In this talk we would like to report on progress on a program towards rebuilding parts of Harmonic Analysis (HA) from scratch, based on methods from elementary linear functional analysis (FA), namely the concepts of Banach spaces and their duals, bounded linear operators and strong convergence of operators. After all, for me, HA has not only been at the cradle of FA, and almost has given birth to it a century ago, but a big junk of HA (including applications to engineering and physics) is concerned with the boundedness of certain linear operators between suitable Banach spaces (of functions or distributions). These are typically infinite dimensional vector spaces, which have to be equipped with suitable (families of semi-) norms in order to allow for infinite sums (series) and convergence in a proper sense.

The historical development of the subject of Fourier Analysis over the last 200 years has led to quite involved concepts, up to the point where experts in pure mathematics derive ``abstract'' results which are so complicated that even they themselves are not able (or willing?) to apply these results to concrete examples. But who else should take the pain of digesting such complicated results which are sometimes derived with very little motivation beyond the point that one gains ``insight at the general level'' and obtains ``answer to questions that nobody would pose unless he/she can provide a complicated and involved answer''. Right, the community is used to many of the complicated concepts and it appears unavoidable that young people have to learn all these techniques in order to make progress and publish a paper. But on the other hand there are so many easy and natural questions which are not covered by the standard mathematical literature, which would help engineers or other applied scientist to make better use of mathematical methods, be they at the structural level (e.g. group theoretical interpretations) or in the mathematical description. Just think of the manifold ``mystifying'' descriptions of the Dirac Delta in articles and even standard introductory books on applied Fourier Analysis or Systems Theory, which leave those poor students with the impression that ``Mathematics is just another form of Black Magic''.

There is another, widespread aspect for the development of mathematical theory: The more we know about a subject the more we identify the most efficient notions, and the better we can describe (and use) mathematical insight. Just think of the famous formula $ exp(2 \pi i) = 1$. Good notations may foster the usability of concepts that appear at first sight esoteric (think of $L^1(R^d)$, based on the Lebesgue integral).

The goal of this presentation will be to illustrate (only) certain aspects of a general idea, called Conceptual Harmonic Analysis (CHA), which aims at rebuilding major parts of HA, with an orientation towards usefulness for the applied sciences, but still with correct and simple mathematical methods (from FA). In its final stage, this concept should provide a basis for developing good algorithms providing quantitative results complementing abstract estimates. These tools should also help to teach the subject properly to young mathematicians and to applied scientists, giving them a solid basis for their research.

More concretely, the methods used to obtain from a strongly continuous representation of a locally compact Abelian group $G$ on a Banach space to the ``integrated group representation'' (i.e. the linearization of the group representation), and the ideas around Banach Gelfand Triples which appear to be relevant here will be addressed. At the inner level often it is a valid idea to replace sums by integrals (in the Riemann sense), just think of the kernel theorem. The expansion to the outer layers then is realized by taking suitable limits. In this way, standard tools (like convolution, Fourier transform, etc.) can be transferred to a rather general setting (e.g. LCA groups). A first outline of this program (which is under further development) has been given
in my course at ETH in the autumn of 2020
(see www.nuhag.eu/ETH20 for the links).