On the rate of uniform approximation by truncated fourier inversion integrals

Abstract

It is well-known that any 2π-periodic continuous function can be approximated by the n-th partial sums of its Fourier series with an order of approximation which is inferior to that by the trigonometric polynomials of best approximation by the factor log n . In the case of approximation of continuous, integrable functions on the real line IR by their truncated Fourier inversion integrals an analogous result is generally accepted to be true. Here we give a proof of such a result in which the singular integrals of de La Vallée Poussin play the role of the elements of best approximation