Divergent Fourier Series: Numerical Experiments

An open problem in the theory of Fourier series is whether there are summable functions such that the partial sums S_n(f, x) diverge faster than log log n, almost everywhere in x. For a class of particularly 'bad' functions Kahane proved that the rate of divergence is faster than o0og log n). We give here a probabilistic interpretation of the Kahane result, which shows that the record values of the sums S_n(f;x) should behave essentially as the record values of a sequence of independent identically distributed random variables, for which we deduce the divergence rate log log n. Numerical computation is in good agreement with the prediction. One can argue that the Kahane examples are in some sense 'optimal', and conclude that, under this assumption, ~(log log n) is the highest possible rate for divergence almost everywhere of the Fourier partial sums for summable functions.