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 rando