maxbaroi said:
I was wondering if anyone has any recommendations for a Fourier Analysis textbook.
I have Stein & Shakarchi's Fourier Analysis textbook, but ideally I'd like to have one that takes advantage of some of the analytic machinery that I know that Stein & Shakarchi doesn't assume. I have a basic graduate level knowledge of Real Analysis: some point-set topology, properties of the Lebesgue measure and measurable functions, some more measure theory, a little functional analysis, topological vector spaces. If anybody has any suggestions I'd really appreciate them.
Thanks!
Have you had a look at Katznelson's "An Introduction to Harmonic Analysis"? It's a nice treatment of classical Fourier analysis that assumes exactly the background you described. There's also Dym and McKean, "Fourier Series and Integrals" at about the same level or maybe a little easier.
Another option that covers a lot more ground that can also be attacked by someone with your background is "Classical and Modern Fourier Analysis" by Grafakos. (I see that this has recently been republished by Springer as two volumes: "Classical Fourier Analysis" and "Modern Fourier Analysis".) This is a more sophisticated book that aims higher and would be a good choice if your aim was to be able to read research papers in this area.
Grafakos covers some major non-trivial modern ground, such as Carleson's 1966 theorem (and Hunt's subsequent extension) that the Fourier series of any L^p function (p > 1) converges almost everywhere.
You may also find it useful to learn about Fourier analysis on distributions (generalized functions); there's some material on that subject in both of the above books, but for more depth you might check out Rudin's "Functional Analysis" or Hörmander's "The Analysis of Linear Partial Differential Operators, Part I: Distribution Theory and Fourier Analysis". The latter is denser and harder, but the paperback edition actually has solutions to all the exercises, a rare resource at this mathematical level!
In ascending order of sophistication, I would rank the above books as follows:
Dym and McKean
Katznelson / Rudin (roughly the same level)
Hörmander
Grafakos
The other option is that of course most books on measure-theoretic real analysis will have some Fourier analysis coverage, including Stein and Shakarchi's third volume, Rudin's "Real and Complex Analysis," Jones' "Lebesgue Integration on Euclidean Space." Whether this is enough depends on how far you want to go.
