For most purposes one may view SIGNALS as (not clearly defined) objects which depend on a parameter, such as time, or location, or even coordinates for a three dimensional space. Just think of audio signals, pictures, or the temperature in a closed room. We can record the audio signal (maybe compress it via MP3), and we can compute the spectrogram (STFT) in real time. We can take pixel images of a scene or measure the room temperature along an array of thermometers in the room. But we could also think of the task of synthesising a
three-dimensional sound field using an array of load-speakers. For all this the theory of $L^p$-spaces does not really help, nor a simplistic derivation of translation invariant linear operators as convolution operators, using the impulse response of the system, or the description as a Fourier multiplier, using the transfer function model for $L^1$-functions. Here and in other places the mysterious Dirac Delta is appearing, usually with a hint to the theory of tempered distributions developed by Laurent Schwartz.

The thesis of this talk is the claim that the best model to-date for general signals is that of mild distributions, i.e. the linear space of objects which have a bounded STFT. They also have a naturally defined Fourier transform. One just has to establish clear rules how to handle these objects, and how to define convergence. In the background is the concept of Banach Gelfand Triples.
We will explain how this leads to the identification of the signal space with the dual of Feichtinger's algebra $\SORd$. The correct view on this is to consider the elements of $\SORd$ as the collection of all reasonable linear measurements! In a similar way the kernel theorem arising in this context provides the insight that general linear operators, namely operators which assign to each test function $f \in \SORd$ some signal $T(f)$ can be described by a kernel, which is simply a mild distribution of $2d$ variables,
in the form of an abstract integral operator. Such operators also have a spreading function and a Kohn-Nirenberg symbol. Furthermore one can show that the Wigner distribution of a signal is a well-defined mild distribution. Various examples of this situation will be listed, emphasizing that all this can be developed without use of either $\L^1(R^d)$ or even $L^2(R^d)$.

Finally the approximation if signals by finite vectors (typically fine regular samples of a continuous function over a sufficiently long interval) will be indicated, and how discretizations (e.g. of the Wigner distribution) can be obtained. Such questions are hard to understand from the point of view of abstract harmonic analysis, but also from a purely computational approach. But isn't a high-resolution TV-set providing a good approximation of \newline images arising in the real world?