# How "big" is harmonic analysis?

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?
• If you accept that Langlands is a big enough program you may want to read Knapp's article that was mentioned in this answer. Commented Mar 10, 2023 at 10:44
• 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. Commented Mar 10, 2023 at 10:51
• Fourier analysis on groups with character theory, part of the development being initated by the famous blind russian mathematician Pontryagin Commented Mar 10, 2023 at 12:32
• 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) Commented Mar 10, 2023 at 15:54
• 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) Commented Mar 10, 2023 at 15:55

Answer to 1: While Fourier analysis is virtually "solved" in the spaces $$L^2([0,2\pi])\quad$$, in the sense that thanks to the orthogonality and completeness of the system $$\{ e^{2\pi i n \cdot} \}_{n=1}^\infty$$ we can assure that the Fourier series of any function $$f\in L^2([0,2\pi])$$ converges, namely,

$$||f-S_N(f)||_{L^2(\mathbb{R})} \longrightarrow 0,$$ where $$S_N(f)(x) := \sum_{n=-N}^N \hat{f}(n)\cdot e^{2\pi i n x}.$$

But, while that is true for $$L^p([0,2\pi])$$ when $$p=2$$, and for many years was believed to be true for a neighbourhood of $$2$$, Fefferman showed in 1971 that for every $$p\neq2$$ there existed a function in $$L^p(\mathbb{R})$$ such that its Fourier series doesn't converge in $$L^p(\mathbb{R})$$. This basicaly means that you can't decompose a given function $$f\in L^p([0,2\pi])$$ into simple frequencies via its Fourier series unless $$p=2$$. In that concern, a major goal in harmonic analysis is to find a way to decompose any given $$f\in L^p([0,2\pi])$$ into simple frequencies regardless of $$p$$. Moreover, even considering $$p=2$$, what happens when you swap the set you are considering your functions is, that is, can you swap $$[0,2\pi]$$ for any other subset of $$\mathbb{R}$$ and have everything work out nicely? The answer is yes in the case of any compact interval, but no in the case of $$\mathbb{R}$$ itself. This leads to the necessity of defining a different method for the non-compact-supported functions $$f \in L^2(\mathbb{R})$$: the Fourier transform:

$$\hat{f}(\xi) := \int_{\mathbb{R}} f(x)e^{-2\pi i x \xi}d\xi$$

Giving it a glance one can easily determine that the definition makes sense in $$L^1(\mathbb{R})$$, but a priori you can't define it the same way for $$p\neq1$$. So,as you can see, just the fact of defining the Fourier transform in the space $$L^p(\mathbb{R})$$ is a challenge when $$p\neq 1$$, let alone determine whether or not the equality

$$f(x) =? \int_{\mathbb{R}} \hat{f}(\xi)e^{2\pi i x \xi}d\xi$$

holds in the sense

$$||f(x) - \int_{\mathbb{|\xi| You can see this as analogous to the series case.

As for question 2:

Fourier analysis interconnects two areas of math hardly related otherwise: analysis and discrete math via the identification

$$f \rightarrow (\hat{f}(n))_{n=1}^\infty$$

that is, f and the sequence of its Fourier coeficients. In fact, in the case $$p=2$$ you get a 1 to 1 identification between functions in $$L^2([0,2\pi])$$ and the sequence space $$\ell^2$$.

Needless to say, harmonic analysis is very useful when it comes to image and sound processing and compressing algorithims.