Taking an axiomatic viewpoint mathematicians avoid to define what
certain ``{\it objects are}''. Instead one starts from certain relations and structures, typically algebraic or analytic, which allow to perform certain computations or carry out some reasoning.


We all know the classical examples: The field of real numbers is made up in such a way that the classical manipulations known for integers or rational numbers, can be performed, with the additional observation that it is also complete with respect to the Euclidean metric. Once we have formulated such axioms we need not (but of course we can) case about the existence of such objects,
e.g. by introducing infinite decimal expressions and their computational rules, but this is just one {\it model}, a completely different one is obtained by Dedekind sections. Of course two such models are isomorphic in a natural sense.

Another well known example is the theory of {\it finite dimensional vector spaces}, as is developed in a typical linear algebra course. Aside from additional properties valid only for concrete examples one does not care, whether the objects are column or row vectors, or polynomials or function inside some infinite dimensional vector space. The common structure can be described using finite bases, and matrices characterizing general {\it linear mappings} using such basis on both the domain and the target space.

In this sense we want to propose a heuristic approach to the world of
signals and operators. At the end we will suggest that so-called
{\it mild distributions}, or the elements of the dual of Feichtinger's algebra are a suitable setting for signals.

The basic idea is that signals are objects (not to be specified here!) which can be ``tested'' by suitable measurements. Although a simplified approach models a signal as a continuous functions, which allows point evaluations as simple measurements, more complex situations are more realistic. Think of the temperature at a given location as a function of time, obtained by placing a thermometer for a little while at a concrete spot and wait for the display to stabilize. This is a typical measurement.
Another very natural object is a sound signal, which can be recorded with the help of a microphone (using your smart-phone or notebook). The digital data obtained in this way are not the exact value of what (?), the air pressure at the microphone, but still one can insert the resulting signal (ideally stored on a CD at the rate of 44100 Hz) into a mathematical software program and analyze the signal in this way.

The underlying reasoning (explained in \citeX{fe24-6} in more detail)
is the assumptions, that both signals and measurements form linear spaces. In fact, the sum are a scalar multiple of a given signal (by amplification) should be signal as well. But also the average of certain (we assume now {\it linear}!) measurements or other linear combinations describe linear mappings defined on the vector space of all signals, thus providing a bilinear mapping defined on the product space, generated from signals and (admissible) measurements. In addition, this setting provides a very natural notion of similarity of signals: Given a finite set $F$ of measurements
and some $\varepsilon > 0$ we say that two signals are close in the
$(F,\varepsilon)$-sense if the difference for each of the measurements of the two signals is at most $\varepsilon$, for all measurements from $F$.

It is plausible that such bilinear (and continuous) pairings can be
realized using spaces of test functions (as measurements) and generalized functions (the dual space) as space of signals. In the concrete scenario we will use the Segal algebra $\SORdN$ (known as Feichtinger's algebra) as the Fourier invariant Banach algebra of test functions, and the dual space, the space of mild distributions, as the space of signals. $\SORdN$ can also be characterized as the subspace of the Schwartz space $\ScPRd$ of tempered distributions with a bounded STFT (short-time Fourier transform).
In fact, for a general $\sigma \in \ScPRd$ and any (real-valued) Schwartz function $g \in \ScRd$ (say a Gaussian), we have
$$ V_g(\sigma)(t,s) = \sigma(exp(2\pi i s \cdot) \cdot g( \cdot - t)), \quad (t,s) \in \TFd. $$

It is also natural to view the ``most general'' linear operators as
those mapping $\SORdN$ into $\SOPRdN$ (in a continuous way). For such
operators we have a so-called kernel theorem: They can be described
as limits of a bounded sequence of integral kernels $K_n(x,y)$ in the
sense of $\SOPsp(\Rtdst)$, where each $K_n$ is represented by a bounded and continuous function. This is a very useful analogue of the Schwartz kernel theorem characterizing continuous linear mappings from $\ScRd$ into $\ScPRd$, by ``kernels'' $K \in \ScPsp(\Rtdst)$.