Estimate of the Lebesgue Function of Fourier Sums in Terms of Modified Meixner Polynomials

The paper is devoted to the study of the approximation properties of Fourier sums in terms of the modified Meixner polynomials $m_{n,N}^\alpha(x)$, $n=0,1,\dots$, which generate, for $\alpha>-1$, an orthonormal system on the grid $\Omega_\delta=\{0,\delta,2\delta,\dots\}$ with weight
$$ \rho_N(x)=e^{-x}\frac{\Gamma(Nx+\alpha+1)}{\Gamma(Nx+1)} (1-e^{-\delta})^{\alpha+1},\qquad \text{where}\quad \delta=\frac{1}{N},\quad N\ge 1. $$
The main attention is paid to the derivation of a pointwise estimate for the Lebesgue function $\lambda_{n,N}^\alpha(x)$ of Fourier sums in terms of the modified Meixner polynomials for $x\in[\theta_n/2,\infty)$ and $\theta_n=4n+2\alpha+2$.