Sampling in spaces of entire functions of exponential type in ${\mathbb{C}}^{n+1}$

In this paper we consider the question of sampling for spaces of entire functions of exponential type in several variables. The novelty resides in the growth condition we impose, that is, that their restriction to a hypersurface is square integrable with respect to a natural measure. The hypersurface we consider is the boundary $b\mathcal{U}$ of the Siegel upper half-space $\mathcal U$anditisfundamentalthat$b\mathcal U$ can be identified with the Heisenberg group ${\mathbb{H}}_n$. We consider entire functions in $\mathbb C^{n+1}$ofexponentialtypewithrespecttothehypersurface$b\mathcal U$whoserestrictionto$b\mathcal U$ are square integrable with respect to the Haar measure on ${\mathbb{H}}_n$. For these functions we prove a version of the Whittaker–Kotelnikov–Shannon Theorem. Instrumental in our work are spaces of entire functions in ${\mathbb{C}}^{n+1}$ of exponential type with respect to the hypersurface $b\mathcal{U}$ whose restrictions to $b\mathcal{U}$ belong to some homogeneous Sobolev space on ${\mathbb{H}}_n$. For these spaces, using the group Fourier transform on ${\mathbb{H}}_n$, we prove a Paley–Wiener type theorem and a Plancherel–P'olya type inequality.