Instead of going on with nonstandard analysis (which I might do later in a new format), I thought I would try another series of posts, on something very near and dear to my heart – Fourier analysis. I have (partly) read many books on Fourier analysis, and even published some work in it. But I have struggled to understand the essence of it – not surprising, since it is a subject both vast and deep. At the moment, I am solving differential equations and having much fun with the process. In doing so, I realized that I build a much deeper understanding when I get my hands dirty, so to speak. This is not a great revelation, but it is a principle I have ignored for too long. To understand Fourier analysis, I shouldn’t be reading the most advanced, abstract books on the subject and looking at the theoretical research, I need to go back to the origin. Why was it formulated this way? Why were kernels introduced? Why do we use Poisson summation? Why is it important that convergence takes place in certain spaces?

In this series, I intend to go back through history, even though I do not intend this as a historical work. It is more about following the evolution of ideas, and so I do not intend it to be 100% chronological. Mathematics is a messy place, and the narrative is not always clear (just like history itself, really). Things are rarely as neat as we make them out to be in our textbooks, and I think we do our students a disservice by not exposing them to these ideas. Fourier techniques are usually presented to the student as if ex nihilo, but there is a fascinating evolution of thought to explore. One of the few books that do not try to hide this is that of Körner, and I’m sure I will be referring to it extensively as I write this.

Let’s go back to the beginning then, and see what Fourier actually did, why he did it and whether anyone had done it before. There is probably no place better to go than Fourier’s “The Analytical Theory of Heat”. Now, I’m obviously not going to read the whole thing, but looking through the table of contents we see that Section II of Chapter 3 starts with “First example of the use of trigonometric series in the theory of heat”, which seems like the kind of thing we’re after.

Perhaps the best place to start would be to discuss Fourier’s heat equation itself:

Here, is some function of distance, , and time. . We’re not going to discuss how this equation came about and are just going to accept it as it is. There’s a good (but short) post on some of the history here. I am only presenting the one-dimensional form of the equation here, but of course the higher-dimensional version can be expressed in terms of the Laplacian. Interestingly, Fourier’s solution of the heat equation was inspired by that of Laplace, which in turn was inspired by work of Poisson.

For a moment, let us think about the content of this equation. What does it mean, and why should it be applicable to heat? Since we have a first derivative in one variable, which denotes rate of change, and the second derivative in another, we know that somehow that change in time is influenced by the curvature of the function, seen as a function of space. Instead of fully exploring this idea myself, I’ll direct you to the video at https://youtu.be/b-LKPtGMdss.

If you have had any courses in differential equations, the solution is quite obvious. It is now completely accepted, and that obfuscates how much of a revolution it actually was. As explained in the post I referred to, mathematicians had certain ideas of what a function should be, and it didn’t look like Fourier’s series conformed to those ideas. One might dismiss this as foolish conservatism by the mathematicians of old, but we must never fall into the trap of thinking our ancestors were ignorant or stupid. Rather, it indicates that Fourier analysis was something truly revolutionary, with tremendous implications for the very foundations of mathematics (more on that later). Even today, simply answering whether a trigonometric series does indeed define a certain kind of function is no simple task. One of the greatest theorems of twentieth-century mathematics is an ostensibly simple question on the convergence of Fourier series of quite well-behaved functions…

Next time, we’ll look at the solutions of the equation.