It is a common method used in many engineering disciplines or in physics to the consider the collection of Dirac measures $\delta_x, x in R^d$ as a ``continuous basis'', e.g. by starting the discussion of time-invariant systems from the ``sifting property of the Dirac delta'', which could be written as
$$f = \delta_0 \ast f [ = T_0 f = Id(f)]. $$

Although sometimes quite intuitive, at least in the context of (continuous) functions which are well described by their pointwise values, the existence of values is almost lost for the Hilbert space $L^2(R^d)$ and completely in the realm of (mild) distributions. Nevertheless there is a kernel theorem for operators from $S_0(R^d)$ to the dual space $S_0^*(R^d)$ of mild distributions, which can be viewed as the representations of the operator, using on both sides (the domain and the target space) the Dirac basis.

The basic properties of the Banach Gelfand triple $(S_0,L_2, S^*_0)(R^d)$ allow to discuss other bases (e.g. by making use of the concept of Fractional Fourier transform) and thus initiate the discussion about the similarity (in the sense of linear algebra) of different representations of this type, which we call "mild bases" (by abuse of language), or "Dirac-like" bases. In this way we can shed some light on what we call the Kirkwood-Dirac Principle
(KDP), which seems to get large attention in the quantum community recently.