The Use of Representations in Applied Harmonic Analysis

The role of unitary group representations in applied mathematics is manifold and has been frequently pointed out and exploited. In this chapter, we first review the basic notions and constructs of Lie theory and then present the main features of some of the most useful unitary representations, such as the wavelet representation of the affine group, the Schrödinger representation of the Heisenberg group, and the metaplectic representation. The emphasis is on reproducing formulae. In the last section we discuss a promising class of unitary representations arising by restricting the metaplectic representation to triangular subgroups of the symplectic group. This class includes many known important examples, like the shearlet representation, and others that have not been looked at from the point of view of possible applications, like the so-called Schrödingerlets.