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