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The Helgason Fourier transform on a class of nonsymmetric harmonic spaces

Published online by Cambridge University Press:  17 April 2009

Francesca Astengo
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca Degli Abruzzi, 24 10129, TorinoItaly
Roberto Camporesi
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca Degli Abruzzi, 24 10129, TorinoItaly
Bianca Di Blasio
Affiliation:
Dipartimento di Matematica, Politecnico di Torino, Corso Duca Degli Abruzzi, 24 10129, TorinoItaly
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Given a group N of Heisenberg type, we consider a one-dimensional solvable extension NA of N, equipped with the natural left-invariant Riemannian metric, which makes NA a harmonic (not necessarily symmetric) manifold. We define a Fourier transform for compactly supported smooth functions on NA, which, when NA is a symmetric space of rank one, reduces to the Helgason Fourier transform. The corresponding inversion formula and Plancherel Theorem are obtained. For radial functions, the Fourier transform reduces to the spherical transform considered by E. Damek and F. Ricci.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1997

References

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