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

@article{AIF_2000__50_1_211_0,
     author = {Tarabusi, Enrico Casadio and Cohen, Joel M. and Colonna, Flavia},
     title = {Range of the horocyclic Radon transform on trees},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {50},
     number = {1},
     year = {2000},
     pages = {211-234},
     doi = {10.5802/aif.1752},
     zbl = {0944.44002},
     mrnumber = {2001g:44003},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2000__50_1_211_0}
}
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/item/AIF_2000__50_1_211_0/

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