The goal of this talk is to point out how some of the soft and heuristic considerations found in engineering books and courses can be described in a mathematically precise way, based on methods from linear {\it functional analysis} and {\it approximation theory}. While the correct term is that of $w^*$-convergence in dual Banach spaces, I will present these concepts in the more concrete setting of so-called {\it mild distributions} over $R^d$.

Together with the Segal algebra $S_0(R^d)$ and the Hilbert space $L^2(R^d)$ the mild distributions form the so-called {\it Banach Gelfand Triple} $(S_0,L^2,S'_0)$, often compared with the number system $(Q,R,C)$, of rational, real and complex numbers. The natural concept of convergence in $S'_0$, in addition to the usual norm convergence, is in fact the $w^*$-convergence, which can be expressed equivalently as the uniform convergence of the spectrograms of such distributions over any compact subset of the {\it time-frequency plane}. Alternatively, one can describe it as the norm convergence for any of the projection onto a finite-dimensional subspace of $S'_0$.

We will illustrate the usefulness of this approach (THE Banach Gelfand Triple, and ``mild convergence'' as I call it now)
for the description of transitions between the different worlds of signals discussed in the engineering world (periodic versus non-periodic, discrete versus continuous, etc.).

Classical Fourier Theory starts from integrable function on the torus, and then extends this notation to functions on the real line or $R^d$ (using Lebesgue integration). Numerical computation of the Fourier transform of a decent function is typically realized with the help of the FFT-algorithm, applied to suitable sampling values of the given function. Finally, an
elegant proof of the Shannon-Sampling Theorem can be given using the fact that the standard Dirac comb on the real line is invariant under the distributional Fourier transform, in the sense of mild distributions. In each of these situations, it makes a lot of sense to ask for the mutual dependence of the different methods, and {\it mild convergence} (another word for distributional convergence) appears as the most suitable unifying concept allowing good answers
to those questions.