Energy-preserving convolution kernel? Is there a function, in continuous or (primarily) discrete time, such that convolving with it preserves the input's energy? For $x$ that is finite-valued,  finitely supported in time, and:

Real or complex?
Real?
Real and non-negative?

where &quot;energy&quot; refers to L2^2:
$$
\text{E}(f(x)) = \int_{a}^{b} |f(x)|^2 dx \ \ \text{or}\
 \ \sum_{n=a}^{b} |f[n]|^2
$$
and following must hold
$$
\text{E}(h \star x) = \text{E}(x)
$$
where $\star$ is linear or circular convolution (either works for me). If  there's no such kernel, can this be proven?

I'm aware of the trivial case of the delta; $h$ should be such that $x$ is transformed non-trivially  (not just a copy, conjugate, rescaling, sign flip, etc).
Idea: we can satisfy Plancherel's formulation per Envidia's answer with a distribution, example being the Dirac delta, but unsure there'd be something one could implement finitely.
I think any allpass filter should do that. Since they only change phase and not amplitude in the frequency domain, the time domain energy remains unchanged per Parseval's theorem.
$$\sum x[n]^2 = \sum |X(k)|^2$$
An all pass only changes the phase of $X(k)$ so total energy is maintained.
You could argue that an allpass is an IIR filter, so it's not suitable to be a convolution kernel (which implies FIR). But some allpass filters are FIR (delays) and you can always truncate an allpass IIR impulse response to the &quot;desired precision&quot;.