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Continuité de la transformation de Fourier nilpotente. (Continuity of the nilpotent Fourier transform). (French) Zbl 0643.43003
Let \({\mathfrak g}\) be the Lie algebra of a real connected simply connected nilpotent Lie group G and let \({\mathfrak g}^*\) be the dual of \({\mathfrak g}\). In [Ann. Phys. 111, 61-110, 111-151 (1978; Zbl 0377.53024 and Zbl 0377.53025.] F. Bayen et. al defined a map \(\theta\) from the space S(G) of rapidly decreasing \(C^{\infty}\) functions on G to the space of rapidly decreasing \(C^{\infty}\) functions on a dense subset of \({\mathfrak g}^*\) and called the map the nilpotent Fourier transform. In this paper the author shows that \(\theta\) is a continuous map in the usual topologies.
Reviewer: S.Sankaran

43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
22E25 Nilpotent and solvable Lie groups
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
70G10 Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics