ISSN 1683-3414 (Print)   •   ISSN 1814-0807 (Online)

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DOI: 10.23671/VNC.2018.3.17961

# Approximative Properties of Special Series in Meixner Polynomials

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 $$0, \delta, \ldots, (r-1)\delta$$.