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Characterization of the range of the Radon transform on homogeneous trees


Authors: Enrico Casadio Tarabusi, Joel M. Cohen and Flavia Colonna
Journal: Electron. Res. Announc. Amer. Math. Soc. 5 (1999), 11-17
MSC (1991): Primary 44A12; Secondary 05C05, 43A85
DOI: https://doi.org/10.1090/S1079-6762-99-00055-4
Published electronically: February 4, 1999
MathSciNet review: 1667635
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Abstract | References | Similar Articles | Additional Information

Abstract: This article contains results on the range of the Radon transform $R$ on the set $\mathcal{H}$ of horocycles of a homogeneous tree $T$. Functions of compact support on $\mathcal{H}$ that satisfy two explicit Radon conditions constitute the image under $R$ of functions of finite support on $T$. Replacing functions on $\mathcal{H}$ by distributions, we extend these results to the non-compact case by adding decay criteria.


References [Enhancements On Off] (What's this?)

  • [A] G. Ahumada Bustamante, Analyse harmonique sur l'espace des chemins d'un arbre, Thesis of Doctorat d'État, Univ. of Paris-Sud (Orsay), 1988.
  • [BC] C. A. Berenstein, E. Casadio Tarabusi, Integral geometry in hyperbolic spaces and electrical impedance tomography, SIAM. J. Appl. Math. 56 (1996), 755-764. MR 97d:53073
  • [BCCP] C. A. Berenstein, E. Casadio Tarabusi, J. M. Cohen, M. A. Picardello, Integral geometry on trees, Amer. J. Math. 113 (1991), 441-470. MR 92g:05066
  • [BFPp] W. Betori, J. Faraut, M. Pagliacci, The horicycles of a tree and the Radon transform, preliminary version of [BFP].
  • [BFP] -, An inversion formula for the Radon transform on trees, Math. Z. 201 (1989), 327-337. MR 90k:22004
  • [BP] W. Betori, M. Pagliacci, The Radon transform on trees, Boll. Un. Mat. Ital. B (6) 5 (1986), 267-277. MR 87h:05073
  • [CCC] E. Casadio Tarabusi, J. M. Cohen, F. Colonna, The range of the horocyclic Radon transform on a homogeneous tree, preprint.
  • [CCP1] E. Casadio Tarabusi, J. M. Cohen, A. M. Picardello, The horocyclic Radon transform on non-homogeneous trees, Israel J. Math. 78 (1992), 363-380. MR 94b:44002
  • [CCP2] -, Range of the X-ray transform on trees, Adv. Math. 109 (1994), 153-167. MR 96a:05043a
  • [CC] J. M. Cohen, F. Colonna, The functional analysis of the X-ray transform on trees, Adv. in Appl. Math. 14 (1993), 123-138. MR 94a:05048
  • [CMS] M. Cowling, S. Meda, A. G. Setti, An overview of harmonic analysis on the group of isometries of a homogeneous tree, preprint.
  • [H] S. Helgason, The Radon transform, Progr. Math., vol. 5, Birkhäuser, Boston, 1980. MR 83f:43012
  • [R] J. Radon, Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten, Ber. Verh. Sächs. Akad. Wiss. Leipzig, Math.-Phys. Kl. 69 (1917), 262-277; reprinted in [H, pp. 177-192].

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Additional Information

Enrico Casadio Tarabusi
Affiliation: Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale A. Moro 2, 00185 Roma, Italy
Email: casadio@alpha.science.unitn.it

Joel M. Cohen
Affiliation: Department of Mathematics, University of Maryland, College Park, MD 20742
Email: jmc@math.umd.edu

Flavia Colonna
Affiliation: Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030
Email: fcolonna@osf1.gmu.edu

DOI: https://doi.org/10.1090/S1079-6762-99-00055-4
Keywords: Radon transform, homogeneous trees, horocycles, range characterizations, distributions
Received by editor(s): October 15, 1998
Published electronically: February 4, 1999
Additional Notes: Supported in part by an Alfred P. Sloan Research Fellowship and NSF grant DMS 95-01056.
Communicated by: Mark Freidlin
Article copyright: © Copyright 1999 American Mathematical Society

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