Introduction to Numerical Methods for Time Dependent Differential Equations

CHAPTER 7

FOURIER SERIES AND INTERPOLATION

Fourier series are among the most useful ideas in mathematics, for many reasons. For the problems we are interested in in this book, Fourier theory is essential to study the stability of numerical approximations and to introduce a family of high-precision methods, the spectral and pseudo-spectral methods. In this chapter we review briefly Fourier series and their ability to represent functions both exactly or as an approximation. We also study Fourier interpolation, which can be thought of as a version of Fourier series for discrete functions.

7.1 Fourier expansion

In this section we consider the expansion of 1-periodic function f(x) into Fourier series. Here f : → is called 1-periodic if f(x + 1) = f(x) for all x. Important examples of 1-periodic functions are the exponentials,

which play a central role in mathematics. One reason is that differentiation results in multiplication by a constant factor,

As a consequence, by the use of exponentials, one can often transform a differential equation into an algebraic equation.

For this reason it ...