Starting from the finite-dimensional situation, very well described by the
linear algebra interpretation of Paul Dirac's bra-ket notation,
it is natural to move first to the setting of Hilbert spaces such as
$ L^2(\mathbb{R}^d)$.
However, already the analysis of pointwise multiplication operators shows
that bounded operators may fail to possess genuine eigenvectors;
instead one is naturally led to objects such as Dirac measures.


This observation motivated the introduction of rigged Hilbert spaces,
typically based on the Schwartz Gelfand triple
$ (\mathcal{S}(\mathbb{R}^d),
L^2(\mathbb{R}^d),
\mathcal{S}'(\mathbb{R}^d)). $
Tempered distributions play a central role in this framework and are widely
used in mathematical physics and engineering, for example in sampling theory
(Dirac combs), in the description of linear operators through distributional
kernels, and in phase-space approaches such as the
Wigner--Weyl, Kohn--Nirenberg, or Kirkwood--Dirac calculi.

The purpose of this talk is to demonstrate that many of these concepts
can be treated within a technically simpler and often more flexible framework
arising from Gabor and time-frequency analysis.
Instead of relying on sophisticated topological vector spaces,
the approach is based on relatively elementary Banach-space methods.

The Banach Gelfand Triple
$ (S_0(\mathbb{R}^d), L^2(\mathbb{R}^d), S_0'(\mathbb{R}^d)), $
consisting of the Fourier invariant Feichtinger algebra $S_0(\mathbb{R}^d)$,
the Hilbert space $L^2(\mathbb{R}^d)$,
and the dual space $S_0'(\mathbb{R}^d)$ of mild distributions, provides a valid alternative.
These spaces admit natural descriptions in terms of Gabor expansions, using coefficients
in $(\ell^1,\ell^2,\ell^\infty)(Z^{2d})$ respectively.
A typical setting is the use time-frequency shifted
Gaussian atoms along a lattice of the form $aZ^{2d}$
for some/any $a < 1$.
With the help of Wilson bases one even obtains a unitary isomorphism
with the sequence-space triple $(\ell^1,\ell^2,\ell^\infty)$.


Within this framework many concepts from physics and engineering, including
Dirac bases, impulse responses, transfer functions, and kernel representations,
can be formulated rigorously while preserving
a close analogy with the finite-dimensional setting.
The resulting viewpoint provides a natural bridge between
harmonic analysis, mathematical physics, and engineering applications.

The setting also opens up the way to replace the
term ``continuous ONB'' for the Fourier basis by an
appropriate setting using Banach frames. There is also
a kernel theorem, allowing to strengthen the analogy
to matrix analysis of linear mappings in this context.
This also opens up the way to structure preserving approximations,
of interest for efficient numerical implementations.