When it comes to the use of the family of Dirac measures on the real line of the Euclidean space R^d physicist and engineers often use weird formulas, involving divergent integrals, which are often manipulated in a formal way, based on the analogy to the finite discrete case, where such manipulations are justified within linear algebra.

The talk is going to describe some ongoing discussion which aims at a mathematically sound description of such formal manipulations in the context of mild distributions. Mild distributions form together with the Hilbert space L2(R^d) and the underlying Banach algebra S_0(R^d) of test functions (the Feichtinger algebra) a so-called Banach Gelfand triple or rigged Hilbert space, comparable with the Schwartz setting leading to (the much larger space of) tempered distributions.

Using this setting the claim that the (continuous) Fourier transform is nothing else but a change of basis, moving from the Dirac basis to the (equivalent) Fourier basis, makes sense and can be very well justified.