Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere
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Knowledge AreaMatemática Aplicada
SponsorsWork partially supported by the MCYT and Fundación Séneca under projects FIS2005-05736-C03-01 and 03100/PI/05.
Bibliographic CitationCALIXTO 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 1069-5869
KeywordsTransformacion rápida de Fourier
Matrices rectangulares de Fourier
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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