In contrast to Abstract Harmonic Analysis which describes signals defined over locally compact groups and makes use of the corresponding Fourier decomposition the treatment of actual signals (audio-recordings, images, movies for streaming) rather views these signals as a collection of localized objects which are provided ``up to some resolution''. The MP3-coding scheme is a good example. It provides full information about a piece of music for the duration of a song, up to 20 kHz, by storing the relevant information for segments of length 512 samples (at the sampling rate of 44100, for HiFi recordings). This extension is different from the approach to a generalized Fourier transform using the Schwartz theory of tempered distributions.

The mathematical theory of mild distributions arose from investigations in time-frequency, specifically Gabor Analysis, i.e. the attempt to put D. Gabor's idea from 1946, claiming that any signal can be described as a double series of time-frequency shifted Gaussians, on solid mathematical grounds. As it turned out, the Segal algebra SO(R^d) (the Feichtinger algebra) is the right object. 

The talk will describe the setting, how it can be used efficiently for the mathematical description of basic problems in engineering. The thesis is simply that ``mild distributions'' (elements of the dual of SO) are the perfect model for signals, with their evaluation on test functions in SO being the possible measurements. Correspondingly all relevant operators have a continuous matrix representation using mild distributions of 2 variables (the kernel theorem in this setting). 

By using appropriate concepts of mild convergence (appearing naturally) one can turn heuristic arguments into more conceptual considerations and derive the form of special version of the Fourier transform (e.g. the classical variant for periodic functions, or the DFT/FFT variant for the case of finite signals) into natural variants of one general scheme (the extended FT for mild distributions).