Suppose we have a rational complex function:
$$G(z)=\frac{a_n z^n+a_{n-1}z^{n-1}+\dots+a_1 z+a_0}{b_m z^m+b_{m-1}z^{m-1}+\dots+b_1 z+b_0}$$
And now we consider only its evaluation on the complex axis $z=i\omega$, $H(\omega)=G(i\omega)$.
$\bullet$ Does exist any other complex function $\hat G(z)$ such that its complex axis evaluation $\hat H(\omega)=\hat G(i\omega)$ satisfies $H(\omega)=\hat H(\omega)$ and $G(z)\neq\hat G(z)$ ?
$\bullet$ If it exists, does $\hat G$ need to be a rational function too? And lastly:
$\bullet$ Has this any relation to the method of analytic continuation? I mean if the fact that we know $G$ on the real axis lets us continuate its domain of definition to all the complex plane (except at its poles).