Range of the horocyclic Radon transform on trees
Annales de l'Institut Fourier, Volume 50 (2000) no. 1, pp. 211-234.

In this paper we study the Radon transform R on the set of horocycles of a homogeneous tree T, and describe its image on various function spaces. We show that the functions of compact support on that satisfy two explicit Radon conditions constitute the image under R of functions of finite support on T. We extend these results to spaces of functions with suitable decay on T, whose image under R satisfies corresponding decay conditions and contains distributions on that are not defined pointwise. We also show that R is one-to-one on these spaces. Formulas are expressed in an invariant fashion in terms of a measure on preserved by the full automorphism group of T.

Dans cet article on étudie la transformation de Radon R sur l’ensemble des horocycles d’un arbre homogène T, et l’on décrit l’image de divers espaces fonctionnels. On montre que l’espace des fonctions à support compact sur qui satisfont à deux conditions de Radon décrites explicitement est égal à l’image par R de l’espace des fonctions à support fini sur T. On étend ces résultats à des espaces de fonctions à décroissance appropriée sur T, dont l’image par R est décrite par des conditions de décroissance et contient des distributions sur qui ne sont pas définies ponctuellement. On montre aussi que R est injective sur ces espaces. Les formules sont exprimées de façon invariante en termes d’une mesure sur qui est préservée par le groupe des automorphismes de T.

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     title = {Range of the horocyclic {Radon} transform on trees},
     journal = {Annales de l'Institut Fourier},
     pages = {211--234},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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     doi = {10.5802/aif.1752},
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Tarabusi, Enrico Casadio; Cohen, Joel M.; Colonna, Flavia. Range of the horocyclic Radon transform on trees. Annales de l'Institut Fourier, Volume 50 (2000) no. 1, pp. 211-234. doi : 10.5802/aif.1752. http://www.numdam.org/articles/10.5802/aif.1752/

[A] G. Ahumada Bustamante, Analyse harmonique sur l'espace des chemins d'un arbre, Thèse de Doctorat d'État, Université de Paris-Sud (Orsay), 1988. | MR

[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 | Zbl

[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 | Zbl

[BFPp] W. Betori, J. Faraut, M. Pagliacci, The horicycles of a tree and the Radon transform, preliminary version of [BFP].

[BFP] W. Betori, J. Faraut, M. Pagliacci, An inversion formula for the Radon transform on trees, Math. Z., 201 (1989), 327-337. | EuDML | MR | Zbl

[BP] W. Betori, M. Pagliacci, The Radon transform on trees, Boll. Un. Mat. Ital. B (6), 5 (1986), 267-277. | MR | Zbl

[CCC] E. Casadio Tarabusi, J.M. Cohen, F. Colonna, Characterization of the range of the Radon transform on homogeneous trees, Electron. Res. Announc. Amer. Math. Soc., 5 (1999), 11-17. | EuDML | MR | Zbl

[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 | Zbl

[CCP2] E. Casadio Tarabusi, J.M. Cohen, A.M. Picardello, Range of the X-ray transform on trees, Adv. Math., 109 (1999), 153-167. | MR | Zbl

[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 | Zbl

[CD] J.M. Cohen, L. De Michele, The radial Fourier-Stieltjes algebra of free groups, Contemp. Math., 10 (1982), 33-40. | MR | Zbl

[CMS] M.G. Cowling, S. Meda, A.G. Setti, An overview of harmonic analysis on the group of isometries of a homogeneous tree, Exposition. Math., 16 (1998), 385-423. | MR | Zbl

[CS] M.G. Cowling, A.G. Setti, The range of the Helgason-Fourier transformation on homogeneous trees, Bull. Austral. Math. Soc., 59 (1999), 237-246. | MR | Zbl

[H] S. Helgason, The Radon transform, Progr. Math., 5, Birkhäuser, Boston, 1980. | MR | Zbl

[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]. | JFM

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