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I don't exactly know how to ask this question, so forgive any lack of clarity.

I understand that harmonic analysis is an abstract generalization of Fourier analysis. But I am having trouble seeing why one would do harmonic analysis, beyond the heuristic "Fourier analysis is cool, so let's do Fourier analysis but abstract".

I wasn't able to boil down my issue to a single question, so I have boiled it down to the 2 following (similar) questions.

  1. When studying "pure" harmonic analysis (i.e for its own sake), what are some big goals, in a first course or in current research?
  2. What are some uses of harmonic analysis in other fields, such as complex analysis, representation theory or really anything else you guys know about?
Harish Chandra Rajpoot
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DevVorb
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    If you accept that Langlands is a big enough program you may want to read Knapp's article that was mentioned in this answer. – Kurt G. Mar 10 '23 at 10:44
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    Prefaces and table of contents of harmonic analysis books/texts is one way to get an idea. Even better, assuming there is a college or university near you, is to browse through harmonic analysis books on the library shelves. – Dave L. Renfro Mar 10 '23 at 10:51
  • @DaveL.Renfro any particular books you would recommend I skim over? – DevVorb Mar 10 '23 at 11:34
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    Fourier analysis on groups with character theory, part of the development being initated by the famous blind russian mathematician Pontryagin – Jean Marie Mar 10 '23 at 12:32
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    I don't know any books that I'd especially recommend, at least not without doing a bit of research into the literature (of which I only know minimally about), but if I were interested in pursuing this, then I'd probably (besides basic googling) just do what I suggested, namely flip through book in harmonic analysis in a university library. I do know, from what I've encountered, that the subject is HUGE and extremely encompassing of many advanced mathematical areas. Some random comments follow, none of which will probably help much with your question, but who knows . . .? (continued) – Dave L. Renfro Mar 10 '23 at 15:54
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    There's a long final chapter in my 1982-1983 graduate real analysis text (.pdf file) that gives an introduction to harmonic analysis. Incidentally, the course was taught by the author of this (latter written) book on harmonic analysis. (continued) – Dave L. Renfro Mar 10 '23 at 15:55
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    Rudin's Fourier Analysis on Groups (mentioned in another comment) and Hewitt's 2-volume treatise Abstract Harmonic Analysis seemed to be mentioned quite a bit when I was an undergraduate and graduate student, and their prefaces and their book reviews (check Bull. Amer. Math. Soc. and JSTOR for reviews of these two books) are probably worth looking over. And I have both Nina Berry's and Zygmund's trigonometric series treatises (each 2 volumes), each of which extensively covers classical topics in harmonic analysis. (continued) – Dave L. Renfro Mar 10 '23 at 15:55
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    Some recent work (last 30 years) has occasionally used certain "smallness notions" that I'm especially familiar with (e.g. see this MSE answer and this google search). – Dave L. Renfro Mar 10 '23 at 15:55
  • Thanks @DaveL.Renfro, cant wait to look all these things up – DevVorb Mar 10 '23 at 23:22

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