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\begin{document}

\title[RANGE OF THE RADON TRANSFORM]
{Characterization of the range\\ of the Radon transform\\ on homogeneous 
trees}
\author{Enrico Casadio~Tarabusi}
\address{Dipartimento di Matematica ``G. Castelnuovo'', 
Universit\`{a} di Roma ``La Sapienza'', 
Piazzale A. Moro 2, 00185~Roma,
Italy}
\email{casadio@alpha.science.unitn.it}
\author{Joel M. Cohen}
\address{Department of Mathematics, 
University of Maryland, 
College Park, MD~20742}
\email{jmc@math.umd.edu}
\author{Flavia Colonna}
\address{Department of Mathematical Sciences, 
George Mason University, 
4400 University Drive, Fairfax, VA 22030}
\email{fcolonna@osf1.gmu.edu}
\keywords{Radon transform, homogeneous trees, horocycles, 
range characterizations, distributions}
\subjclass{Primary 44A12; Secondary 05C05, 43A85}
\thanks{Supported in part by an Alfred P. Sloan Research
Fellowship and  NSF grant DMS 95-01056.}
\issueinfo{5}{02}{}{1999}
\dateposted{February 4, 1999}
\pagespan{11}{17}
\PII{S 1079-6762(99)00055-4}
\def\copyrightyear{1999}
\copyrightinfo{1999}{American Mathematical Society}
\commby{Mark Freidlin}
\date{October 15, 1998}
\begin{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 {\em 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.
\end{abstract}
\maketitle


 
\section{Introduction}

  We study the Radon transform $R$ on the set $\Hc $ of
horocycles of a homogeneous tree $T$, and describe its image on various
function spaces.  We show that the functions of compact support on $\Hc $
that satisfy two explicit {\em Radon conditions\/} constitute the image
under $R$ of functions of finite support on $T$.  Larger domains and 
ranges
are described by adding decay criteria to the domain and range,
although we show that functions on $\Hc $ need to be replaced by
distributions.  

	The {\bf Radon transform\/} ({\bf RT\/} for short), in its original 
formulation by Radon \cite{R}, associates to each (sufficiently nice) 
function on $\R ^{2}$ its one-dimensional Lebesgue integrals along all 
affine
straight lines. This transform has been receiving considerable 
attention for its highly applicable nature and intrinsic interest, 
leading to a variety of generalizations.  

	In $\Hb ^{2}$ lines correspond to two essentially different kinds of 
one-dimensional submanifolds: geodesics and horocycles, giving rise to 
two different RTs (cf. \cite{H}).

	Homogeneous trees are widely regarded as discrete
counterparts of $\Hb ^{2}$, as well as objects of thorough study in
harmonic analysis in their own right.  Exactly like $\Hb ^{2}$, they 
feature two distinct kinds of RTs, namely the {\bf geodesic RT\/} 
(a.k.a\@. the {\bf X-ray transform\/}, since it is reminiscent of the 
CAT-scan procedure (cf. \cite{BC})), and the {\bf horocyclic RT}.  
Several of the standard RT issues in this
setting have been investigated over time by various authors: e.g.,
\cite{BCCP}, \cite{A} for injectivity and inversion, \cite{CCP2} for 
range
characterization, and \cite{CC} for function space setting for the 
geodesic
RT; \cite{BP}, \cite{BFP}, \cite{CCP1} for injectivity and inversion of 
the
horocyclic RT (part of the results therein are rewritten in~\cite{CMS} 
for
the  Abel transform, which is a  multiple of the RT). Another related 
transform has been  studied recently by Cowling and Setti.
 
  In this work, we pursue a description of the range of the
horocyclic RT $R$ on a homogeneous tree $T$ of degree $q+1$ with $q\ge 
2$. We first state two natural explicit relations (one of which had 
already been observed in \cite{BFPp} and \cite{BFP} for radial 
functions) for functions on the space $\Hc $ of
horocycles of $T$.  We then show that among compactly supported functions
on $\Hc $, these conditions completely characterize the range of $R$ on 
finitely supported functions on (the set of vertices of) $T$.  Similar 
descriptions are valid for the range of $R$ on larger function spaces, 
although distributions on $\Hc $ need then to be taken into account.
 All the results with complete proofs can be found in \cite{CCC}.	

