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.