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 functional analysis and 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 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. A natural concept of convergence in $S'_0$ 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 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.