Mean Periodicity on Phase Space and the Heisenberg Group

Abstract

AbstractIn the previous chapters the theory of mean periodicity on the translation groups ℝn, n≥1, was developed. The most natural generalization of the translation groups are nilpotent groups. The Heisenberg group Hn is a principal model for nilpotent groups, and the results obtained for Hn may suggest results that hold more generally for this important class of Lie groups. In the case of the Heisenberg group it is very hard to study mean periodicity for functions of arbitrary growth. On the other hand, one can obtain interesting results for functions satisfying certain growth conditions. The main point of the chapter is to consider functions on Hn which are 2π-periodic in the t variable. They arise as functions on reduced Heisenberg group H nred . The study of mean periodicity on H nred is reduced to the case of the phase space ℂn by taking the Fourier transform in the t variable. The first section of the chapter contains some preliminary results concerning mean periodic functions on ℂn. The second section is devoted to phase-space analogues of John’s uniqueness theorem and related questions. In the last section the kernel of the twisted convolution operator f→f⋆T is studied. In particular, it is shown that, for a broad class of distributions T, any smooth function in the kernel has an expansion in terms of eigenfunctions of the special Hermite operator \(\mathfrak{L}\) satisfying the same convolution equation.KeywordsPhase SpacePeriodic FunctionUniqueness TheoremHeisenberg GroupNilpotent GroupThese 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.