  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 \emph{time-frequency
analysis} or \emph{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.
