Periodic Functions

Abstract

Fourier analysis has its roots in Fourier’s work on the theory of heat where he found it necessary to express any periodic function by a trigonometric series. The issues of how such expansions are to be interpreted and when they are possible are surprisingly deep and have motivated much mathematics since Fourier’s initial contribution. This chapter, an introduction to this topic, will begin with the formal definition of Fourier series of periodic functions as well as a review of the various function spaces essential to a proper study of the convergence of Fourier series. We will then proceed to study the convergence of the Fourier series expansions of functions in these functions spaces as well as the relationship of the smoothness of functions to the decay of the series coefficients.