Sampling Theorem and Discrete Fourier Transform on the Hyperboloid

TL;DR

This paper develops a sampling theorem and discrete Fourier transform framework for holomorphic functions on the hyperboloid using coherent-state techniques, enabling reconstruction from samples and analyzing undersampling effects.

Contribution

It introduces a new sampling theorem for functions on the hyperboloid, including a reconstruction formula and conditions for partial recovery under undersampling.

Findings

Reconstruction formula with sinc-type kernel for bandlimited functions

Discrete Fourier transform from N samples on the hyperboloid

Approximation accuracy improves as N approaches infinity

Abstract

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 $N\to\infty$.