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The range of the Helgason-Fourier transformation on homogeneous trees

  • Michael Cowling (a1) and Alberto G. Setti (a2)
Abstract

Let be a homogeneous tree, o be a fixed reference point in , and be the closed ball of radius N in centred at o. In this paper we characterise the image under the Helgason–Fourier transformation ℋ of , the space of functions supported in , and of , the space of rapidly decreasing functions on . In both cases our results are counterparts of known results for the Helgason–Fourier transformation on noncompact symmetric spaces.

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References
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[1]Betori W., Faraut J. and Pagliacci M., ‘An inversion formula for the Radon transform on trees’, Math. Zeit. 201 (1989), 327337.
[2]Tarabusi E. Casadio, Cohen J.M. and Colonna F., ‘The range of the horocyclical Radon transform on homogeneous trees’, preprint.
[3]Cowling M., Meda S. and Setti A.G., ‘An overview of harmonic analysis on the group of isometries of a homogeneous tree’, Exposit. Math. 16 (1998), 385424.
[4]Figà-Talamanca A. and Nebbia C., Harmonic analysis and representation theory for groups acting on homogeneous trees, London Math. Soc. Lecture Notes Series, 162 (Cambridge Univ. Press, Cambridge, 1991).
[5]Figá-Talamanca A. and Picardello M., Harmonic analysis on free groups (Dekker, New York, 1983).
[6]Helgason S., Geometric analysis on symmetric spaces, Math. Surveys and Monog. (Amer. Math. Soc, Providence, R.I., 1994).
[7]Mantero A. M. and Zappa A., ‘The Poisson transform and representations of a free group’, J. Funct. Anal. 51 (1983), 372399.
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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