	We thank Hillel Furstenberg for many useful conversations and for his 
insights into the problem.
\section{Preliminaries}
  
	The boundary $\MYO $ of $T$ is the set of equivalence classes of 
infinite 
paths under the relation $\path {v}{0}{1}\simeq \path {v}{1}{2}$. For 
any vertex 
$u$, we denote by $[u,\myo )$ the (unique) path starting at $u$ in the 
class $\myo $. Then $\MYO $ can be identified with the set of paths 
starting 
at $u$. Each $\myo \in \MYO $ induces an orientation on the edges of 
$T$: 
$[u,v]$ is positively oriented if $v\in [u,\myo )$. 

	For $\myo \in \MYO $, and $u,v\in T$, define the {\bf horocycle 
index\/} 
$\myk _{\myo }(u,v)$ as the number of positively oriented edges minus the 
number of negatively oriented edges in the path from $u$ to $v$. Given 
$u\in T$ and $\myo \in \MYO $, the {\bf horocycle through $u$ touching 
$\myo $} 
is the set $\{v:\,\myk _{\myo }(u,v)=0\}$. More generally, for any $n\in 
\Z $, 
the {\bf horocycle of index $n$ touching $\myo $ with respect to $u$} is 
$h^{u}_{\myo ,n}=\{w\in T\colon \myk _{\myo }(u,w)=n\}.$ Then the set of 
vertices may be 
decomposed as $\coprod _{n\in \Z }h_{\myo ,n}^{u}.$

	For $u$ fixed, the map $(n,\myo )\mapsto h^{u}_{\myo ,n}$ is a 
one-to-one 
correspondence between $\Z \times \MYO $ and the set $\Hc $ of 
horocycles.  

\begin{definition}\label{def:1} The {\bf $L^{1}$-horocyclic Radon transform\/} 
$R$ on $T$ is given by $Rf(h)\!=\sum_{v\in h}f(v)$ for $f\in 
L^{1}T$, and $h\in \Hc $.
\end{definition}


	For $u,v\in T$, set $S(u,v)=\{h\in \Hc : \exists \myo \in \MYO \text{ 
s.t. 
}h=h^{u}_{\myo ,0}, v\in [u,\myo )\}.$ The topology generated by the 
sets 
$S(u,v)$ makes $\Hc $ totally disconnected.  Then $\Hc $ is homeomorphic 
to $\Z \times \MYO $, where $\MYO $ is endowed with the compact topology 
generated by $I^{u}_{v}=\{\myo \in \MYO : v\in [u,\myo )\}$. For any 
$u\in T$, there 
is a measure $\m ^{u}$ on $\MYO $: $\m ^{u}(I^{u}_{v}) = 1/c_{d(u,v)}.$

	The family of horocycles through a fixed $\myo $ does not depend on the 
choice of the reference vertex $u$, but indices do: 
$h^{v}_{\myo ,n}=h^{u}_{\myo , n + \myk _{\myo }(u,v)}.$ 

	 For simplicity of notation, we fix a root $e$ throughout, and set 
$\hon =h^{e}_{\myo ,n}$, $\m =\m ^{e}$, $d\myo =d\m ^{e}(\myo )$, 
$k(v,\myo ) = \myk _{\myo }(e,v)$, 
and $I_{v}=I^{e}_{v}$. Notice that $d\m ^{v}(\myo )=q^{k(v,\myo 
)}\,d\myo $.   

	For $\myo \in \MYO $, let $\myo _{n}\in [e,\myo )$ be the vertex of 
length $n$. For 
$v\in T$, and $0\le n \le |v|$, let $v_{n}\in [e,v]$  be the vertex of 
length $n$.  For $v\in T$ and $n\ge |v|$, the set $D_{n}(v)=\{u:\ |u|=n 
\text{ and } u_{|v|}=v\}$ is the set of {\bf descendants} of $v$ of 
length $n$.

\begin{definition}\label{def:2} For a function $\f $ on $\myH $, we define the {\bf 
Radon conditions} as follows:

$(R_{1})$\  $\sum_{n}\f (h_{\myo ,n}^{v})$ is independent of 
$v$ and $\myo $.

$(R_{2})$\  For any $v\in T$, $n\in \Z $, 
\begin{equation*}\int _{\MYO }\f (h_{\myo ,n}^{v})d\m ^{v}\myo = 
q^{-n}\int _{\MYO }\f (h_{\myo ,-n}^{v})d\m ^{v}\myo .\end{equation*}
\end{definition}


\begin{proposition}\label{prop:1}  If $f\in L^{1}T$, then $Rf$ is a continuous 
function satisfying the Radon conditions.\end{proposition}


There are, however, continuous functions satisfying the Radon 
conditions that are of the form $Rf$ for $f\notin L^{1}T$. 

Proposition \ref{prop:1} is proved by showing first that the Radon conditions are 
satisfied for the function $\f =R\myc _{u}$, where $\myc _{u}$ is the 
characteristic function of $\{u\}$, and then extending linearly.   

