The area of Fourier analysis connected to signal processing theory has undergone a rapid development in the last two decades. The aspect of this development that has received the most publicity is the theory of wavelets and their relatives, which involves expansions in terms of sets of functions generated from a single function by translations and dilations. However, there has also been much progress in the related area known as time-frequency analysis or Gabor analysis, which involves expansions in terms of sets of functions generated from a single function by translations and modulations. In this area there are some questions of a concrete and practical nature whose study reveals connections with aspects of harmonic and functional analysis that were previously considered quite pure and perhaps rather exotic. In this expository paper, I give a survey of some of these interactions between the abstruse and the applicable. It is based on the thematic lectures which I gave at the Ninth Discussion Meeting on Harmonic Analysis at the Harish-Chandra Research Institute in Allahabad in October 2005.
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