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It's occurred to me that I really need to learn fourier analysis, but I'm not quite sure what book to use. For context, I've taken a very good analysis course and am quite comfortable with the measure theory you might see in a typical graduate course. I haven't, however, seen PDEs yet.

This leaves me in a bit of a quandary. I'd prefer to use a book that assumes measure theory and some functional analysis, so that rules out things like Stein and Shakarchi's book, which seems to be a staple for introductory Fourier. On the other hand, I do want to learn Fourier with a hands on approach, and from what I've seen the books that assume the kind of background that I'm talking about kind of gloss over the essential techniques of the matter to move on to more advanced material.

So, there are pros and cons to both types of books. What would you all recommend? Is there a book that fully develops the techniques and heart of Fourier analysis, while also assuming a sufficient mathematical maturity level?

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    $\begingroup$ take a look a Dym and McKean's classic Fourier Series And Integrals and the comments about it in this answer. It seems a strong book about the matter. $\endgroup$ – Masacroso Aug 3 '17 at 5:23
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    $\begingroup$ Have you looked at Katznelson? $\endgroup$ – Bungo Aug 3 '17 at 5:38
  • $\begingroup$ Oh, these are both excellent actually. I think I'll go with Katznelson, and will perhaps supplement it with Dym and McKean. Thank you both! $\endgroup$ – David Bowman Aug 3 '17 at 17:48
  • $\begingroup$ @DavidBowman a fellow Chicagoan shouldn't miss out on Schlag's 2 volume series on classical harmonic analysis. It's a great book with loads of difficult exercises. $\endgroup$ – user217285 Dec 13 '17 at 6:52
  • $\begingroup$ @Nitin I was planning on taking Schlag's harmonic course (in which we'll presumably use those texts) this spring! They look great. $\endgroup$ – David Bowman Dec 13 '17 at 17:42

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