First some introduction. The Fourier transform of function $f(t)$ is define as:
$$\hat{f}(\xi) = \int\limits_{-\infty}^\infty f(t) e^{-i 2 \pi \xi t} dt$$
If we think of $f(t)$ as being a signal we can think of the square magnitude of the signal integrated over time as being its total energy:
$$\int\limits_{-\infty}^\infty |f(t)|^2 dt = E$$
We can say that the Fourier transform conserves this energy (Plancherel theorem) meaning:
$$\int\limits_{-\infty}^\infty |f(t)|^2 dt = E = \int\limits_{-\infty}^\infty |\hat{f}(\xi)|^2 d\xi$$
Now for my actual question. What if the total energy of the signal isn't finite but the average energy per time unit is? By this I mean that:
$$\frac{\int\limits_{a}^b |f(t)|^2 dt}{b-a}\to E$$
When $a\to-\infty$ and $b\to\infty$. My question is, is this quantity in some sense conserved by the Fourier transform, i.e. can one somehow also calculate this $E$ for $\hat{f}(\xi)$?
I can think of some special cases. If e.g. $f(t)$ is periodic (with period $P$), i.e. $f(t+P) = f(t)$ then the Fourier transform is composed of Dirac delta functions at regular intervals:
$$\hat{f}(\xi) = \sum\limits_{n=-\infty}^\infty c_n \delta(\xi-n/P)$$
and per Parseval's identity:
$$\sum\limits_{n=-\infty}^\infty |c_n|^2 = E$$
However, it would be interesting to know if you can do something similar for a wider class of functions and how would that be defined rigorously.