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Fourier transform of Schwartz functions on the Heisenberg group. (English) Zbl 1277.43012
The authors consider the Schwartz space $$\mathcal S(H_1)$$ on the (three-dimensional) Heisenberg group $$H_1$$. They characterise $$\mathcal S (H_1)$$ via sequences of Schwartz functions defined on $$\mathbb R^2$$. The definition of these sequences relies on the decomposition given by the representation theory of the group $$H_1$$ (Plancherel formula) as well as certain special features of the action of the torus $$\mathbb T_1$$ on functions of $$H_1$$. This generalises the results in [J. Funct. Anal. 251, No. 2, 772–791 (2007; Zbl 1128.43009); ibid. 256, No. 5, 1565–1587 (2009; Zbl 1167.43008)] by the same authors in the case of $$\mathbb T_1$$-invariant functions in $$\mathcal S(H_1)$$. The proofs are based on the results in [loc. cit.] as well as Whitney’s extension properties and a deep understanding of the Gelfand spectrum in this context.

##### MSC:
 43A80 Analysis on other specific Lie groups 22E25 Nilpotent and solvable Lie groups
##### Keywords:
Fourier transform; Schwartz space; Heisenberg group
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