Approximation of functions defined on the grid $\{0, \delta, 2\delta, \ldots\}$ by Fourier-Meixner sums

The present paper is devoted to the study of approximation properties of partial sums of the Fourier series in the modified Meixner polynomials $M_{n,N}^\alpha(x)=M_n^\alpha(Nx)$ $(n=0, 1, \dots)$ which for $\alpha>-1$ constitute an orthogonal system on the grid $\Omega_{\delta}=\{0, \delta, 2\delta, \ldots\}$, where $\delta=\frac{1}{N}$, $N>0$ with weight $w(x)=e^{-x}\frac{\Gamma(Nx+\alpha+1)}{\Gamma(Nx+1)}$. The main attention is paid to obtaining an upper estimate for the Lebesgue function of these partial sums.