Asymptotic properties of polynomials, orthogonal in Sobolev sence and associated with the Jacobi polynomials

I. I. Sharapudinov

Daghestan Electronic Mathematical Reports, Tome 6 (2016), pp. 1-24.

Résumé

We consider polynomials $p_{r,n}^{\alpha,\beta}(x)$ $(n=0,1,\ldots)$, generated by classical Jacobi polynomials $p_{n}^{\alpha,\beta}(x)$ and forming orthonormal system with respect to Sobolev-type inner product \begin{equation*} ,g>=\sum_{\nu=0}^{r-1}f^{(\nu)}(-1)g^{(\nu)}(-1)+\int_{-1}^{1}f^{(r)}(t)g^{(r)}(t)\rho(t) dt, \end{equation*} where $\rho(x)=(1-x)^\alpha(1+x)^\beta$ – Jacobi weight function. The explicit \linebreak representations for polynomials $p_{r,n}^{\alpha,\beta}(x)$ are obtained and using these ones the asymptotic properties of polynomials $p_{r,n}^{\alpha,\beta}(x)$ are investigated.

Détail

Keywords: orthogonal polynomials, Sobolev orthogonal polynomials, Jacobi polynomials, Chebyshev polynomials of the first kind, Legendre polynomials.

Bibliographie

[1] Kwon K.H., Littlejohn L.L., “The orthogonality of the Laguerre polynomials $\{L_n^{(-k)}(x)\}$ for positive integers $k$”, Ann. Numer. Anal., 1995, no. 2, 289–303 | MR | Zbl

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

[3] Marcellan F., Alfaro M. and 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

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

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

[6] Lopez G. Marcellan F. Vanassche W., “Relative Asymptotics for Polynomials Orthogonal with Respect to a Discrete Sobolev Inner-Product”, Constructive Approximation, 11:1 (1995), 107–137 | DOI | MR | Zbl

[7] Marcellan F., Yuan Xu, “On Sobolev orthogonal polynomials”, Expositiones Mathematicae, 33:3 (2015), 308–352 | DOI | MR | Zbl

[8] Sharapudinov I.I., “Sistemy funktsii, ortogonalnye po Sobolevu, porozhdennye ortogonalnymi funktsiyami”, Sovremennye problemy teorii funktsii i ikh prilozheniya, Materialy 18-i mezhdunarodnoi Saratovskoi zimnei shkoly, OOO «Izdatelstvo «Nauchnaya kniga», Saratov, 2016, 329–332

[9] Gonchar A.A., “O skhodimosti approksimatsii Pade dlya nekotorykh klassov meromorfnykh funktsii”, Matematicheskii sbornik, 97(139):4(8) (1975), 607–629 | MR

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

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

[12] Solodovnikov V.V., Dmitriev A.N., Egupov N.D., Spektralnye metody rascheta i proektirovaniya sistem upravleniya, Mashinostroenie, Moskva, 1986 | MR

[13] Pashkovskii S., Vychislitelnye primeneniya mnogochlenov i ryadov Chebysheva, Nauka, Moskva, 1983 | MR

[14] Arushanyan O.B., Volchenskova N.I., Zaletkin S.F., “Priblizhennoe reshenie obyknovennykh differentsialnykh uravnenii s ispolzovaniem ryadov Chebysheva”, Sib. elektron. matem. izv., 7 (1983), 122–131 | MR

[15] Arushanyan O.B., Volchenskova N.I., Zaletkin S.F., “Metod resheniya zadachi Koshi dlya obyknovennykh differentsialnykh uravnenii s ispolzovaniem ryadov Chebysheva”, Vych. met. programmirovaniya, 14:2 (2013), 203–214 | MR

[16] Arushanyan O.B., Volchenskova N.I., Zaletkin S.F., “Primenenie ryadov Chebysheva dlya integrirovaniya obyknovennykh differentsialnykh uravnenii”, Sib. elektron. matem. izv., 11 (2014), 517–531 | MR | Zbl

[17] Lukomskii D.S., Terekhin P.A., “Primenenie sistemy Khaara k chislennomu resheniyu zadachi Koshi dlya lineinogo differentsialnogo uravneniya pervogo poryadka”, Sovremennye problemy teorii funktsii i ikh prilozheniya, Materialy 18-i mezhdunarodnoi Saratovskoi zimnei shkoly, OOO «Izdatelstvo «Nauchnaya kniga», Saratov, 2016, 171–173

[18] Magomed-Kasumov M.G., “Priblizhennoe reshenie obyknovennykh differentsialnykh uravnenii s ispolzovaniem smeshannykh ryadov po sisteme Khaara”, Sovremennye problemy teorii funktsii i ikh prilozheniya (Saratov: OOO «Izdatelstvo «Nauchnaya kniga»), Materialy 18-i mezhdunarodnoi Saratovskoi zimnei shkoly, 2016, 176–178

[19] Sharapudinov I.I., “Priblizhenie funktsii s peremennoi gladkostyu summami Fure–Lezhandra”, Matematicheskii sbornik, 191:5 (2000), 143–160 | DOI | MR | Zbl

[20] Sharapudinov I.I., “Approksimativnye svoistva operatorov ${\cal Y}_{n+2r}(f)$ i ikh diskretnykh analogov”, Matematicheskie zametki, 72:5 (2002), 765–795 | DOI | MR | Zbl

[21] Sharapudinov I.I., Smeshannye ryady po ortogonalnym polinomam, Izdatelstvo Dagestanskogo nauchnogo tsentra, Makhachkala, 2004

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

[23] Sharapudinov I.I., “Approksimativnye svoistva smeshannykh ryadov po polinomam Lezhandra na klassakh $W^r$”, Matematicheskii sbornik, 197:3 (2006), 135–154 | DOI | MR | Zbl

[24] Sharapudinov I.I., “Approksimativnye svoistva srednikh tipa Valle-Pussena chastichnykh summ smeshannykh ryadov po polinomam Lezhandra”, Matematicheskie zametki, 84:3 (2008), 452–471 | DOI | MR | Zbl

[25] Sharapudinov I.I., “Smeshannye ryady po ultrasfericheskim polinomam i ikh approksimativnye svoistva”, Matematicheskii sbornik, 194:3 (2003), 115–148 | DOI | MR | Zbl

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

[27] Sharapudinov I.I., Muratova G.N., “Nekotorye svoistva r-kratno integrirovannykh ryadov po sisteme Khaara”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 9:1 (2009), 68–76

[28] Gonchar A.A., “O skhodimosti approksimatsii Pade dlya nekotorykh klassov meromorfnykh funktsii”, Matematicheskii sbornik, 97(139):4(8) (1975), 607–629 | MR

[29] Sege G., Ortogonalnye mnogochleny, Fizmatgiz, Moskva, 1962

[30] Fikhtengolts G.M., Kurs differentsialnogo i integralnogo ischisleniya, Fizmatgiz, Moskva, 2001, 810 pp.

[31] Sharapudinov I.I., “Nekotorye spetsialnye ryady po obschim polinomam Lagerra i ryady Fure po polinomam Lagerra, ortogonalnym po Sobolevu”, Dagestanskie elektronnye matematicheskie izvestiya, 4 (2015), 31–73

[32] Gasper G., “Positiviti and special function”, Theory and appl. Spec. Funct., ed. Richard A. Askey, 1975, 375–433 | DOI | MR | Zbl