On the uniform convergence of the Fourier series by the system of polynomials generated by the system of Laguerre polynomials

R. M. Gadzhimirzaev

Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, Tome 20 (2020) no. 4, pp. 416-423.

Résumé

Let $w(x)$ be the Laguerre weight function, $1\le p\infty$, and $L^p_w$ be the space of functions $f$, $p$-th power of which is integrable with the weight function $w(x)$ on the non-negative axis. For a given positive integer $r$, let denote by $W^r_{L^p_w}$ the Sobolev space, which consists of $r-1$ times continuously differentiable functions $f$, for which the $(r-1)$-st derivative is absolutely continuous on an arbitrary segment $[a, b]$ of non-negative axis, and the $r$-th derivative belongs to the space $L^p_w$. In the case when $p=2$ we introduce in the space $W^r_{L^2_w}$ an inner product of Sobolev-type, which makes it a Hilbert space. Further, by $l_{r,n}^\alpha(x)$, where $n = r, r + 1, \dots$, we denote the polynomials generated by the classical Laguerre polynomials. These polynomials together with functions $l_{r,n}^\alpha(x)=\frac{x^n}{n!}$, where $n=0, 1, r-1$, form a complete and orthonormal system in the space $W^r_{L^2_w}$. In this paper, the problem of uniform convergence on any segment $[0,A]$ of the Fourier series by this system of polynomials to functions from the Sobolev space $W^r_{L^p_w}$ is considered. Earlier, uniform convergence was established for the case $p=2$. In this paper, it is proved that uniform convergence of the Fourier series takes place for $p>2$ and does not occur for $1\le p2$. The proof of convergence is based on the fact that $W^r_{L^p_w}\subset W^r_{L^2_w}$ for $p>2$. The divergence of the Fourier series by the example of the function $e^{cx}$ using the asymptotic behavior of the Laguerre polynomials is established.

Bibliographie

[1] Pérez T. E., Piñar M. A., Xu Y., “Weighted Sobolev orthogonal polynomials on the unit ball”, J. Approx. Theory, 171 (2013), 84–104 | DOI | MR

[2] Marcellán F., Xu Y., “On Sobolev orthogonal polynomials”, Expositiones Math., 33:3 (2015), 308–352 | DOI | MR | Zbl

[3] Fernández L., Marcellán F., Pérez T. E., Piñar M. A., Xu Y., “Sobolev orthogonal polynomials on product domains”, J. Comput. and Appl. Math., 284 (2015), 202–215 | DOI | MR | Zbl

[4] Delgado A. M., Fernandez L., Lubinsky D. S., Pérez T. E., Piñar M. A., “Sobolev orthogonal polynomials on the unit ball via outward normal derivatives”, J. Math. Anal. and Appl., 440:2 (2016), 716–740 | DOI | MR | Zbl

[5] Sharapudinov I. I., “Sobolev-orthogonal systems of functions associated with an orthogonal system”, Izv. Math., 82:1 (2018), 212–244 | DOI | DOI | MR | Zbl

[6] Magomed-Kasumov M. G., “A Sobolev orthogonal system of functions generated by a Walsh system”, Math Notes, 105:3–4 (2019), 543–549 | DOI | DOI | MR | Zbl

[7] Sharapudinov I. I., Gadzhieva Z. D., Gadzhimirzaev R. M., “Systems of functions orthogonal with respect to scalar products of Sobolev type with discrete masses generated by classical orthogonal systems”, Daghestan Electronic Mathematical Reports, 2016, no. 6, 31–60 (in Russian) | DOI

[8] Sharapudinov I. I., Magomed-Kasumov M. G., “On representation of a solution to the Cauchy problem by a Fourier series in Sobolev-orthogonal polynomials generated by Laguerre polynomials”, Diff. Equat., 54:1 (2018), 49–66 | DOI | DOI | MR | MR | Zbl

[9] Gadzhimirzaev R. M., “Recurrence relations for polynomials orthonormal on Sobolev, generated by Laguerre polynomials”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 18:1 (2018), 17–24 (in Russian) | DOI | MR | Zbl

[10] Szego G., Orthogonal Polynomials, AMS Colloq. Publ., 23, 1939, 440 pp. | MR