Recall that given a series $x(k)$, the Z-transform $\mathcal{Z}$ is defined as:
$$\mathcal Z(x(k)) = \sum_{k =0}^{\infty} x(k) z^{-k}$$ where $x(k)$ satisfies $|x(k)| \leq M\rho^k$, $M, \rho$ real constants
Does anyone know what are the function spaces of this mapping (i.e. conditions on x(k) and $\mathcal Z(x(k))$ as defined by their spaces) and their notations?