Why are $L^2$ spaces preferred over $L^1$ spaces in some applications?

Question

In various applications where we are dealing with electromagnetic or acoustic waves, such as medical imaging, we use the space $L^2$ or the related Sobolev space $W^{1,2}$. I have never seen a reason given for this. Does anyone know why these spaces are preferred over say $L^1$ spaces?

Answer 1

In Quantum Mechanics the inner product between two functions $\phi_1$ and $\phi_2$ is given by

$$\langle \phi_1(x) | \phi_2(x) \rangle = \int dx \, \phi_1^*(x) \phi_2(x) \, ,$$

and this inner product has a very special place dedicated to it in the theory. Namely, the equation

$$ \langle \phi_1(x) | \phi_1(x) \rangle = \int dx \, |\phi_1^*(x)|^2 =1 $$

must be true for any wavefunction that makes sense (i.e, wavefunctions must be normalizable). This shows already the importance of $L^2$ spaces in QM, let aside its importante in Fourier analysis and harmonic theory. Mathematically, $L^2$ spaces are actually the only $L^p$ spaces that are also Hilbert spaces, where you can apply the extremely important Riesz Representation Theorem and say that to every ket there's a unique bra (modulo the usual equivalence class of $L^p$ spaces). Albeit some non-normalizable wavefunctions (like position and momentum, which require rigged spaces) are also very important to the theory, most of QM is built upon the notion of $L^2$ spaces.

Answer 2

The two spaces you mentioned initially, differently from the third one, are Hilbert spaces so that one can exploit the spectral theory, i.e., a canonial well-established way to handle mode decomposition methods which are of central relevance for the theory of waves.

Also, second-order hyperbolic equations imply conservation laws based on natural quadratic forms on Cauchy data giving rise to Hilbert-Sobolev space structures.

Answer 3

It's actually more a matter of convenience. It allows one to apply Hilbert space methods, which yields fast algorithms. But such methods do not yield the best results for most image processing problems. E.g., as pointed out in this paper, a method involving the $L^1$ norm will work much better to eliminate impulse noise. In general one can say that image processing done in an optimal way is nothing more than a Bayesian inverse problem, i.e. given the raw sensor data, what is the most likely image?