What is the goal of harmonic analysis?

Question

I am taking a basic course in harmonic analysis right now. Going in, I thought it was about generalizing Fourier transform / series: finding an alternative representation of some function where something works out nicer than it did before.

Now, having taken the first few weeks of this, it is not at all about Fourier analysis but about the Hardy-Littlewood-Maximal-operator, interpolation theorems, Stein's theorem/lemma and a lot of constants which we try to improve constantly in some bounds. We are following Stein's book on singular integrals, I guess.

Can anyone tell me where this is leading? Why are we concerned with this kind of operators and in which other areas are the results helping?

Answer 1

Harmonic analysis is a very powerful tool to deal with many mathematical problems: i restrict myself on non-linearities in Partial differential equations (since it is my domain of research and you can adapt mutatis mutandis for any other domain because only technicalities will be discussed) for three important reasons:

1. It is based on divide and conquer philosophy: it uses particular ways and methods (Fourier series/transform, Littlewood Paley decomposition) of decomposing the phase plane which takes a single function and writes it as a superposition of a countably infinite family of functions of varying frequencies.

2. Other advantage of harmonic analysis methods is that a partial differential equation (those who involves more than one variable $t$: time and $x$: space for example) will be seen as an ordinary differential equation. In fact, derivatives in phase space became multiplications in frequency space $$\mathcal{F}(\nabla f)(\xi)=2\pi i \xi\hat{f}(\xi)$$ where by $\hat{f}$ and $\mathcal{F}(\nabla f)$ we refer to Fourier transform of $f$ and $\nabla f$ respectively. Then the answer to your question is: the goal of harmonic analysis is to serve as a generous bag of tools and powerful tricks.

3. A final modest information that i allow myself to add and it may contribute as a part of the answer: The idea of fractional derivative also becomes clear in the context of harmonic analysis. In fact, by applying formally the inverse Fourier transform on $(2\pi i \xi)^{\gamma}\hat{f}(\xi)$ to obtain the derivative of order $\gamma$ of $f(x)$ for all $\gamma\in \Bbb{R}.$ Some operations can be performed only in Frequeny space, namely when we apply the Fourier transform and then take the inverse to go back to phase space. As for operators and the whole theory, we must always remember that harmonic analysis is a "huge bag" that contains the fruit of contributions of many mathematicians developped according to necessity.

Answer 2

Ultimately it helps one to prove theorems (like existence and uniqueness) of partial differential equations.

Answer 3

You may enjoy reading this survey:

Harmonic analysis as the exploitation of symmetry–a historical survey, by George W. Mackey, in Bull. Amer. Math. Soc. 3 (1980), 543–698.

The one-page introduction gives the goals of harmonic analysis in abstract but familiar terms.

Answer 4

With that set-up I would expect to see boundedness of operators such as Hilbert transform, Reisz transform, etc. Expect to see fractional powers of Laplacian $\Delta$. Of course it ultimately depends on the instructor. There are an array of topics she/he could choose from.