[Math] Fourier Analysis textbook recommendation

fourier analysisreference-request

I am taking a fourier analysis course at the graduate level and I am unhappy with the textbook (Stein and Shakarchi). What I am looking for is a book that is less conversational and more to the point. Further, I am not terribly interested in applications and would rather be exposed to how Fourier Analysis fits into the broader framework of analysis.

For background, I used Baby Rudin for a one-year course in advanced calculus, I am currently taking a course from Kolmogorov and Fomin's Introductory Real Analysis and I have taken complex analysis (using Conway's text, Functions of One Complex Variable) as well as topology (using Munkres as well as Engelking) at the graduate level, but I have not yet been introduced to the Lebesgue integral.

Best Answer

You mentioned graduate level. So you really should first learn Lebesgue integration. (Stein and Shakarchi volume 3 is not bad, as are many of the usual suspects -- Big Rudin and Royden's book on measure theory, just to name a couple.)

Then I would recommend any/all of the following:

  • Stein and Weiss, Introduction to Fourier Analysis on Euclidean Spaces (after that you may also be interested in Stein's Singular Integrals and Differentiability Properties of Functions and Harmonic Analysis)
  • Grafakos, Classical and Modern Fourier Analysis (which has been republished in the GTM series as two separate books; you should start with the Classical Fourier Analysis volume).
  • Sogge, Fourier Integrals in Classical Analysis

For one aspect of how Fourier analysis fits into the broader framework of analysis, I also recommend studying some distribution theory, and theory of partial/pseudo/para-differential operators. Some interesting texts in that regard include:

  • Friendlander and Joschi, Introduction to the theory of distributions
  • Hörmander, Analysis of Linear Partial Differential Operators, volumes 1-4 (the first volume includes a quick "review" of the parts of Fourier analysis used; I put the word in quotes because, well, it is Hörmander...)
  • Alinhac and Gérard, Pseudo-differential Operators and the Nash-Moser Theorem (and if you read French, you should consider looking at the French original)
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