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In signal processing and pure mathematics it is useful to
represent complicated signals or functions  in
terms of linear combinations of simpler building blocks  ; the
classical case, where is
assumed to be an orthonormal basis for a Hilbert
space  to
which the relevant signals belong, leads to the representation  .
However, the assumption that forms
an ONB is quite restrictive, and it limits the other properties one can
expect from .
We discuss concrete cases from time-frequency analysis, where desirable
properties of can
not be combined with being
an ONB. It turns out that much more freedom can be attained by assuming
that is
a frame rather than an ONB; however, until recently, the concrete
applications of frames have been limited by certain computational
difficulties. In the talk we discuss recent constructions of frames
that overcome these difficulties, with particular focus on Gabor frames and
wavelet frames.
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