Mean Periodicity on Phase Space and the Heisenberg Group

  • Valery V. VolchkovEmail author
  • Vitaly V. Volchkov
Part of the Springer Monographs in Mathematics book series (SMM)


In the previous chapters the theory of mean periodicity on the translation groups ℝ n , n1, was developed. The most natural generalization of the translation groups are nilpotent groups. The Heisenberg group H n is a principal model for nilpotent groups, and the results obtained for H n 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 H n which are 2π-periodic in the t variable. They arise as functions on reduced Heisenberg group H red n . The study of mean periodicity on H red n is reduced to the case of the phase space ℂ n by taking the Fourier transform in the t variable. The first section of the chapter contains some preliminary results concerning mean periodic functions on ℂ n . 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 ffT 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.


Phase Space Periodic Function Uniqueness Theorem Heisenberg Group Nilpotent Group 
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Copyright information

© Springer-Verlag London Limited 2009

Authors and Affiliations

  1. 1.Mathematical DepartmentDonetsk National UniversityDonetskUkraine

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