applications of Plancherel formulae

Question

I've learned a few things about harmonic analysis on semisimple Lie groups recently and the amount of effort that goes into the proof of the Plancherel formula seems overwhelming. Of course it has led to great discoveries in representation theory, but I was wondering whether there are some direct applications of the Plancherel formula. And now when I'm thinking about it, I don't even recall any applications of the classical Plancherel formula.

Answer 1

The most basic and useful application of the classical Plancherel formula is to define the Fourier transform on $L^2$ by density. It is then customary to illustrate Plancherel formula by computing a few basic integrals. For example, from the easy fact that $$ \widehat{{\bf 1}_{[-1, 1]}}(\xi) = \int_{-1}^1 e^{-i\xi x} \, dx = 2 \, {\sin \xi \over \xi}, $$ we compute the not totally obvious yet classical integral $$ \int_{\bf R} \Bigl({\sin \xi\over \xi}\Bigr)^2 d\xi = {\pi \over 2} \int_{\bf R} ({\bf 1}_{[-1, 1]}(x))^2\,dx = \pi. $$ From the easy $$ \widehat{e^{iax}{\bf 1}_{[0,\infty[}(x)}= {-i \over \xi -a} $$ and the standard formula for the derivative, we get the not so easy Wallis integral $$ \int_{\bf R} {dx\over (1+x^2)^n} = \biggl\Vert {1\over (x-i)^{n}}\biggr\Vert_2^2 = {2\pi\over (n-1)!^2} \ \Bigl\Vert \xi^{n-1}\,e^{\xi}\,{\bf 1}_{{\bf R}_-}(\xi)\Bigr\Vert_2^2 % = {4\pi^2\over (n-1)!^2} \int_0^\infty \xi^{2n-2} e^{-\xi} d\xi = {(2n-2)!\over 2^{2n-2}(n-1)!^2} \, \pi $$ and so on.

For the Lie group case, see the thesis of R. Gomez

Answer 2

The key fact which underlies the Plancheral formula (say for real semi-simple Lie groups) is Harish--Chandra's theorem describing the discrete series (when they exist, what their infinitesimal characters are, what their Harish-Chandra characters are, and so on). (See my answer here for an elaboration on this.)

The results on the existence and basic properties of discrete series are fundamental in all of the theory of automorphic forms, Shimura varieties, the trace formula, and so on. It's impossible to overstate this.

Answer 3

The standard way to decompose $L^2(G)$ (where $G$ can be a wide variety of things) is to decompose a dense subspace (usually Schwartz or compactly supported), then prove the Plancherel formula, which implies that the decomposition extends to $L^2$. So it's foundational. If you already believe the result, the work you have to do might seem excessive, though.

People don't seem to do so much "applied" harmonic analysis on, say, semisimple Lie groups, so you (by which I mean "I") don't see many applications. But you can use the (noncommutative) Fourier transform to solve certain differential equations (e.g. when the operator comes from the center of the universal enveloping algebra), and you can talk about Sobolev spaces, so you can prove things about smoothness of solutions. The spectral decomposition of automorphic forms is used a lot (especially applied to analytic number theory).

Answer 4

The Plancherel formula does almost all the work when you show that the continuous wavelet transform or the short-time Fourier transform are isometries (up to a constant).

Answer 5

The Plancherel Transform does get complicated especially for non commutative groups, and semi simple Lie groups happen to be the worse ones. A fairly simple application of the Plancherel transform on $\mathbb R$ is that it takes complicated convolutions to simple pointwise multiplication on the frequency domain.

Also in wavelets analysis and representation theory, complicated operators get simplified by intertwining them with the Fourier transform. For example consider the coefficient operator $W_f:L^2(\mathbb R)\to L^2(\mathbb R)$ such that $W_{g}(f)(x) = \langle f,L(x)g \rangle $ if you try to prove that the Left regular representation $L$ of $\mathbb R$ is not admissible in term of continuous wavelets, you will need to take the Fourier (plancherel in this case) transform of $W_{g}f$ in order to prove that it is not square integrable.

Vignon S. Oussa