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