		Fix $v\in T$. For $0\le t\le |v|$, let $I_{v}^{t}=\{\myo \in \MYO 
:k(v,\myo ) = 
2t-|v|\}$. Then for $t\ne |v|$, $I_{v}^{t}=I_{v_{t}}-I_{v_{t+1}}$, 
$I_{v}^{|v|}=I_{v}$, and $\MYO =\coprod_{t=0}^{|v|}I_{v}^{t}.$ 
Using the 
relations 
$\hon ^{v}=h_{\myo ,n+k(v,\myo )}$ and $d\m ^{v}\myo =q^{k(v,\myo 
)}\,d\myo $, condition 
$(R_{2})$ may be rewritten as 
\begin{equation*}(R_{2}')\quad \sum \limits _{t=0}^{|v|}q^{2t-|v|}\int 
_{I_{v}^{t}}\f (h_{\myo ,n+
2t-|v|})
=q^{-n}\sum \limits _{t=0}^{|v|}q^{2t-|v|}\int _{I_{v}^{t}}\f (h_{\myo 
,-n+
2t-|v|})\,d\myo .\end{equation*}

	In \S 3, we characterize the range of the RT on the set of functions on 
$T$ of finite support, and then in \S 4, after defining $Rf$ as a 
distribution on $\Hc $, we obtain a similar characterization for the 
case of $f$ of infinite support.

\section{Functions of compact support}

\begin{theorem}\label{thm:1} The image of $R$ on the space of functions on $T$ 
of finite (i.e. compact) support is the space of functions on 
$\mathcal{H}$ 
of compact support satisfying the Radon conditions. \end{theorem}


	The proof is based on the use of a generalization of radiality:

\begin{definition}\label{def:3} Let $N$ be a non-negative integer.

(1) A function $f$ on $T$ is {\bf $N$-radial\/} if for all 
$v\in T$ with $|v|\ge N$, $f(v)$ depends only on $v_{N}$ and $|v|$.

(2) $f$ has {\bf radius} $N$ if $\{v: |v|\le N\}$ is the 
smallest disk centered at $e$ containing the support of $f$ (so 
$f(v)=0$ for $|v|>N$).

(3) A function $\f $ on $\mathcal{H}$ is {\bf $N$-radial\/} if 
$\f (\hon )$ depends only on $\myo _{N}$ and $n$.

(4) $\f $ has {\bf radius} $N$ if $[-N,\dots ,N]\times \MYO $ 
is 
the smallest such set containing the support of $\f $ (so $\f (\hon )=0$ 
for $|n|>N$). 
\end{definition}


	In particular, a $0$-radial function on $T$ is what is generally 
called {\em radial}. 

We actually prove a more precise version of Theorem \ref{thm:1}, specifically 
that the image under $R$ of the set of functions on $T$ of radius less 
than or equal to $N$ is the set of continuous functions on $\Hc $ of 
radius less than or equal to $N$ satisfying the Radon conditions. This 
result is established by means of Propositions \ref{prop:2} and \ref{prop:3}, 
whose proofs are outlined below.

	For $N\ge 0$, let $E^{N}$ be the set of $N$-radial functions on $\Hc $ 
of 
radius less than or equal to $N$ satisfying $(R_{1})$ and $(R_{2})$.  

\begin{proposition}\label{prop:2} $E^{N} = E^{N-1} \oplus \bigoplus _{|v|=N} 
\BigC R\myc _{v}.$\end{proposition}


	It follows by induction that $E^{N}$ is the image under $R$ of the set 
of functions of radius less than or equal to $N$.

\begin{proposition}\label{prop:3} If $\f $ is a function on $\Hc $ of compact 
support satisfying the Radon conditions, then there exists $N$ such 
that $\f \in E^{N}$.\end{proposition}


	 Let $\{v^{1},\dots ,v^{c_{N}}\}$ be an enumeration of the vertices of 
length $N$.  If $v\in T$, $|v|\le N$, let $A_{v}^{t} = \{j: 
I_{v^{j}}\subset I_{v}^{t}\}$.  Thus $I_{v}^{t}=\coprod_{j\in 
A_{v}^{t}}I_{v^{j}}$. If $j_{0}$ is 
the index such that $v=v^{j_{0}}$, then $A_{v}^{N} = \{j_{0}\}$. 
Observe that 
$\{1,2,\dots ,c_{N}\} = \coprod_{t=0}^{|v|}A_{v}^{t}$ and 
recall that 
$\MYO =\coprod_{t=0}^{|v|}I_{v}^{t}$. Let $\f \in E^{N}$, and 
set 
$a_{n,j}=\f (h_{\myo ,n})$ for $\myo _{N} = v^{j}$.  Then $(R_{2}')$ 
becomes
\begin{equation*}(R_{2}'')\quad \sum_{t=0}^{M} q^{2t} \sum_{j\in A_{v}^{t}} 
a_{n+2t-M,j} = q^{-n}\sum \limits _{t=0}^{M} q^{2t} \sum \limits _{j\in 
A_{v}^{t}} a_{-n+2t-M,j},\end{equation*}
for $|v|=M\le N$.

