Reddit mentions of Fourier Analysis: An Introduction (Princeton Lectures in Analysis)

You want to read http://www.amazon.com/Fourier-Analysis-Introduction-Princeton-Lectures/dp/069111384X, literally starts from the beginning talking about this- but I'm still reading it so I'll let someone who knows more actually answer you ;)

If you're interested in Fourier series in general, I'd recommend a couple of different books. They all contain these results (some contain more constructive versions than others).


[Stein and Shakarchi's Fourier Analysis: An Introduction] (http://www.amazon.com/Fourier-Analysis-Introduction-Princeton-Lectures/dp/069111384X) is probably the most accessible book I can think of. It doesn't assume much analysis background, and it's a pretty easy read. It contains all the classical goodies you should see on Fourier analysis and Fourier series without having to use any measure theory. It also springboards into the 3rd volume in this series, which is on measure theory.



Sticking with the classical camp but adding in a bit of measure theory and functional analysis, there's Katznelson's An Introduction to Harmonic Analysis and the infamous Zygmund Trigonometric Series. Zygmund is an exceedingly comprehensive introduction to Fourier series at the beginning graduate level. And I do mean comprehensive. It was published in 1935, and it's a fair bet that it captured close to everything that was known about convergence results concerning Fourier series at that time.


The last way I'd go (and I wouldn't really look at it until you have some background in the above) is Javier Duoandikoetxea's Fourier Analysis. The book makes very free use of measure theory and functional analysis. It also assumes a pretty good working familiarity with the theory of distributions (which it introduces at rapid speed).

I know that in the long run competition math won't be relevant to graduate school, but I don't think it would hurt to acquire a broader background.

That said, are there any particular texts you would recommend? For Algebra, I've heard that Dummit and Foote and Artin are standard texts. For analysis, I've heard that Baby Rudin and also apparently the Stein-Shakarchi Princeton Lectures in Analysis series are standard texts.

Assuming you know analysis up to the Riemann integral and some basic stuff on uniform convergence of functions, then I think almost everything I mentioned is covered in chapters 2 and 3 of Fourier Analysis: An Introduction by Stein and Shakarchi. The only exception is Carleson's Theorem, which is very hard and if you really do need it then you'd be better off treating it as a black box.

To piggy back off of danielsmw's answer...

> Fourier analysis is used in pretty much every single branch of physics ever, seriously.

I would phrase this as, "partial differential equations (PDE) are used in pretty much every single branch of physics," and Fourier analysis helps solve and analyze PDEs. For instance, it explains how the heat equation works by damping higher frequencies more quickly than the lower frequencies in the temperature profile. In fact Fourier invented his techniques for exactly this reason. It also explains the uncertainty principle in quantum mechanics. I would say that the subject is most developed in this area (but maybe that's because I know most about this area). Any basic PDE book will describe how to use Fourier analysis to solve linear constant coefficient problems on the real line or an interval. In fact many calculus textbooks have a chapter on this topic. Or you could Google "fourier analysis PDE". An undergraduate level PDE course may use Strauss' textbook whereas for an introductory graduate course I used Folland's book which covers Sobolev spaces.

If you wanted to study Fourier analysis without applying it to PDEs, I would suggest Stein and Shakarchi or Grafakos' two volume set. Stein's book is approachable, though you may want to read his real analysis text simultaneously. The second book is more heavy-duty. Stein shows a lot of the connections to complex analysis, i.e. the Paley-Wiener theorems.

A field not covered by danielsmw is that of electrical engineering/signal processing. Whereas in PDEs we're attempting to solve an equation using Fourier analysis, here the focus is on modifying a signal. Think about the equalizer on a stereo. How does your computer take the stream of numbers representing the sound and remove or dampen high frequencies? Digital signal processing tells us how to decompose the sound using Fourier analysis, modify the frequencies and re-synthesize the result. These techniques can be applied to images or, with a change of perspective, can be used in data analysis. We're on a computer so we want to do things quickly which leads to the Fast Fourier Transform. You can understand this topic without knowing any calculus/analysis but simply through linear algebra. You can find an approachable treatment in Strang's textbook.

If you know some abstract algebra, topology and analysis, you can study Pontryagin duality as danielsmw notes. Sometimes this field is called abstract harmonic analysis, where the word abstract means we're no longer discussing the real line or an interval but any locally compact abelian group. An introductory reference here would be Katznelson. If you drop the word abelian, this leads to representation theory. To understand this, you really need to learn your abstract/linear algebra.

Random links which may spark your interest:

• Fourier analysis also finds applications in number theory, for example in the Hardy-Littlewood circle method.
  
• Solving cubic equation using discrete Fourier transform
  
• The Wavelet transform
  
• Fourier optics
  
• Monstrous moonshine
  
• Fractional Laplacian/calculus

I've heard that Fourier Analysis: An Introduction by Stein and Shakarchi is an excellent text. Actually all four books in that series have been given glowing reviews by the people I've spoken to. I plan to pick up all of them once I find them cheap enough...

It may interest you that one of the authors, Elias Stein, is a leading figure in the field of harmonic analysis.