What are the implications of:
If $x(t)$ is real and $x(-t) = x^*(t)$, then $X(-j\omega) = X^*(j\omega)$ and $X(j\omega)$ is real.
I am trying to understand it and I would like to research it further (what's the theorem that guarantees this?).
I'd like to understand why it's important aside from the fact that it guarantees a symmetric Fourier transform or frequency response if we're talking about that of the impulse response.