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 under-sampling. 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.

26 latex pages. Final version published in J. Fourier Anal. Appl

References in corpus (1)

• Canonical representation of spherical functions: Sylvester's theorem, Maxwell's multipoles and Majorana's sphere

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