Approximative Properties of Special Series in Meixner Polynomials

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.23671/VNC.2018.3.17961 Approximative Properties of Special Series in Meixner Polynomials Gadzhimirzaev R. M. Vladikavkaz Mathematical Journal 2018. Vol. 20. Issue 3. Abstract: 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 \(0, \delta, \ldots, (r-1)\delta\). Keywords: Meixner polynomials, approximative properties, Fourier series, special series, Lebesgue function. Language: Russian Download the full text For citation: Gadzhimirzaev R. M. Approximative Properties of Special Series in Meixner Polynomials, . Vladikavkazskij matematicheskij zhurnal [Vladikavkaz Math. J.], vol. 20, no. 2, pp. 21-36. DOI 10.23671/VNC.2018.3.17961 + References 1. Nikiforov A. F., Suslov S. K., Uvarov V. B. Klassicheskie ortogonalnye mnogochleny diskretnoj peremennoj [Classical Orthogonal Polynomials of a Discrete Variable], Moscow, Nauka, 1985, 216 p. (in Russian). 2. Sharapudinov I. I. Mnogochleny, ortogonalnye na setkah [Polynomials Orthogonal on Grids], Mahachkala, Izd-vo DGPU, 1997, 255 p. (in Russian). 3. Bateman H., Erdelyi A. Higher Transcendental Functions , Vol. 2, N. Y.-Toronto-London, McGraw-Hill Book Company, Inc. 1953. 4. Gadzhieva Z. D. Mixed Series by Meixner Polynomials. PhD thesis , Saratov, Saratov State. Univ., 2004 (in Russian). 5. Sharapudinov I. I. Mixed Series by Classical Orthogonal Polynomials, Dagestanskie ehlektronnye matematicheskie izvestiya [Dagestan Electronic Mathematical Reports], 2015, no. 3, pp. 1-254 (in Russian). DOI: 10.31029/demr.3.1. 6. Sharapudinov I. I. Some Special Series by General Laguerre Polynomials and Fourier Series by Laguerre Polynomials, Orthogonal in Sobolev Sense, Dagestanskie ehlektronnye matematicheskie izvestiya [Dagestan Electronic Mathematical Reports], 2015, no. 4, pp. 32-74 (in Russian). DOI: 10.31029/demr.4.4. 7. Gadzhimirzaev R. M. The Fourier Series of the Meixner Polynomials Orthogonal with Respect to the Sobolev-Type Inner Product, Izv. Sarat. Un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika [Izv. Saratov Univ. (N. S.), Ser. Math. Mech. Inform.], 2016, Vol. 16, no. 4, pp. 388-395 (in Russian). DOI: 10.18500/1816-9791-2016-16-4-388-395. 8. Gadzhimirzaev R. M. The Fourier Series of the Meixner Polynomials Orthogonal with Respect to the Sobolev-Type Inner Product, Sovremennye problemy teorii funkcij i ih prilozheniya: Materialy 18-j mezhdunarodnoj Saratovskoj Zimnej Shkoly [XVIII International Saratov Winter School "Modern Problems of Function Theory and Their Applications"], 2016, p. 102-104 (in Russian). The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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