The proof of Propositions \ref{prop:2} and \ref{prop:3} is 
based on repeated applications of 
$(R_{2}'')$ for various values of $n$ and $M$. For instance, if we set 
$M=N$ and $n=2N$, the left-hand side of $(R_{2}'')$ reduces to 
$\sum_{j\in A_{v}^{0}}a_{N,j}$, since $n+2t-M>N$ except for 
$t=0$. 
On the right-hand side, $a_{-n+2t-M}=0$, except for $t=M=N$, leaving 
just $\sum_{j\in A_{v}^{N}}a_{-N,j}$, which is $a_{-N,j_{0}}$, 
where 
$v=v^{j_{0}}$. Thus $\sum_{j\in 
A_{v}^{0}}a_{N,j}=a_{-N,j_{0}}q^{N}.$ In 
particular, if $a_{N,j}=0$ for all $j$, then $a_{-N,j}=0$ for all $j$.

	If $\f \in E^{N}$, then the function $\tilde {\f }=\f - 
\sum_{j=1}^{c_{N}} a_{M,j}R(\myc _{v^{j}})$ has the property 
that 
$\tilde {\f }(\hon )=0$ for $n=N$ as well as for $|n|>N$. Hence 
$\tilde {a}_{N,j}=0$ for all $j$, and so, by what we just proved, 
$\tilde {a}_{-N,j}=0$ for all $j$. Thus $\tilde {\f }\in E^{N-1}$, 
proving 
Proposition \ref{prop:2}.

	Now let $\f $ be a function with compact support satisfying the Radon 
conditions. Since topologically $\mathcal{H}\simeq \BigZ \times \MYO $ 
with $\MYO $ 
compact, there is some positive integer $N$ such that the support of 
$\f $ is contained in $[-N,N]\times \MYO $, i.e. $\f (\hon )=0$ for 
$|n|>N$. 
Then $\f $ has radius less than or equal to $N$. Again using 
$(R_{2}'')$, 
it is possible to show that $\f $ is $N$-radial. Thus $\f \in E^{N}$, 
proving Proposition \ref{prop:3}, and hence Theorem \ref{thm:1}.

\section{Non-compact support}

	In this section we develop a parallel theory for distributions on 
$\Hc $ and define certain {\bf decay conditions\/} for functions on $T$ 
and distributions on $\Hc $.

	For $r>0$, define $\mathcal{A}_{r}$ as the class of all functions 
$f:T\to \BigC $ satisfying the {\bf decay condition\/}: 
\begin{equation*}\sum_{n=|v|}^{\infty }t^{n}\left |\sum 
\limits _{u\in D_{n}(v)}f(u)\right |<\infty \quad \forall t\in [0,r),\ 
\forall v\in T.\end{equation*} 

	Observe that $L^{1}T\subset \mathcal{A}_{1}$, since for $f\in L^{1}T$ 
and $0\le t 
< 1$,
\begin{equation*}\sum \limits _{n=|v|}^{\infty }t^{n} \left | \sum 
\limits _{u\in D_{n}(v)}f(u)\right |\le \sum \limits _{n=|v|}^{\infty 
}\sum \limits _{u\in D_{n}(v)}|f(u)|\le \sum _{u\in T}|f(u)|= 
\|f\|_{1}.\end{equation*} 
	
	The elementary measurable sets in $\Hc $ can be generated by all sets 
of the form $\{\hon \in \Hc : \myo \in I_{v}\}$, which may be 
identified with 
$\{n\}\times I_{v}$. A {\bf distribution\/} on $\mathcal{H}$ is an 
element of 
the dual of the vector space generated by the characteristic functions 
of the elementary measurable sets of $\myH $. Thus, since 
$I_{v}=\coprod_{u^{-}=v}I_{u}$, we may think of a distribution 
on 
$\Hc $ as a function $\f $ on the sets $\{n\}\times I_{v}$ satisfying 
the 
property
\begin{equation*}\f (\{n\}\times I_{v})=\sum \limits _{u^{-}=v}\f 
(\{n\}\times I_{u}).\end{equation*}

