The Heisenberg Group: A Different Point of View

Abstract

Time-frequency shifts are the fundamental operators in time-frequency analysis. In this chapter we change the point of view and consider the collection of time-frequency shifts {TxMw: (x,w)ℝ2d} as an object of independent interest. The Heisenberg group and its representation theory emerge as the underlying structure. The existence of a non-commutative group as the background of time-frequency analysis is the very reason for the rich and beautiful mathematical structures and the wide arsenal of methods that appear in time-frequency analysis. The Heisenberg group adds new insights even to the classical harmonic analysis on ℝd, as is shown in the enlightening article “On the role of the Heisenberg group in harmonic analysis” by R. Howe [152]. In time-frequency analysis the Heisenberg group is omnipresent, and it is not an exaggeration to claim that time-frequency analysis is an aspect of the theory of the Heisenberg group. An exposition of time-frequency analysis cannot be considered complete without a brief excursion into the non-commutative harmonic analysis of the Heisenberg group. This chapter offers a pedestrian approach to some of its aspects. As a reward we will not only gain a deeper insight into the mathematical foundations of time-frequency analysis but also new results about Gabor frames. For a more thorough treatment of the Heisenberg group the reader should consult [104] or [238].KeywordsUnitary RepresentationHeisenberg GroupRiesz BasisIrreducible Unitary RepresentationGabor FrameThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.