Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere
Knowledge Area
Matemática AplicadaSponsors
Work partially supported by the MCYT and Fundación Séneca under projects FIS2005-05736-C03-01 and 03100/PI/05.Publication date
2008-04Publisher
SpringerBibliographic Citation
CALIXTO MOLINA, Manuel, GUERRERO GARCÍA, Julio, SÁNCHEZ MONREAL, Juan Carlos. Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere. Journal of Fourier Analysis and Applications, 14 (4): 538-567, Abril 2008. ISSN 1069-5869Keywords
Transformacion rápida de FourierEsferas Riemann
Armónicos esféricos
Funciones Majorana
Matrices circulantes
Matrices rectangulares de Fourier
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.
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