双曲面上的采样定理和离散傅里叶变换
Sampling Theorem and Discrete Fourier Transform on the Hyperboloid
摘要：
利用相干态（CS）方法，证明了伪酉群SU（1，1）的齐次空间上的双曲面（或其在开单位圆盘$\mathbb D_1$上的赤平投影）上的全纯函数的一个采样定理.我们提供了一个带限函数的重建公式，通过一个sinc型内核，并从适当选择的$N$样本的离散傅立叶变换。我们还研究了欠采样的情况下，带无限的功能和条件下，从$N$样本的部分重建仍然是可能的和近似的精度，这往往是准确的限制$N\to\infty$。
英文摘要：
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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