6 - Fourier series and randomness

Summary

In this chapter we apply some probabilistic tools developed in Chapter 5 to Fourier series. Our main goal here is to be able to answer, using probabilistic arguments, questions which in and of themselves are deterministic. For example, in the following section we address the problem of a.e. convergence of the partial sums of the Fourier series of L¹(T) functions. By means of a combination of deterministic and probabilistic ideas we shall show that L¹ functions exist for which the partial sums of their Fourier series diverge almost everywhere. Section 6.2 is more probabilistic from the start, as it deals with the problem of deciding the convergence of Fourier series with random and independent coefficients. This is a classic topic, going back to the work of Paley and Zygmund in the 1930s; they introduced randomness into Fourier series as a tool for answering deterministic questions.

Fourier series on L¹(T): pointwise questions

This section is devoted to the question of the almost everywhere convergence of Fourier series for L¹(T) functions. From the previous chapters we know that this is not an elementary question, since the partial sums S_N are given by convolutions, with the Dirichlet kernels D_N, which do not form an approximate identity. Note also that we have already encountered an a.e. convergence question that is not associated with an approximate identity, namely, that for the Hilbert transform; see Proposition 3.18.