Gabor Analysis over Quasicrystals meets Operator Algebras

Petter Nyland (NTNU Trondheim)

Thursday 30th January 16:00-17:00 Maths 311B

Abstract

Two fundamental families of operators on the Hilbert space of square-integrable functions are time shifts and frequency shifts. These are the central players in time-frequency analysis. The subfield of Gabor analysis is concerned with frames that arise from time-frequency shifts of a fixed window function, so-called Gabor frames. Typically, these time-frequency shifts range over a lattice in the time-frequency plane. Some 10 years ago, F. Luef connected the work of M. Rieffel from the 80’s on non-commutative tori to Gabor frames over lattices. Thus providing a bridge between operator algebras and time-frequency analysis, which has yielded new results in both fields. Recently, results on Gabor frames over more general point-sets than lattices, such as quasicrystals, have appeared. It is then natural to ask whether one can find a similar connection to operator algebras in this more general setting. I will give a selective overview of this theory, and try to indicate what is different when we pass from lattices to quasicrystals.

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