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Buy Classical Fourier Analysis (Graduate Texts in Mathematics) on ✓ FREE SHIPPING on qualified orders. Loukas Grafakos (Author). out of. book, which is a sequel to GTM Classical Fourier Analysis, 3rd Edition. This L. Grafakos, Modern Fourier Analysis, Graduate Texts in. The primary goal of this text is to present the theoretical foundation of the field of Fourier analysis. This book is mainly addressed to graduate.

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By using our site, you acknowledge that you have read and understand our Cookie PolicyPrivacy Policyand our Terms of Service. 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. My intention is to read this book classicl 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.

Classical Fourier Analysis – Loukas Grafakos – Google Books

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 cpassical 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 cladsical into one’s learning of harmonic analysis are welcome.

Classical Fourier Analysis

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 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.


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: 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 classical 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 anlaysis 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. 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.

Loomis’ “An introduction to abstract harmonic analysis”. For connections with unitary representations: To my mind, the classical subject is quite different from the modern, evolved form of the subject.

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. One of my favourite books for harmonic analysis is Fourier analysis by Javier Duoandikoetxea. That is a terrific reference for background regardless of what you want to do with harmonic analysis.

I would tackle this before moving onto Elias Stein’s book “harmonic analysis: As mentioned above, it really depends on what type of harmonic analysis you are interested in, but I would certainly recommend those as well as harmonic analysis by Katznelson, the two volume books by Grafakos, both of Stein’s books on “introduction to Fourier analysis on Euclidean spaces” and “singular integrals and differentiability properties of functions” are useful for singular integrals.

I’d also recommend a treatise on trigonometric series by Bary. Zygmund’s two volume books on trigonometric series are good, but I would tackle a few other books on harmonic analysis before going for it.


If you like abstract harmonic analysis, go for “principles of harmonic analysis” by Anton Deitmar.

Terence Tao’s website is great for lecture notes all academic resources on his website are great! Finally, “lectures on nonlinear wave equations” by Christopher Sogge and “nonlinear dispersive equations” by Terence Tao are great books that have a focus on dispersive PDEs using techniques from harmonic analysis such as Littlewood-Paley theory.

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Home Questions Tags Users Unanswered. Learning roadmap for harmonic analysis Ask Question. Thank you very much! I think this should be community wiki.

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Keep some standard reference books like Zygmund, etc clzssical, wikipedia, google, MO, math. Dear Peter, the link to the ANU course notes appears to be broken.

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 analydis know what is covered; unfortunately, I can’t seem to find anything on the ANU maths webpage about it. The lecture notes aanalysis the entire course.

Pascal was a classcial 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. 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! If you’re looking at Fourier analysis on groups, there are a couple great books that are quite recent: Peter,Mark Dietmar’s books and especially Folland’s,are both excellent suggestions.

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: Sign up or log in Sign up using Google.

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