Codes and Expansions (CodEx) Seminar
Hartmut Führ (RWTH Aachen)
Frames of translates and the bandwidth intuition
Frames of translates have been the object of study in various different contexts. They arise by letting suitable subsets of a locally compact group act on one or several "generators", i.e., functions in the \(L^2\)-space of the group. One of the basic questions in this area is which subspaces of this \(L^2\)-space have frames of translates, and what sort of requirements the existence incurs on the set(s) of translations employed in the construction of the frame. The fundamental example in this respect is provided by the Shannon sampling theorem for bandlimited functions on the reals, where there is a well-understood and rather rigid inverse proportionality between bandwidth (length of Fourier support) on the one hand and sufficient and/or necessary density conditions for the sets of translates generating the frame.
This talk studies different versions of the problem and present instances where the bandwidth intuition is essentially valid, as well as examples showing where it is not. We consider the general settings of generalized shift-invariant systems over locally compact abelian groups, and frames of translates over non-abelian groups.
The talk is based on joint results with Jakob Lemvig and Vignon Oussa.