Abstract: One of the basic questions in classical signal recovery asks for a set of reasonable conditions under which a signal f: {\mathbb Z}_N^d \to {\mathbb C}, sent via its Fourier transform with frequencies {\{\widehat{f}(m)\}}_{m \in S} missing, can be recovered exactly and uniquely. Matolcsi and Szucs (1973) and Donoho and Stark (1989) laid a mathematical foundation for this problem and showed its fundamental connection with the Fourier uncertainty principle. In this talk, we are going to discuss a variety of aspects of this wonderful problem, an approach using classical restriction theory and Rudin/Bourgain theory of \Lambda_p sets, and some analytic underpinnings of these theories. The interplay between discrete, continuous, probabilistic, and arithmetic methods will be emphasized throughout.