	If $f\in L^{1}T$, then $Rf$ is defined on each horocycle and is 
bounded. 
By abuse of notation, we define $Rf$ as the distribution given by 
\begin{equation*}Rf(\{n\}\times I_{u})=\int _{I_{u}}Rf(\hon )d\myo 
.\end{equation*} 
Now for a larger class of functions on $T$, this leads to the following 
definition of the Radon transform as a distribution:
	
\begin{definition}\label{def:4} For a function $f$ on $T$, let 
\begin{equation*}Rf(\{n\}\times I_{u})=\sum \limits _{m=0}^{\infty 
}\sum \limits _{|v|=m}f(v)R\myc _{v}(\{n\}\times I_{u}),\end{equation*} 
if this is defined for all $u\in T,$ and all $n\in \BigZ $.
\end{definition}


This definition is consistent with the previous formula, since 
\[f= 
\sum \limits _{m=0}^{\infty }\sum \limits _{|v|=m}f(v)\myc _{v}.
\]
We extend the Radon conditions to the case of distributions as follows:

$(R_{1})\ \sum_{n\in \Z }\f (\{n\}\times I_{v})/\m 
(I_{v})$ is 
independent of $v$.

$(R_{2})\ \text{For all\  }v\in T,\  n\in \Z $,
\begin{equation*}\sum \limits _{t=0}^{|v|}q^{2t-|v|}\f (\{n+
2t-|v|\}\times I_{v}^{t})
=q^{-n}\sum \limits _{t=0}^{|v|}q^{2t-|v|}\f (\{-n+2t-|v|\}\times 
I_{v}^{t}).\end{equation*}

For $r>0$, define $\mathcal{B}_{r}$ as the class of all distributions 
$\f $ on 
$\mathcal{H}$ satisfying the {\bf decay condition\/}: 
\begin{equation*}\sum \limits _{n=|v|}^{\infty }t^{n}q^{n}\left |\f 
(\{n\}\times I_{v})\right |<\infty \quad \text{ for all } t\in [0,r), 
v\in T.\end{equation*}

\begin{theorem}\label{thm:2} For $r>1/\sqrt q$, $R(\mathcal{A}_{r})$ is the set of 
all 
$\f \in \mathcal{B}_{r}$ satisfying the Radon conditions.\end{theorem}


	A distribution $\f $ on $\mathcal{H}$ is {\bf $N$-radial} if $\f 
(\{n\}\times I_{v})$ depends only on $n$ and $v_{N}$. 

	The proof of Theorem \ref{thm:2} is based on the use of $N$-radial functions and 
$N$-radial distributions. Given a positive number $r$, and a 
non-negative integer $N$, let $\mathcal{A}_{r}^{N}$ be the space of 
$N$-radial 
functions in $\mathcal{A}_{r}$, and let $\mathcal{B}_{r}^{N}$ be the 
space of 
$N$-radial distributions in $\mathcal{B}_{r}$. The key result in proving 
Theorem \ref{thm:2} is the following 

\begin{proposition}\label{prop:4} For $r>1/\sqrt q$, the image of the Radon 
transform on $\mathcal{A}_{r}^{N}$ is the set of all $\f \in 
\mathcal{B}_{r}^{N}$ 
satisfying the Radon conditions.\end{proposition}


	The following example shows that the use of distributions is necessary:
	
\begin{definition5}  Let $\l _{1},\dots ,\l _{q}$ be complex numbers of 
absolute value one, such that $\sum_{j=1}^{q}\l _{j}=2/3$, and 
set 
$\l _{q+1} = \l _{1}$.  Label the vertices as follows:  let 
$x_{1},\dots ,x_{q+1}$ be the vertices of length 1.  If $v\ne e$ has 
already been labeled, write the immediate descendants of $v$ as 
$vx_{1},\dots ,vx_{q}$.  Thus a typical vertex $v$ of length $N$ is 
labeled as $x_{i_{1}}\dots x_{i_{N}}$, where the $i_{j}$ are between 1 
and 
$q$, except for $i_{1}$ which can also be $q+1$.  Then define $f(v)$ as 
$\l _{i_{1}}\dots \l _{i_{N}}\left (\frac{4}{3}\right )^{N},\ 
f(e)=1$. Thus 
\begin{equation*}\left |\sum \limits _{u\in D_{n}(v)}f(u)\right | = 
|f(v)|(8/9)^{n-N}=\left (\frac{8}{9}\right )^{n}\left 
(\frac{3}{2}\right )^{N}.\end{equation*}

	If $0