The uncertainty principle, restriction theory, signal recovery and sampling on manifolds

University Of Rochester, Rochester NY

Investigators

Abstract

In this project, the PI will investigate the problem of recovering the missing values of a signal using effective and mathematically justified algorithms. The PI will investigate both the discrete case, which has potential applications to the imputation of missing values in data science, and the continuous case, with potential applications in fire detection and image reconstruction. The discrete mechanisms studied in the project lend themselves to numerical investigations where undergraduate students play an important role, contributing to the wide dissemination of the underlying ideas and training of the next generation of mathematicians. In this project, the PI will study a broad-based approach to Fourier restriction that seeks to unify results in discrete, Euclidean, and Riemannian manifold settings. We begin by proposing a restriction theory-based approach to Fourier uncertainty signal recovery in a discrete setting. The PI will employ variants of Bourgain's Lambda(q) theorem to cover the cases that are currently out of reach. This will lead the PI naturally to the study of annihilating pair inequalities where the ideas in the project lead to results in discrete, continuous, and manifold settings. The PI will engage in a systematic study of the uncertainty principle on Riemannian manifolds where we circle back to the probabilistic bounds that arise in the discrete signal recovery part of the proposal. The uncertainty principle on manifolds leads the PI to study random sampling on manifolds where the strong unique continuation property of the Laplace-Beltrami operator plays the key role. The question of the stability of the proposed sampling result leads the PI to study the smallest singular values of the resulting matrices, which have natural applications to the signal recovery part of the proposal. Finally, the PI will develop a comprehensive approach to the spectral synthesis problem previously studied by Agmon, Agranovsky, Hormander, Narayanan, and others. This approach is based on restriction theory ideas and leads to a natural signal recovery question where properties of the classical Helmholtz equation play the key role. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.