This fall I will be teaching our senior seminar in Analysis, and I have decided to focus on Fourier Analysis. So I was delighted to receive a review copy of this little book: Fourier Analysis in 146 pages, including applications, sounded like something I could use to prepare.
It got more interesting when I read the introduction:
Showing a Fourier transform to a physics student generally produces the same reaction as showing a crucifix to Count Dracula. This may be because the subject tends to be taught by theorists who themselves use Fourier methods to solve otherwise intractable differential equations. The result is often a heavy load of mathematical analysis.
Hmm. I’m certainly a “theorist” but my interest in Fourier Analysis has more to do with representation theory than PDEs. It got a little stranger when, a few lines later, the author said that in practice “the transforms are done digitally and there is a minimum of mathematics involved.” It seems that by “mathematics” and “mathematical analysis” what James means is computation. This is reinforced a little further on: “In spite of the forest of integration signs throughout the book there is in fact very little integration done and most of that is at high-school level.” And here I always thought that it was the physicists who liked to compute things.
Anyway, despite the initial rhetoric this is a very nice book. The author focuses on the interpretation of Fourier transforms as finding the components of the initial function that correspond to oscillations at various frequencies. When the initial function is periodic, this yields Fourier series, since only a discrete set of frequencies is involved. When it is not periodic, one gets a continuous range of frequencies, sums turn into integrals, and we have the Fourier transform. All this is presented intuitively and without proofs, as is proper for a book of this kind.
In order to be able to include Fourier series as a special case of the Fourier transform, the author introduces Dirac’s delta “function” and even “delta combs,” which are infinite linear combinations of delta functions. No hint is given as to how to conceive of the delta function in a way that makes mathematical sense. In fact, the author even allows himself to give the delta function as an example of an unbounded function that nevertheless has a Fourier transform. Only occasionally is there a hint that there might be something to worry about, as when a footnote on page 31 cautions the reader not to apply the theorem about the Fourier transform of a derivative to the delta function.
A mathematician reading this book sometimes feels himself in the Twilight Zone, as the author assumes that arguments about signals and their detection are easier to understand than the underlying mathematics. Sometimes that is true even for me, but more often I have to read the mathematical result in order to understand the physical motivation!
Weird as this is, reading this book is a very useful discipline for anyone who is going to teach Fourier Analysis: it helps us understand how some of the users of the mathematics think about it, and includes several very nice applications. My students, most of whom are not physicists, would hate it.
Fernando Q. Gouvêa is Carter Professor of Mathematics at Colby College in Waterville, ME.
1. Physics and Fourier transforms
2. Useful properties and theorems
3. Applications 1: Fraunhofer diffraction
4. Applications 2: signal analysis and communication theory
5. Applications 3: spectroscopy and spectral line shapes
6. Two-dimensional Fourier transforms
7. Multi-dimensional Fourier transforms
8. The formal complex Fourier transform
9. Discrete and digital Fourier transforms