Reconstruction of Paley-Wiener functions on the Heisenberg group

TL;DR

This paper generalizes Paley-Wiener functions to Riemannian manifolds with bounded geometry, providing a reconstruction formula from discrete samples and extending classical Fourier analysis concepts.

Contribution

It introduces a new framework for Paley-Wiener functions on manifolds and generalizes the Whittaker-Shannon sampling formula to this setting.

Findings

Paley-Wiener functions are uniquely determined by discrete samples.

A reconstruction formula from discrete data is established.

The results extend classical Fourier analysis to Riemannian manifolds.

Abstract

Let $M$ be a Riemmanian manifold with bounded geometry. We consider a generalization of Paley-Wiener functions and Lagrangian splines on $M$. An analog of the Paley-Wiener theorem is given. We also show that every Paley-Wiener function on a manifold is uniquely determined by its values on some discrete sets of points. The main result of the paper is a generalization of the Whittaker-Shannon formula for reconstruction of a Paley-Wiener function from its values on a discrete set. It is shown that every Paley- Wiener function on $M$ is a limit of some linear combinations of fundamental solutions of the powers of the Laplace-Beltrami operator. The result is new even in the one-dimentional case.