This survey talk presents a conceptual framework, both a mindset and a collection
of mathematical tools, which facilitates the transition from discrete models to
continuous-variable formulations, as they arise naturally in physics and
engineering.

Many methods originating in finite-dimensional linear algebra, where they are
well understood and computationally accessible, are routinely transferred to
infinite-dimensional settings using tools from functional analysis. In this way,
Hilbert spaces, Lebesgue spaces such as \(L^2(\mathbb{R}^d)\) or
\(L^1(\mathbb{R}^d)\), and the theory of tempered distributions have become
standard tools in Fourier analysis, time-frequency analysis, abstract harmonic
analysis, and engineering applications.

The aim of this talk is to promote a slightly different perspective, inspired by
time-frequency analysis and, in particular, by Gabor analysis. This viewpoint is
also in the spirit of Paul Dirac's bra-ket formalism, emphasizing a flexible and
operational handling of signals and operators.

A central role is played by the space of \emph{mild distributions}, i.e.,
tempered distributions with uniformly bounded spectrogram. These objects provide
a mathematically robust yet intuitively appealing model for generalized signals
and linear systems. They form a Banach space, identified as the dual of the Segal
algebra \(S_0(G)\), also known as the Feichtinger algebra, over a locally compact
Abelian group \(G\).

Together with the Hilbert space \(L^2(G)\), this leads to the structure of a
\emph{Banach Gelfand Triple}
\[
S_0(G) \subset L^2(G) \subset S_0'(G).
\]
This triple may be compared with familiar hierarchies such as
\[
\mathbb{Q} \subset \mathbb{R} \subset \mathbb{C}.
\]

This framework is not only natural for Gabor analysis and the study of
pseudo-differential operators, but also provides useful insight into classical
questions such as summability theory and the theory of generalized stochastic
processes. One of the key structural facts is the existence of a kernel theorem,
which can be understood as an infinite-dimensional analogue of matrix
representations of linear mappings.

Overall, the talk offers a glimpse into what may be called \emph{conceptual
harmonic analysis}: an approach integrating abstract harmonic analysis with
computational and applied harmonic analysis, while maintaining close connections
to the needs of physics, engineering, and the applied sciences.