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In short, I am interested to know of the various approaches one could take to learn modern harmonic analysis in depth. However, the question deserves additional details. Currently, I am reading Loukas Grafakos' "Classical Fourier Analysis" (I have progressed to chapter 3). My intention is to read this book and then proceed to the second volume (by the same author) "Modern Fourier Analysis". I have also studied general analysis at the level of Walter Rudin's "Real and Complex Analysis" (first 15 chapters). In particular, if additional prerequisites are required for recommended references, it would be helpful if you could state them.

My request is to know how one should proceed after reading these two volumes and whether there are additional sources that one could use that are helpful to get a deeper understanding of the subject. Also, it would be nice to hear suggestions of some important topics in the subject of harmonic analysis that are current interests of research and references one could use to better understand these topics.

However, I understand that as one gets deeper into a subject such as harmonic analysis, one would need to understand several related areas in greater depth such as functional analysis, PDE's and several complex variables. Therefore, suggestions of how one can incorporate these subjects into one's learning of harmonic analysis are welcome. (Of course, since this is mainly a request for a roadmap in harmonic analysis, it might be better to keep any recommendations of references in these subjects at least a little related to harmonic analysis.)

In particular, I am interested in various connections between PDE's and harmonic analysis and functional analysis and harmonic analysis. It would be nice to know about references that discuss these connections.

Thank you very much!

Additional Details: Thank you for suggesting Stein's books on harmonic analysis! However, I am not sure how one should read these books. For example, there seems to be overlap between Grafakos and Stein's books but Stein's "Harmonic Analysis" seems very much like a research monograph and although it is, needless to say, an excellent book, I am not very sure what prerequisites one must have to tackle it. In contrast, the other two books by Stein are more elementary but it would be nice to know of the sort of material that can be found in these two books but that cannot be found in Grafakos.

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I think this should be community wiki. –  Spencer Jun 3 '11 at 9:45
    
Made CW for reasons explained in the FAQ - mathoverflow.net/faq#communitywiki –  François G. Dorais Jun 3 '11 at 10:53
    
There is also Representation Theory and Noncommutative Harmonic Analysis I and II, by Kirillov, Soucek and Neretin. –  Max Muller Jun 8 '11 at 17:06
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4 Answers 4

It depends very much on what areas of harmonic analysis you're interested in, of course. Grafakos' books are excellent and really quite advanced, and if you wish to continue in that style of harmonic analysis, then there's not much else you can do other than start reading many of the articles that he cites. On the other hand, there are interesting areas in harmonic analysis not covered by Grafakos. I'd recommend a couple of textbooks by Stein: Singular Integrals and Differentiability Properties of Functions and Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals. There are probably some other interesting textbooks on singular integral operators that might be useful (though I can't think of any off the top of my head). One other interesting (and very modern) area is wavelets: Mayer's book Wavelets and Operators is probably the place to start there. Other useful resources are lecture notes or survey articles about harmonic analysis available online. For example, Pascal Auscher taught a course at ANU on harmonic analysis using real-variable methods last year, and one of the students in the class typed up notes, which are available here. Similarly, Terry Tao taught a course a few years ago, and he has lecture notes here and here. Finally, if you want to learn about harmonic analysis with an operator-theoretic bent, there are useful lecture notes here and here.

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Dear Peter, the link to the ANU course notes appears to be broken. –  Amitesh Datta Jun 3 '11 at 9:51
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@Amitesh: should be fixed now, was just some shoddy hyperlinking on my part. –  Peter Humphries Jun 3 '11 at 9:59
    
Dear Peter, thanks! I was just wondering, in case you knew, do these notes cover the entire course or do they only cover a part of the course? I was thinking of taking this course at some point in the future and it would be helpful to know what is covered; unfortunately, I can't seem to find anything on the ANU maths webpage about it. –  Amitesh Datta Jun 3 '11 at 10:59
    
The lecture notes cover the entire course. Pascal was a visiting professor last year, but he is back in France now, and in any case the course was a special topics course, which vary from year to year, so I can't see it being taught again any time soon. Your best bet is to talk to Alan McIntosh and organise a reading course. –  Peter Humphries Jun 3 '11 at 11:20
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Since you have read Rudin's "Real and complex analysis", you are ready to attack Rudin's "Fourier analysis on groups", which is equally pleasant reading.

Still valuable for the link with Banach algebra theory, is L.H. Loomis' "An introduction to abstract harmonic analysis".

For connections with unitary representations: the 2nd half of Dixmier's "C*-algebras", or better R. Howe and E.C. Tan "Non-abelian harmonic analysis (applications of $SL_2(\mathbb{R})$)" (everything is in the subtitle!)

If you want to see connections with number theory, I recommend Weil's "Basic number theory".

Now you can guess my age from this list of references!

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If you're looking at Fourier analysis on groups, there are a couple great books that are quite recent: Deitmar's A First Course in Harmonic Analysis (which is quite simple), then Deitmar and Echterhoff's Principles of Harmonic Analysis (which looks more at nonabelian groups). –  Peter Humphries Jun 3 '11 at 10:46
    
Dear Alain, thanks! –  Amitesh Datta Jun 3 '11 at 11:09
    
Folland's A Course in Abstract Harmonic Analysis (1995) is another excellent recent text. –  Mark Kim Jun 3 '11 at 11:09
    
@Peter,Mark Dietmar's books and especially Folland's,are both excellent suggestions. In fact,I'd recommend ANY of Folland's textbooks for graduate students of analysis. –  Andrew L Jun 4 '11 at 3:18
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To my mind, the classical subject is quite different from the modern, evolved form of the subject

I started on the classical side with Yitzhak Katznelson's An Introduction to Harmonic Analysis: This is in the classical camp: Lots on Fourier Series. Very clear; very nice proofs. You will learn lots of gems about trigonometric series. In this classical camp, Zygmund's treatise Trigonometric Series (two volumes) deserves a mention. This is also a very beautiful book.

For 'harmonic analysis' as a modern field, you ought to get your hands on a copy of Stein's books as in Peter's answer. The late Tom Wolff has a very useful set of notes in this regard, available (I think, still) from Izabella Laba's homepage.

I also second the recommendation to look at Tao's old dvi/pdf notes on his website and later on on his blog. For example, I remember finding his post on interpolating $L^p$ spaces very nice.

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Dear Spencer, thanks! –  Amitesh Datta Jun 3 '11 at 11:10
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you can also look as a primer lecture notes on topological groups Higgins, london Mathematical society very easy to read

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