Overview

We are ready to discuss the accuracy of spectral methods. As stated in chapter 1, the typical convergence rate is O( N ^{? m}) for every m for functions that are smooth (fast!) and O( c ^N) (0 < c > 1) for functions that are analytic (faster!). Such behavior is known as spectral accuracy.

To derive these relationships we shall make use of the Fourier transform in an argument consisting of two steps. First, a smooth function has a rapidly decaying transform. The reason is that a smooth function changes slowly, and since high wavenumbers correspond to rapidly oscillating waves, such a function contains little energy at high wavenumbers. Second, if the Fourier transform of a function decays rapidly, then the errors introduced by discretization are small. The reason is that these errors are caused by aliasing of high wavenumbers to low wavenumbers.

We carry out the argument for the real line ; similar reasoning applies in the periodic case. The following theorem collects four statements relating smoothness of u and decay of . Each condition on the smoothness of u is stronger than the last and implies a correspondingly faster decay rate for . This theorem makes use of standard mathematical ideas (differentiability, analyticity, complex plane) that some readers may be less familiar with than they would like. Rest assured at least that the book does not get more technical than this! The "big O" symbol is used...