Overview

In the preceding chapters, we introduced several kinds of Fourier series: the Fourier sine series, cosine series, quarter-wave sine series, quarter-wave cosine series, and the full Fourier series. These series were primarily used to represent the solution to differential equations, and their usefulness was based on two facts:

1. Each is based on an orthogonal sequence with the property that every continuous function can be represented in terms of this sequence.

2. The terms in the series represent eigenfunctions of certain simple differential operators (under various boundary conditions). This accounts for the fact that it is computationally tractable to determine a series representation of the solution to the corresponding differential equation.

In this chapter we will go deeper into the study of Fourier series. Specifically, we will consider the following questions:

1. What is the relationship among the various kinds of Fourier series?

2. How can a partial Fourier series be found and evaluated efficiently?

3. Under what conditions and in what sense can a function be represented by its Fourier series?

4. Can the Fourier series method be generalized to more complicated differential equations (including nonconstant coefficients and/or irregular geometry)?

Our discussion will justify many of the statements we made earlier in the book concerning the convergence of Fourier series. It also introduces the fast Fourier transform (FFT), which is an exciting and recent development (from the last half of the twentieth century ^[63]) in the long history of Fourier series. The calculation of N Fourier coefficients would appear to require