A united approach to accelerating trigonometric expansions

The Fourier series of a smooth function on a compact interval usually has slow convergence due to the fact that the periodic extension of the function has jumps at the interval endpoints. For various symmetry conditions polynomial interpolation methods have been developed for performing a boundary correction. The resulting variants of Krylov approximants are a sum of a correction polynomial and a Fourier sum of the corrected function [1–8]. In this paper, we review these methods and derive estimates in the maximum norm. We further show that derivatives of the Krylov approximants are again Krylov approximants of derivatives of the considered function. This enables us to give a unified treatment of the problem of simultaneous approximation.