  In this paper we study spaces of holomorphic functions on the Siegel upper
half-space $\mathcal U$ and prove Paley-Wiener type theorems for such spaces.
The boundary of $\mathcal U$ can be identified with the Heisenberg group
$\mathbb H_n$. Using the group Fourier transform on $\mathbb H_n$, Ogden-Vagi
proved a Paley-Wiener theorem for the Hardy space $H^2(\mathcal U)$.
  We consider a scale of Hilbert spaces on $\mathcal U$ that includes the Hardy
space, the weighted Bergman spaces, the weighted Dirichlet spaces, and in
particular the Drury-Arveson space, and the Dirichlet space $\mathcal D$. For
each of these spaces, we prove a Paley-Wiener theorem, some structure theorems,
and provide some applications. In particular we prove that the norm of the
Dirichlet space modulo constants $\dot{\mathcal D}$ is the unique Hilbert space
norm that is invariant under the action of the group of automorphisms of
$\mathcal U$.
