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Given a real-valued analytic function $f$ defined on the whole real line, and its Hilbert transform ${\cal H}f$, it seems that the quantity $f(x)^2+{\cal H}f(x)^2$ should have some kind of importance as an energy measure. It is the square of the complex modulus of the analytic extension of $f$ to the complex plane: $f(x)+i{\cal H}f(x)$. What are keywords (or textbooks, or even academic papers) that focus on this quantity?

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Engineers often call real valued functions "signals". Denoting Hilbert transform $\mathcal H\{\cdot\}$, there exists a concept called Analytic signal denoted $\mathcal A\{\cdot\}$, where for a real valued signal $f(x)$:

$$\mathcal{A}\{f\}(x) = f(x) + i\mathcal{H}\{f\}(x)$$

Then, since $|a+ib|^2=a^2+b^2$ for any pair of reals $a,b$ we can see that $$|\mathcal A\{f\}(x)|^2=f(x)^2+(\mathcal{H}\{f\}(x))^2$$

As you suspect it is a kind of energy measure. In electrical engineering and signal processing the complex absolute value above is often called envelope of a signal. And if we square it (which is just application of a monotonic growing nonlinear function) we get the function you investigated.

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  • $\begingroup$ Thank you. I think this is what I was looking for. So I could call it "the square of the envelope of the analytical representation of the function $f$"? It is a bit of a mouthful but I guess that's the best we've got. $\endgroup$ – Oliver Jan 7 at 1:39
  • $\begingroup$ I'm afraid I don't know any shorter name for it, but maybe there exists some shorter name. Actually for engineers "analytic signal" would be better. Mathematicians may confuse it with "complex analytical" which is another concept involving Cauchy-Riemann equations en.wikipedia.org/wiki/Cauchy%E2%80%93Riemann_equations. $\endgroup$ – mathreadler Jan 7 at 8:04

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