Can a complex signal be one-sided (causal in time and positive only spectrum in frequency) in both domains?
I understand that a function can't have finite support in both domains, but what if both domains extend to infinity in just the positive direction?
The analytic signal $x_a = x + j \hat x$, where $\hat x$ is the Hilbert Transform of $x$, will be one-sided in the other domain. The impulse response of the Hilbert transform itself is non-causal and extends to $\pm \infty$, so even if $x$ is causal, the analytic signal itself cannot be causal. So we can rule out analytic signals, but these are not the only signals that can be one-sided in the other domain. (Pass any analytic signal through an all-pass signal and you will no longer have the Hilbert relationship between the real and imaginary components required of an analytic signal).
Generally any one-sided signal can be decomposed into even and odd components with symmetry such that when added only the positive half remains, and the even components if real will be real in the other domain, and the odd components if real will be imaginary in the other domain (and vice versa). So we seek to show that a more generalized complex one-side waveform which can be reduced into even and odd components does or can't possibly have transformed components that are also even and odd in the other domain. I can't yet get past this point to either find such a case that does exist, or clear proof that it can't possibly exist (with no restriction to real signals only etc).
Or is there a clearer proof or example of the ability to have a one-sided signal in both domains?
Clearly we can approximate obtaining single-side waveforms in both domains sufficiently for any practical use, to the same extent we are able to practically use Hilbert Transforms through delay and truncation and ignoring the error- so I'm not looking for that answer but the theoretical proof that it can't be done within the constraints of finite delay (or the example that can be).