  Using Coherent-State (CS) techniques, we prove a sampling theorem for
holomorphic functions on the hyperboloid (or its stereographic projection onto
the open unit disk $\mathbb D_1$), seen as a homogeneous space of the
pseudo-unitary group SU(1,1). We provide a reconstruction formula for
bandlimited functions, through a sinc-type kernel, and a discrete Fourier
transform from $N$ samples properly chosen. We also study the case of
undersampling of band-unlimited functions and the conditions under which a
partial reconstruction from $N$ samples is still possible and the accuracy of
the approximation, which tends to be exact in the limit $N\to\infty$.
