Pointwise convergence of Fourier series. Carleson’s theorem

Abstract

In this project we study the pointwise convergence of Fourier series. Our main goal is the proof of Carleson’s theorem, which states, roughly speaking, that the Fourier series of any periodic and square integrable function converges to the function almost everywhere. The proof will be based on that presented in the article Pointwise convergence of Fourier series, by Charles Fefferman (see [4]). The structure and the notations will be similar to those of the article, but the proofs and the concepts will be explained in much more detail. In Chapter 1 we revise the history of Fourier series until the proof of Carleson’s theorem by Fefferman [1] [3]. We also explain the structure of the project in detail. In Chapter 2 we relate the convergence problem of Fourier series to the boundedness of an operator. In the third chapter, using dyadic grids, we decompose the mentioned operator in simpler operators. In the fourth chapter we handle some technicalities concerning the dyadic grids chosen. In Chapter 5 we give the intuition for the proof of Carleson’s theorem and we specify the main goal. In the sixth chapter the main lemmas of the project are proved, which give as a consequence the proof of Carleson’s theorem in the seventh chapter.