Overview

We now derive our first spectral method, as given by the doubly infinite matrix of (1.4). This scheme applies to a discrete, unbounded domain, so it is not a practical method. However, it does introduce the mathematical ideas needed for the derivation and analysis of the practical schemes we shall see later.

Our infinite grid is denoted by , with grid points x _j = jh for , the set of all integers:

We shall derive (1.4) by various methods based on the key ideas of the semidiscrete Fourier transform and band-limited sinc function interpolation. Before discretizing, we review the continuous case [DyMc86, Kat76, K r90]. The Fourier transform of a function u ( x), , is the function ( k) defined by

The number ( k) can be interpreted as the amplitude density of u at wavenumber k, and this process of decomposing a function into its constituent waves is called Fourier analysis. Conversely, we can reconstruct u from by the inverse Fourier transform: ^[*]

This is Fourier synthesis. The variable x is the physical variable, and k is the Fourier variable or wavenumber.

We want to consider x ranging over rather than . Precise analogues of the Fourier transform and its inverse exist for this case. The crucial point is that because the spatial domain is discrete, the wavenumber k will no longer range over all of