# Unique characterization of the Fourier transform and generalized transforms

Presenter: **Roy Oste**, Ghent University, Belgium

Joint work with **Hendrik De Bie, Joris Van der Jeugt**

The Fourier transform (FT) is of crucial importance in a whole range of
areas such as harmonic analysis and signal processing as it has many
interesting properties.
A natural question is: Which properties are sufficient to uniquely
characterize the FT?
Additionally, given a specific set of properties of the FT, one can
inquire whether there are any other transforms that satisfy these
properties.
In order to answer these questions, we work in the framework of
representation theory of the Lie algebra $\mathfrak{sl}_2$, and its
refinement the Lie superalgebra $\mathfrak{osp}(1|2)$.
This natural generalization brings us to the setting of Clifford
analysis, where we obtain generalized Fourier transforms that act on
functions taking values in a Clifford algebra.