Approximative properties of Fourier–Meixner sums

We consider the problem of approximation of discrete functions $f=f(x)$ defined on the set $\Omega_\delta= \{0,\, \delta,\, 2\delta, \,\ldots\}$, where $\delta=\frac{1}{N}$, $N>0$, using the Fourier sums 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 $\Omega_{\delta}$ with the weight function $\displaystyle w(x) = e^{-x}\frac{\Gamma(Nx+\alpha + 1)}{\Gamma(Nx + 1)}$. We study the approximative properties of partial sums of Fourier series in polynomials $M_{n, N}^\alpha(x)$, with particular attention paid to estimating their Lebesgue function.