Frames for spaces of Paley-Wiener functions on Riemannian manifolds

TL;DR

This paper develops a stable reconstruction method for Paley-Wiener functions on Riemannian manifolds using frames, generalizing classical exponential frame results and applying to hyperbolic space.

Contribution

It introduces a new frame-based reconstruction approach for Paley-Wiener functions on manifolds, extending classical results to curved spaces.

Findings

Stable reconstruction from inner products with compactly supported distributions.

Generalization of Duffin-Schaeffer exponential frames to Riemannian manifolds.

Application to hyperbolic space in Poincare upper half-plane model.

Abstract

It is shown that Paley-Wiener functions on Riemannian manifolds of bounded geometry can be reconstructed in a stable way from some countable sets of their inner products with certain distributions of compact support. A reconstruction method in terms of frames is given which is a generalization of the classical result of Duffin-Schaeffer about exponential frames on intervals. All results are specified in the case of the two-dimensional hyperbolic space in its Poincare upper half-plane realization.