Some special series by general Laguerre polynomials and Fourier series by Laguerre polynomials, orthogonal in Sobolev sense

I. I. Sharapudinov

Daghestan Electronic Mathematical Reports, Tome 4 (2015), pp. 31-73.

Résumé

Some special series on Laguerre polynomials are considered and their approximative properties are investigated. In particular, the upper estimate for the Lebesgue function of introduced special series by Laguerre polynomials is obtained. The polynomials $ l_{r,k}^\alpha(x)$ $(k=0,1,\ldots)$ , generated by classical orthogonal Laguerre polynomials $L_k (x) (k = 0; 1; \ldots)$ and orthonormal with respect to the Sobolev-type inner product \begin{equation*} ,g>=\sum_{\nu=0}^{r-1}f^{(\nu)}(0)g^{(\nu)}(0)+\int_0^\infty f^{(r)}(t)g^{(r)}(t)t^\alpha e^{-t}dt, \end{equation*} are introduced and investigated. The representations of these polynomials in the form of certain expressions containing Laguerre polynomials $L_n^{\alpha-r}(x)$ are\linebreak obtained. An explicit form of the polynomials $ l^\alpha_{r,k+r}(x)$ which is an expansion in powers of $x^{r+l}$ with $l=0,\ldots,k$ is established. These results can be used in the study of asymptotic properties of polynomials $l^\alpha_{r,k}(x)$ when $k\to\infty$ and in the study of approximative properties of partial sums of Fourier series by these polynomials. It is shown that Fourier series by polynomials $l^\alpha_{r,k}(x)$ coincides with the mixed series by Laguerre polynomials introduced and studied earlier by the author. Besides it is shown if $\alpha=0$, then mixed series on Laguerre polynomials and, as a corollary, the Fourier series by polynomials $l^0_{r,k}(x)$ represents the particular cases of special series, introduced in present paper.

Détail

Keywords: Laguerre polynomials, mixed series on Laguerre polynomials, special series, Laplas transform, Sobolev orthogonal polynomials, Lebesgue inequality.

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., 2 (1995), 289–303 | MR | Zbl

[2] 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

[3] 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, North-Holland | DOI | MR | Zbl

[4] 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

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

[6] Marcellan F., Yuan Xu, On Sobolev orthogonal polynomials, 25 Mar 2014, 40 pp., arXiv: 1403.6249v1 [math.CA]

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

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

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

[10] 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

[11] Sharapudinov I.I., Smeshannye ryady po ortogonalnym polinomam, Izdatelstvo Dagestanskogo nauchnogo tsentra, Makhachkala, 2004, 176. pp.

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

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

[14] 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

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

[16] Askey R., Wainger S., “Mean convergence of expansions in Laguerre and Hermite series”, Amer. J. Mathem., 87 (1965), 698–708 | MR

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

[18] Krylov V.I., Skoblya N.S., Metody priblizhennogo preobrazovaniya Fure i obrascheniya preobrazovaniya Laplasa, Nauka, Moskva, 1974