Introduction

In this chapter we study Fourier series. We use Fourier series to represent or approximate functions defined on a finite interval. In this sense Fourier series are similar to polynomials or power series. However, Fourier series are in other ways both better and more general. Fourier series are one example of a closed infinite orthonormal system in an inner product space. They are an application of the general theory presented in the previous chapter. Fourier series also have various specific properties of their own and we shall study some of them. Fourier series were first defined, not too surprisingly, by Jean Baptiste Joseph Fourier (1768-1830) about 200 years ago. That they are an “old” topic does not detract from their importance. Fourier was a mathematician and an engineer who developed these series in order to solve certain problems in partial differential equations. In the last section of this chapter, we present one application of this kind. (Fourier was a participant in the French Revolution. He was with Napoleon in the Egyptian campaign of 1798 and was considered one of the “savants” who accompanied Napoleon in this campaign. He was, for a time, governor of lower Egypt, and later Prefect of Is&re (at Grenoble).)