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

Abstract

The paper is devoted to the study of the approximation properties of Fourier sums in terms of the modified Meixner polynomials m ^α _n,N (x), n = 0,1,..., which generate, for α > -1, an orthonormal system on the grid Ω_δ = {0, δ, 2δ,...} with weight

$${\rho _N}(x) = {e^{ - x}}\frac{{\Gamma (Nx + \alpha + 1)}}{{\Gamma (Nx + 1)}}{(1 - {e^{ - \delta }})^{\alpha + 1}},\;\;\;\;\text{where}\;\;\delta = \frac{1}{N},\;N \geq 1.$$

The main attention is paid to the derivation of a pointwise estimate for the Lebesgue function λ ^α _n,N (x) of Fourier sums in terms of the modified Meixner polynomials for x ∈ [θ_n/2, ∞) and θ_n = 4n + 2α + 2.