One of the basic problems of exact signal recovery is to consider a signal $f: {\mathbb Z}_N^d \to {\mathbb C}$ and transmit its Fourier transform such that the frequencies ${\{\widehat{f}(m)\}}_{m \in S}$ are missing. The question is, under what reasonable assumptions can the original signal be reconstructed exactly and uniquely? We will discuss the role of Fourier restriction in this problem, as well as applications to thoroughly practical problems like the imputation of missing values in time series.