Strong Approximation and the Behaviour of Fourier Series

Abstract

Since 1963, a work of G. Alexits and D. Kralik [1], the so called strong approximation of Fourier series has developed very rapidly. The difference between ordinary and strong approximation is that the latter examines means of type $$ {\left\{ {\sum\limits_k {{t_{{nk}}}{{\left| {{s_k} - f} \right|}^p}} } \right\}^{{1/p}}}\,\left( {{t_{{nk}}} \geqslant 0,\,p > 0} \right) $$ where sk(x) =sk(f;x) is the k-th partial sum of the Fourier series of the 2π periodic function f, or of even more general types. In this work we apply mostly known strong approximation results for proving theorems concerning the behaviour of Fourier series. For our purposes the case p = 1 will be sufficient, therefore, the cited results are presented only in this particular case. In the first two paragraphs we prove two approximation theorems, and in the last two ones we estimate σ n α (f) - f, α >-1/2, for “almost all n”.