A Reconstruction Method for Band-Limited Signals on the Hyperbolic Plane

TL;DR

This paper introduces a method to reconstruct band-limited functions on the hyperbolic plane using an iterative algorithm based on the Helgason-Fourier transform, ensuring convergence with dense sampling.

Contribution

It presents a novel reconstruction algorithm for band-limited functions on the hyperbolic plane leveraging Helgason-Fourier analysis, with proven convergence guarantees.

Findings

Convergence rate is geometric for dense sampling.

Reconstruction is possible from countable sample sets.

Sampling density is critical for accurate reconstruction.

Abstract

A notion of band limited functions is considered in the case of the hyperbolic plane in its Poincare upper half-plane $\mathbb{H}$ realization. The concept of band-limitedness is based on the existence of the Helgason-Fourier transform on $\mathbb{H}$. An iterative algorithm is presented, which allows to reconstruct band-limited functions from some countable sets of their values. It is shown that for sufficiently dense metric lattices a geometric rate of convergence can be guaranteed as long as the sampling density is high enough compared to the band-width of the sampled function.