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Spherical harmonics, the Weyl transform and the Fourier transform on the Heisenberg group. (English) Zbl 0596.46034
Let H be an unbounded operator defined on a dense subset of a Hilbert space such that \([H^*,H]=\lambda I\) where \(\lambda\) \(\neq 0\). The first part of this work is devoted to the definition and properties of an operator f(H) where f is a suitable function. Such an operator is called the Weyl correspondant of f and it is shown that for f an harmonic polynomial the definition of f(H) agrees with ”any reasonable definition” of a polynomial function of H. These results are then used in the computation of an exact formula for the Fourier transforms of a certain regular homogeneous distribution on the Heisenberg group and to the study of singular convolution operators on the Heisenberg group.
Reviewer: J.Lacroix

46F12 Integral transforms in distribution spaces
47A60 Functional calculus for linear operators
22E30 Analysis on real and complex Lie groups
43A30 Fourier and Fourier-Stieltjes transforms on nonabelian groups and on semigroups, etc.
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