Fourier Series

This chapter is not a mini-course on Fourier series. If it were it would aim to cover all the basics of this topic, that is, all the basic results and their proofs, or at least give the links to them. We can say that since it is not, this chapter is a concise introduction only to certain properties of Fourier series. We are not going to cover or even mention all the features and peculiarities of this machinery. We are traveling along the path that leads to the appearance of the Fourier transform not as a non-periodic extension of the Fourier series but as an indispensable tool for clarifying or solving certain periodic problems. Of course, just a scheme is outlined, some topics are included either because they are really related, or just because they are dear to the author’s taste or memory. In other words, here the reader will find either the facts and results the author dealt with or those that impressed the author in the very beginning of his mathematical biography or later on.