On the measure of approximation for some linear means of trigonometric Fourier series

Many approximation methods in C-2 pi may be generated via a certain function phi is an element of C-[0,C-1] with phi(0)=1, phi(1)=0. The function phi(j)(t)=cos(j-1/2)pi t (j is an element of N) generates the Rogosinski approximation method [N. K. Bari, ''A Treatise on Trigonometric Series,'' Vols. I, II, Pergamon Press, New York, 1964]. Our idea consists in representing phi by the orthogonal system phi(j) to extend results previously known for the Rogosinski method to arbitrary approximation methods. We illustrate this by proving two asymptotic estimates for the measure of approximation. (C) 1997 Academic Press.