Rogosinsky--Bernstein Polynomial Method of Summation of Trigonometric Fourier Series

R. M. Trigub

Matematičeskie zametki, Tome 111 (2022) no. 4, pp. 592-605

Résumé

General Rogosinsky–Bernstein linear polynomial means $R_n(f)$ of Fourier series are introduced and three convergence criteria as $n\to\infty$ are obtained: for convergence in the space $C$ of continuous periodic functions and for convergence almost everywhere with two guaranteed sets (Lebesgue points and $d$-points). For smooth functions, the rate of convergence in norm of $R_n(f)$, as well as of their interpolation analogues, is also studied. For approximation of functions in $C^r$, the asymptotics is found along with the rate of decrease of the remainder term.

Détail

Keywords: series and Fourier transforms, Hardy's inequality, Riesz means, modulus of smoothness, linearized modulus of smoothness, Jackson's theorem, conjugate function, entire functions of exponential type, comparison principle, Marcinkiewicz's inequality and discretization.
Mots-clés : Lebesgue points ($l$-points) and $d$-points, Vallée-Poussin polynomial