Time-Frequency Localization and Sampling of Multiband Signals

This paper begins with a review of some classical work of Landau, Slepian, Pollak and Widom concerning essentially time- and bandlimited signals and ends reviewing some recent work of Candes, Romberg and Tao that places specific but probabilistic limitations on essential time-and bandlimiting for finite signals and their discrete Fourier transforms. In between we outline some conceptual bridges from the continuous, singleband setting to the finite, multiband setting and pose a number of open problems whose solutions would solidify the connections outlined. These connections involve time-frequency localization of multiband signals and sampling theory for such signals.