A Universal Sampling Method for Reconstructing Signals with Simple Fourier Transforms

Reconstructing continuous signals based on a small number of discrete samples is a fundamental problem across science and engineering. We are often interested in signals with "simple" Fourier structure - e.g., those involving frequencies within a bounded range, a small number of frequencies, or a few blocks of frequencies i.e., bandlimited, sparse, and multiband signals, respectively. More broadly, any prior knowledge on a signal's Fourier power spectrum can constrain its complexity. Intuitively, signals with more highly constrained Fourier structure require fewer samples to reconstruct.