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DOI: 10.23671/VNC.2018.3.17961

Approximative Properties of Special Series in Meixner Polynomials

Gadzhimirzaev R. M.
Vladikakazian 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
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