., Rezola M. L., “Orthogonal polynomials on Sobolev spaces: old and new directions”, Journal of Computational and Applied Mathematics, 48 (1993), 113–131 | DOI | MR | Zbl

[3] Meijer H. G., “Laguerre polynomials generalized to a certain discrete Sobolev inner product space”, J. Approx. Theory, 73 (1993), 1–16 | DOI | MR | Zbl

[4] Kwon K. H., Littlejohn L. L., “The orthogonality of the Laguerre polynomials $\{L_n^{(-k)}(x)\}$ for positive integers $k$”, Ann. Numer. Anal., 2 (1995), 289–303 | MR | Zbl

[5] Kwon K. H., Littlejohn L. L., “Sobolev orthogonal polynomials and second-order differential equations”, Rocky Mountain J. Math., 28 (1998), 547–594 | DOI | MR | Zbl

[6] Marcellan F., Yuan Xu, On Sobolev orthogonal polynomials, 2014, 40 pp., arXiv: 1403.6249v1 [math.CA] | MR

[7] Sharapudinov I. I., “Smeshannye ryady po polinomam Chebysheva, ortogonalnym na ravnomernoi setke”, Matem. zametki, 78:3 (2005), 442–465 | DOI | Zbl

[8] Sharapudinov I. I., Sharapudinov T. I., “Smeshannye ryady po polinomam Yakobi i Chebysheva i ikh diskretizatsiya”, Matem. zametki, 88:1 (2010), 116–147 | DOI | Zbl

[9] Trefethen L. N., Spectral methods in Matlab, SIAM, Philadelphia, 2000 | MR | Zbl

[10] Trefethen L. N., Finite difference and spectral methods for ordinary and partial differential equation, Cornell University, 1996

[11] Magomed-Kasumov M. G., “Priblizhennoe reshenie obyknovennykh differentsialnykh uravnenii s ispolzovaniem smeshannykh ryadov po sisteme Khaara”, Sovremennye problemy teorii funktsii i ikh prilozheniya, Materialy 18-i mezhdunarodnoi Saratovskoi zimnei shkoly (27 yanvarya–3 fevralya 2016), 2016, 176–178

[12] Chebyshev P. L., “Ob interpolirovanii velichin ravnootstoyaschikh $(1875)$”, Poln. sobr. soch., v. 2, Izd. AN SSSR, M., 1947, 66–87 | MR

[13] Sharapudinov I. I., Mnogochleny, ortogonalnye na diskretnykh setkakh, Izd-vo Dagestansk. gos. ped. un-ta, Makhachkala, 1997

[14] Sharapudinov I. I., “Ob asimptotike mnogochlenov Chebysheva, ortogonalnykh na konechnoi sisteme tochek”, Vestn. MGU. Ser. 1, 1 (1992), 29–35

[15] Sharapudinov T. I., “Approksimativnye svoistva smeshannykh ryadov po polinomam Chebysheva, ortogonalnym na ravnomernoi setke”, Vestn. Dagestansk. nauchn. tsentra RAN, 29 (2007), 12–23 | Zbl