Using coherent-state techniques, we prove a sampling theorem for Majorana's (holomorphic) functions on the Riemann sphere and we provide an exact reconstruction formula as a convolution product of N samples and a given reconstruction kernel (a sinc-type function). We also discuss the effect of over-and undersampling. Sample points are roots of unity, a fact which allows explicit inversion formulas for resolution and overlapping kernel operators through the theory of Circulant Matrices and Rectangular Fourier Matrices. The case of band-limited functions on the Riemann sphere, with spins up to J , is also considered. The connection with the standard Euler angle picture, in terms of spherical harmonics, is established through a discrete Bargmann transform. Mathematics Subject Classification (2000)32A10 · 42B05 · 94A12 · 94A20 · 81R30 Communicated by Thomas Strohmer.M. Calixto ( ) · J.C. Sánchez-Monreal

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 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 → ∞. Mathematics Subject Classification (2000)32A10 · 42B05 · 94A12 · 94A20 · 81R30 Communicated by T. Strohmer.M. Calixto ( ) · J.C. Sánchez-Monreal

We provide (partial) reconstruction formulas and discrete Fourier transforms for wave functions in standard Fock-Bargmann (holomorphic) phase-number representation from a finite number N of phase samples {θ k = 2πk/N } N −1 k=0 for a given mean number p of particles. The resulting Coherent State (CS) subsystem S = {|z k = p 1 2 e iθ k } is complete (a frame) for truncated Hilbert spaces (finite number of particles) and reconstruction formulas are exact. For an unbounded number of particles, S is "almost complete" (a pseudo-frame) and partial reconstruction formulas are provided along with an study of the accuracy of the approximation, which tends to be exact when p < N and/or N → ∞.MSC: 81R30, 81R05, 42B05, 30H20, 42C15