In a celebrated paper from over 30 years ago, Donoho and Stark proved that the uncertainty principle can be used to show that a signal for which some of the frequencies remain unobserved can be recovered exactly provided that the number of missing frequencies is sufficiently small and the signal is sufficiently sparse. We are going to investigate this problem from the point of view of restriction theory, prove several associated uncertainty principles, and explore some simple signal recovery mechanisms. This is joint work with Azita Mayeli.