Sampling Theorem and Discrete Fourier Transform on the Hyperboloid

Abstract

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

Extended MacMahon–Schwingerʼs Master Theorem and conformal wavelets in complex Minkowski space

“…The next step should be the discretization problem. References [34,35,36] give us the general guidelines to construct discrete (wavelet) frames on the sphere and the hyperboloid and [37] on the Poincaré group. The conformal group is much more involved, though in principle the same scheme applies.…”

Quantum statistical properties of multiphoton hypergeometric coherent states and the discrete circle representation

“…This gives z c as a function of (k, α, β). For example, for canonical and Perelomov SU(1,1) cases, we can explicitly compute this critical value, which results in z 2 c = k and z 2 c = (k − 1)/(k + 2s − 2), respectively [69,70]. For the case of Perelomov SU(1,1), this step-function behavior is sharper and sharper for higher values of k and s.…”

“…Eigenstates ofâ k can be seen as a discretization of the previous integral and therefore as an approximation to number states |n by CS superpositions on the circle. This fact was exploited in [68][69][70] to formulate sampling theorems and discrete Fourier transforms on the sphere, hyperboloid and complex plane, by using the circle representation of SU (2), SU (1, 1) and canonical CS. In this article we want to extend all these interesting constructions to general hypergeometric-like CS.…”

Multiresolution Analysis on Compact Riemannian Manifolds

“…Applications of frames on manifolds to scattering theory, to statistics and cosmology can be found in [1], [2] [19], [20], [22], [27]- [29]. There is also a number of papers in which different kind of wavelets and frames are developed on non-compact homogeneous manifolds and in particular on Lie groups, see, e.g., [5], [6]- [9], [12]- [14], [30], [31], [37].…”