Spectral Derivatives
How to replace derivatives by multiplication
Taking derivatives is rather easy. At least, if you do it by hand. Well, it may be cumbersome and tedious, but there is only a handful of rules, and all you need is time, pencil, and paper. For numerical work, the analytic rules are no longer helpful, instead, there is a range of numerical methods one can use. One is finite differences, which I discussed in this article. Another class of methods for taking derivatives is the spectral method. The name spectral refers to the methods using Fourier analysis, which is also used to compute the spectrum of audio signals, for example. This is what today’s article is about. But before we discuss fully numerical examples, we’ll talk about the Fourier way of taking derivatives of non-numerical, non-discrete functions.
Fourier Series
Consider the function
on the interval 𝑥∈[−𝜋,𝜋]. It looks like that:
