Sampling Theorem and Discrete Fourier Transform on the Hyperboloid

Details Sampling Theorem and Discrete Fourier Transform on the Hyperboloid Sampling Theorem and Discrete Fourier Transform on the Hyperboloid

2009-04-29

arxiv.org/abs/0904.4716v2

J Fourier Anal Appl (2011) 17: 240--264

Physics

Mathematical Physics

22 pages, 2 figures. Final version published in J. Fourier Anal. Appl

Scientific paper

10.1007/s00041-010-9142-5

Using Coherent-State (CS) techniques, we prove a sampling theorem for holomorphic functions on the hyperboloid (or its stereographic projection onto the open unit disk $\mathbb D_1$), seen as a homogeneous space of the pseudo-unitary group SU(1,1). We provide a reconstruction formula for bandlimited functions, through a sinc-type kernel, and a discrete Fourier transform from $N$ samples properly chosen. We also study the case of undersampling of band-unlimited functions and the conditions under which a partial reconstruction from $N$ samples is still possible and the accuracy of the approximation, which tends to be exact in the limit $N\to\infty$.

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