Bulletin of the Australian Mathematical Society

Research Article

The range of the Helgason-Fourier transformation on homogeneous trees

Michael Cowlinga1 and Alberto G. Settia2

a1 School of Mathematics, University of New South Wales, Sydney NSW 2052, Australia e-mail: m.cowling@unsw.edu.au

a2 Università dell'Insubria—Polo di Como, Facoltà di Scienze, via Lucini 3, I-22100 Como, Italy e-mail: setti@fis.unico.it


Let S0004972700032858_inline1 be a homogeneous tree, o be a fixed reference point in S0004972700032858_inline1, and S0004972700032858_inline2 be the closed ball of radius N in S0004972700032858_inline1 centred at o. In this paper we characterise the image under the Helgason–Fourier transformation xs210B of S0004972700032858_inline3, the space of functions supported in S0004972700032858_inline2, and of S0004972700032858_inline4, the space of rapidly decreasing functions on S0004972700032858_inline1. In both cases our results are counterparts of known results for the Helgason–Fourier transformation on noncompact symmetric spaces.

(Received July 14 1998)