Approximative properties of special series in Meixner polynomials

R. M. Gadzhimirzaev

Vladikavkazskij matematičeskij žurnal, Tome 20 (2018) no. 3, pp. 21-36

Résumé

In this article the new special series in the modified Meixner polynomials $M_{n,N}^\alpha(x)=M_n^\alpha(Nx)$ are constructed. For $\alpha>-1$, these polynomials constitute an orthogonal system with a weight-function $\rho(Nx)$ on a uniform grid $\Omega_{\delta}=\{0, \delta, 2\delta, \ldots\}$, where $\delta=1/N$, $N>0$. Special series in Meixner polynomials $M_{n,N}^\alpha(x)$ appeared as a natural (and alternative to Fourier–Meixner series) apparatus for the simultaneous approximation of a discrete function $f$ given on a uniform grid $\Omega_\delta$ and its finite differences $\Delta^\nu_\delta f$. The main attention is paid to the study of the approximative properties of the partial sums of the series under consideration. In particular, a pointwise estimate for the Lebesgue function of mentioned partial sums is obtained. It should also be noted that new special series, unlike Fourier–Meixner series, have the property that their partial sums coincide with the values of the original function in the points $0, \delta, \ldots, (r-1)\delta$.

Bibliographie

[1] Nikiforov A. F., Suslov S. K., Uvarov V. B., Classical Orthogonal Polynomials of a Discrete Variable, Nauka, M., 1985, 216 pp. (in Russian) | MR

[2] Sharapudinov I. I., Polynomials Orthogonal on Grids, Izd-vo DGPU, Mahachkala, 1997, 255 pp. (in Russian)

[3] Bateman H., Erdelyi A., Higher Transcendental Functions, v. 2, McGraw-Hill Book Company, Inc., N. Y.–Toronto–London, 1953 | MR | MR

[4] Gadzhieva Z. D., Mixed Series by Meixner Polynomials, PhD thesis, Saratov State. Univ., Saratov, 2004 (in Russian)

[5] Sharapudinov I. I., “Mixed Series by Classical Orthogonal Polynomials”, Dagestan Electronic Mathematical Reports, 2015, no. 3, 1–254 (in Russian) | DOI | MR

[6] Sharapudinov I. I., “Some Special Series by General Laguerre Polynomials and Fourier Series by Laguerre Polynomials, Orthogonal in Sobolev Sense”, Dagestan Electronic Mathematical Reports, 2015, no. 4, 32–74 (in Russian) | DOI

[7] Gadzhimirzaev R. M., “The Fourier Series of the Meixner Polynomials Orthogonal with Respect to the Sobolev-Type Inner Product”, Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform., 16:4 (2016), 388–395 (in Russian) | DOI | MR

[8] Gadzhimirzaev R. M., “The Fourier Series of the Meixner Polynomials Orthogonal with Respect to the Sobolev-Type Inner Product”, XVIII International Saratov Winter School «Modern Problems of Function Theory and Their Applications» (2016), 102–104 (in Russian)