Davide Barbieri (Università Autonoma di Madrid)

We give an survey of some recent results concerning the structure of bases and frames generated by unitary group orbits in Hilbert spaces.
Invariant subspaces can be characterized, by means of Fourier intertwining operators, as modules whose rings of coefficients are given by group von Neumann algebras. It can be shown that these modules are naturally endowed with an unbounded operator-valued pairing which defines a noncommutative Hilbert structure. Roughly speaking, each orbit defines a point in such a Hilbert module, and the noncommutative pairing defines the analogous of a scalar product. Frames and bases obtained by countable families of orbits can then be characterized in terms of new notions of noncommutative reproducing systems, for which a full theory of linear expansions has been developed.
Motivations for this study come from problems in approximation theory, concerning group generalizations of wavelets and multiresolution analysis, and issues of regular sampling in shift-invariant spaces such as generalized Paley Wiener spaces.