I’d like to give a very strong recommendation to a book that James Grimmelmann suggested I read on the subject of orthogonal functions. The book is T.W. Körner’s Fourier Analysis. It’s spectacular.
In my experience, the biggest failing in math textbooks is that they start too far down the stack. What one wants is a book that says, “Here’s the problem we’re trying to solve. Now, we have this pre-existing tool that seems like it might be helpful toward solving this problem, so let’s see if it helps here. The tool needs to be tweaked somewhat, so how should we tweak it? We need to know that we can use this particular technique, so we’ll need a theorem that says Here’s some intuition about how we’d prove that. All that we need now is a little bit of work to fill in the details.” Körner’s book takes exactly this approach, and it’s glorious.
Contrast this with most math books. In my experience, most start at axioms and build up, until toward the end we find the interesting theorems. I’m by no means a practicing mathematician, but my sense of the history of math is that normally solutions to problems have started in the middle: we have a problem we want to solve, so we prove some theorems that are pretty far along in the chain (not close to the axioms, that is), and over time work backwards to more-rigorously justify the theorems with more fundamental axioms. Then also move forward, and prove theorems that are generalizations of the middle-level theorems proved earlier. Certainly much math began with, say, problems in physics; Fourier series are a good example of this, having arisen in Fourier’s Théorie Analytique de la Chaleur (Analytical Theory of Heat). The theorem doesn’t start with the axioms; it starts much closer to a physical system, then works both backwards and forwards.
I’ve been searching for years for a book whose pedagogy matches the history of the subject. I’ve wanted this not only because I have a taste for the history of math, but because I believe there’s something natural and easy about teaching a subject the way that society developed it. The same approach would be lovely in, say, a physics textbook: start with the simplest models that physicists developed in the 1500’s, show where they failed, and show how they moved to fix them. Few books do this. One that does — and it’s a historical treatise rather than a textbook — is Richard Rhodes’s The Making of the Atomic Bomb. Now if only we could get more textbooks to use this technique.
Körner falters a little bit when he tries to be a straightahead history book rather than a textbook informed by history. One wouldn’t necessarily notice his weaknesses in this respect if Donald Knuth’s The Art of Computer Programming (3 vols.) didn’t exist. Knuth is a master mathematician, an epoch-making computer scientist, and a peerless scientific historian.
That doesn’t even count as a gripe, though: Körner has succeeded brilliantly in exactly the endeavours he sought to master. I’d recommend his book to anyone who wants to understand Fourier series, their use, and their history.
P.S. (22 February 2007): I bought a used copy of this book off the Amazon Marketplace — specifically from proquobooks, aka internationalbooks. I got it for a grand total of $13.26, including shipping. It was a pristine copy, virtually indistinguishable from a brand-new one. A brand-new copy on Amazon, assuming you opt for free Super-Saver Shipping, costs $54.60. That puts a big smile on my face. Googling for ‘proquobooks’ gives a negative review near the top, but my experience with them has been quite positive. Even the mailer in which they shipped the book was great, and it arrived within three days of my ordering — certainly better than Super-Saver Shipping. I’d buy from proquobooks again in a heartbeat.
