Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

Details Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere Sampling Theorem and Discrete Fourier Transform on the Riemann Sphere

2006-12-15

arxiv.org/abs/math-ph/0612046v2

J. Fourier. Anal. Appl. 14, 538-567 (2008)

Physics

Mathematical Physics

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

Scientific paper

10.1007/s00041-008-9027-z

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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