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

R. M. Gadzhimirzaev

Matematičeskie zametki, Tome 106 (2019) no. 4, pp. 519-530

Résumé

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 $\theta_n=4n+2\alpha+2$.

Détail

Keywords: Meixner polynomials, Fourier series
Mots-clés : Lebesgue function.