What is the physical interpretation of $H_2$ norm?

Question

Two years ago a younger me asked whether if there is some physical interpretation of $L_1$ and $L_2$ norm. Much insights has been gained since then, and I have recognized that it is widely accepted that the $L_2$ norm represents the energy of a signal $f(t)$.

Is there a similar physical (physics motivated) interpretation of $H_2$ norm? $$\sup_{0<r<1}\left(\frac{1}{2\pi} \int_0^{2\pi}\left|f \left (re^{i\theta}\right )\right|^2 \; \mathrm{d}\theta\right)^\frac{1}{2}<\infty$$ https://en.wikipedia.org/wiki/Hardy_space

The problem is that the signal now must be complex. Should we treat $f(x+iy)$ be some sort of signal with a complex and real part i.e. electromagnetic wave? So $H_2$ space is the space of finite energy electromagnetic waves $E(x,t) = E_0\exp[i(kx −ωt −θ)]$?

Answer 1

From a Math only point-of-view, $f(z) =\sum_{n=0}^{\infty}a_n z^n$ is in $H^2$ iff $\sum_{n=0}^{\infty}|a_n|^2 =\|f\|_{H^2}^2 < \infty$. So this space consists of all power series with square sumable coefficients. I'm not sure how you want to interpret the sum of squares of the coefficients for a particular application, but $H^2(D)$ is the same as $\ell^2(\mathbb{Z}^+)$.

Answer 2

Here's what I learned from the robust control course that I have taken a while ago (might be wrong though, it's been quite a long time).

When we are dealing with signals, i.e. functions of time which is a real variable, we describe them by the Lebesgue space, $\mathcal L$. But naturally in the signal analysis and control systems, we take the Laplace transform of signals and transfer functions. In this case they become a function of $s$, which is complex. So we switch to the Hardy space and denote them by being a member of, for example, $\mathcal H^2$.