Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

Abstract

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

Sampling Theorem and Discrete Fourier Transform on the Hyperboloid

“…This case will be named oversampling, since there are more data than unknowns, and will be discussed in Sect. 4.1 (see also [5]). In other contexts, when (2.10) holds, the set Q is said to be sampling for the space H [14].…”

“…The most common example is the Bargmann-Fock space of analytical functions on C, where one can find rectangular lattices which are sampling (and therefore A is invertible), or which are interpolating (and thus B is invertible), but not both simultaneously [14,23]. Examples of critical sampling are given by the space of band limited functions on R and the set Z, which is both sampling and interpolating, and the space of functions on the Riemann sphere (or rather its stereographic projection onto the complex plane) with fixed angular momentum s and the set of N th -roots of unity, with N = 2s + 1 [5].…”

“…Usually, this pre-factor is absorbed into the integration measure in (3.13). 4 In the case of the sphere [5] the sampling points were the N th -roots of unity, which formed an abelian subgroup Z N of SU (2), and the main advantage of this was that B was a circulant matrix. In that case the sampling was regular [14].…”

“…where F denotes the Rectangular Fourier Matrix (see [5]) given by F kn = 1 √ N e −i2πkn/N , k = 0, . .…”

“…In a previous article [5], we proved sampling theorems and provided discrete Fourier transforms for holomorphic functions on the Riemann sphere, using the machinery of Spin CS related to the special unitary group SU (2), which is the double cover of the group SO(3) of motions of the sphere S 2 . Here we study similar discretization problems for its noncompact counterpart SO(2, 1) (the group of motions of the Lobachevsky plane) or, more precisely, for its double cover SU (1,1).…”