On the divergence of Fourier series

Abstract

1. By a well known theorem of Kolmogoroff there is a function whose Fourier series diverges almost everywhere. Actually, Kolmogoroff's proof was later generalized so that the Fourier series diverged everywhere [2, p. 175]; but we shall be concerned only with the almost everywhere theorem here. The proof involves rather severe restrictions on the orders of the partial sums which are shown to diverge. The following problem connected with this theorem was suggested to the author by Professor A. Zygmund. Given a sequence I p, } of positive integers increasing to oo, can an integrable function f on (0, 2r) be constructed so that the partial sums of its Fourier series of order p, diverge almost everywhere ? The object of our paper is to give an affirmative answer to this question. Let sp(x; f) denote the pth partial sum of the Fourier series of the function f at the point x.