Abstract Harmonic Analysis deals with Fourier expansions of functions in terms of pure frequencies, which are the natural eigenvectors of the corresponding translation operators. At an operator level a similar situation appears in the context of the Wigner-Weyl calculus relevant for pseudo-differential operators or the treatment of time-variant linear systems. Here the building blocks are the time-frequency shifts which allow to represent Hilbert-Schmidt operators in a similar way (using building blocks which are outside the given Hilbert space).
These examples show already the short-comings of the pure Hilbert spaces setting.

The purpose of this talk is to indicate how the popular notion of a ``continuous orthonormal basis'' which is often used in a physics or engineering context for the family of Dirac measures, can be put on solid mathematical concepts, making use of the Banach Gelfand Triple $(\SOsp,\Lsp_2,\SOPsp)$. In the background of these considerations stands the characterization of these spaces using Gabor expansions, with coefficients in $(\ell^1,\ell^2,\ell^\infty)$, and the fact that $\SOPsp$, the space of mild distributions, can be viewed as a good model for ``signals'' and ``operators'', in analogy two finite vectors and matrix representations of linear mappings in linear algebra.
Using such a view-point one can consider the (fractional) Fourier transform as a change of bases. In analogy with the DFT (discrete Fourier transform), where the rows and columns contain the (Fourier) basis vectors similar considerations apply for the corresponding continuous kernels describing these integral transforms.