Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group pp 545-557 | Cite as

# Mean Periodicity on Phase Space and the Heisenberg Group

## Abstract

In the previous chapters the theory of mean periodicity on the translation groups ℝ^{ n }, *n**≥*1, 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 2*π*-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 *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.

## Keywords

Phase Space Periodic Function Uniqueness Theorem Heisenberg Group Nilpotent Group## Preview

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