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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Communicated by Thomas Strohmer.
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Calixto, M., Guerrero, J. & Sánchez-Monreal, J.C. Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere. J Fourier Anal Appl 14, 538–567 (2008). https://doi.org/10.1007/s00041-008-9027-z
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DOI: https://doi.org/10.1007/s00041-008-9027-z