Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

TL;DR

This paper develops a sampling theorem and reconstruction method for holomorphic functions on the Riemann sphere using coherent states, Fourier analysis, and matrix theory, with applications to band-limited functions and spherical harmonics.

Contribution

It introduces an exact sampling and reconstruction framework for functions on the Riemann sphere, utilizing roots of unity and circulant matrices, extending classical Fourier techniques to this setting.

Findings

Explicit inversion formulas for sampling operators

Effective reconstruction of band-limited functions

Analysis of over- and under-sampling effects

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.