There is no doubt that the Fourier transform is one of the crucial methods for electrical engineering, as it provides the basis for systems theory (using impulse response and transfer function) or digital signal processing (sampling, periodization, etc.). Continuous signals are approximated by discrete ones and vice versa (Dirac sequences approximate the Dirac Delta).

In this talk the author will describe a setting which allows to give a mathematically correct but still practically useful approach to a (limited) theory of distributions. The approach is based on methods from time-frequency methods. Within the theory of tempered distributions one could characterize the ``mild distributions'' as those which have a uniformly bounded spectrogram (e.g. with respect to a Gaussian window). Distributional convergence corresponds then simply to uniform convergence of spectrograms over compact subsets of the time-frequency plane (resp. of phase spaces).

We will shortly describe the setting, which requires only basic results from linear functional analysis (but no Lebesgue integration theory of topological vector spaces, just Riemann integrals). In particular, we can indicate how a new syllabus for teaching these foundations of application oriented Fourier Analysis to engineers or physists could look like. These suggestions are based on teaching experience of the author at various international universities.