Approximative properties of special series in Meixner polynomials

In this article the new special series in the modified Meixner polynomials Mαn,N(x)=Mαn(Nx) are constructed. For α>-1, these polynomials constitute an orthogonal system with a weight-function ρ(Nx) on a uniform grid Ωδ={0,δ,2δ,…}, where δ=1/N, N>0. Special series in Meixner polynomials Mαn,N(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 Ωδ and its finite differences Δνδf. 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,δ,…,(r-1)δ.