This is a presentation to a general audience of undergraduate students at the Dept. of Mathematics, University of Edinburgh.

Given a recorded piece of music it is easy to imagine, to listen to the melody, i.e. the change of frequencies over time. When we try to teach the computer to do this we need the (fast) Fourier transform. Local Fourier Analysis (also called Gabor Analysis) is the mathematical theory for such ``non-orthogonal representations of vectors/signals''. While thought to be necessarily computational intensive and instable, modern mathematics has revealed a very nice underlying structure that allows for efficient implementations. Such things are at the basis of CD recording or MP3 players, but similar methods can be used to desribe the transmittion of signals between the basis station and your mobile phone. As time permits simple examples/applications from image analysis will be provided.