01 February 2013

Fourier Theory, a great mathematical poem

Fourier Series

The greatest inventions of mankind are music and mathematics, and they meet in the theory of Fourier series. A beautiful mathematical theory, which at the same time reveals the interplay between the fundamental frequency and its harmonics, i.e., the mystery of timbre which distinguishes music from sound. The theory originates with

 (Analytical Theory of Heat, full text here or here). 

In this book, Fourier establishes mathematical models for the dissipation of heat, at the same time developing new mathematical tools for their solution. Quite an achievement, not unlike Newton's. From a purely mathematical viewpoint, the most surprising statement is, that 'any' periodic function can be written as a series of sines and cosines. Fourier's argument left much to be desired though, and the first convincing proof was this one (also here), by Dirichlet, 1829. Many different formulations, generalizations and proofs are known by now. In the framework of Riemann integration the theorem, stripped to the bone, can be stated as follows.

Fourier could not have imagined a form so neat and general, because the first sound theory of integration (Riemann, 1854) was still three decades ahead. This or this single page (not going beyond basic integration) suffices for our proof. Add half a page to obtain the familiar form
valid if the function is (besides mildly regular) also periodic and continuous. In this form we see the fundamental frequency and its infinite number of harmonics displayed. If you want to hear how it sounds, click this image

and play around with the first thirty-two coefficients (blue for cosines, green for sines).

A mathematician's eulogy

As any book on the subject will teach you: everything about Fourier series is interesting and there are no easy, let alone trivial, results. Pointwise convergence, considered above, is just the first of many, many fascinating aspects. The marvel of the subject was realized early on. In this paper from 1864 Lord Kelvin, the great physicist, describes Théorie de la Chaleur as

Fourier's great mathematical poem

(fourth line of paragraph 4). The quote is frequently attributed to Maxwell, but he doesn't come any closer than "Fourier, in his great work on the conduction of heat..." (The scientific letters and papers of James Clerk Maxwell, ed. by P.M. Harman, Cambridge University Press, 1990-2002, Vol. II, p. 358). But it is true that Maxwell read Fourier's book when he was seventeen, and was deeply impressed. I would have preferred the quote to have been Maxwell's, because Maxwell was a talented poet himself. (Speaking of physicists writing poems: Oppenheimer also did.)