ELECTRONIC RESEARCH ANNOUNCEMENTS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 5, Pages 11–17 (February 4, 1999) S 1079-6762(99)00055-4

CHARACTERIZATION OF THE RANGE OF THE RADON TRANSFORM ON HOMOGENEOUS TREES

ENRICO CASADIO TARABUSI, JOEL M. COHEN, AND FLAVIA COLONNA

(Communicated by Mark Freidlin)

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

1. Introduction

We study the Radon transform R on the set H of horocycles of a homogeneous tree T, and describe its image on various function spaces. We show that the functions of compact support on H that satisfy two explicit 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 H need to be replaced by distributions.

The Radon transform (RT for short), in its original formulation by Radon [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 H^{2} lines correspond to two essentially different kinds of one-dimensional submanifolds: geodesics and horocycles, giving rise to two different RTs (cf. [H]).

Homogeneous trees are widely regarded as discrete counterparts of H^{2}, as well as objects of thorough study in harmonic analysis in their own right. Exactly like H^{2}, they feature two distinct kinds of RTs, namely the geodesic RT (a.k.a. the X-ray transform, since it is reminiscent of the CAT-scan procedure (cf. [BC])), and the horocyclic RT. Several of the standard RT issues in this setting have been investigated over time by various authors: e.g., [BCCP], [A] for injectivity and inversion, [CCP2] for range characterization, and [CC] for function space setting for the geodesic RT; [BP], [BFP], [CCP1] for injectivity and inversion of the horocyclic

\begin{matrix}c \\ ◯ \end{matrix}1999 American Mathematical Society

Received by the editors October 15, 1998.

1991 Mathematics Subject Classification. Primary 44A12; Secondary 05C05, 43A85.

Key words and phrases. Radon transform, homogeneous trees, horocycles, range characterizations, distributions.

Supported in part by an Alfred P. Sloan Research Fellowship and NSF grant DMS 95-01056.

RT (part of the results therein are rewritten in [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 ≥ 2. We first state two natural explicit relations (one of which had already been observed in [BFPp] and [BFP] for radial functions) for functions on the space H of horocycles of T. We then show that among compactly supported functions on H, 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 H need then to be taken into account. All the results with complete proofs can be found in [CCC].

We thank Hillel Furstenberg for many useful conversations and for his insights into the problem.

2. Preliminaries

The boundary Ω of T is the set of equivalence classes of infinite paths under the relation [v_{0}, v_{1}, …] ≃ [v_{1}, v_{2}, …]. For any vertex u, we denote by [u, ω) the (unique) path starting at u in the class ω. Then Ω can be identified with the set of paths starting at u. Each ω ∈ Ω induces an orientation on the edges of T: [u, v] is positively oriented if v ∈ [u, ω).

For ω ∈ Ω, and u, v ∈ T, define the horocycle index κ_{ω}(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 ∈ T and ω ∈ Ω, the horocycle through u touching ω is the set {v : κ_{ω}(u, v) = 0}. More generally, for any n ∈ ℤ, the horocycle of index n touching ω with respect to u is h_{ω,n}^{u} = {w ∈ T: κ_{ω}(u, w) = n}. Then the set of vertices may be decomposed as ∐_{n∈ℤ} h_{ω,n}^{u}.

For u fixed, the map (n, ω) ↦ h_{ω,n}^{u} is a one-to-one correspondence between ℤ×Ω and the set H of horocycles.

Definition 1. The L^{1}-horocyclic Radon transform R on T is given by Rf(h)= ∑_{v∈h} f(v) for f ∈ L^{1}T, and h ∈ H.

For u, v ∈ T, set S(u, v) = {h ∈ H : ∃ω ∈ Ω s.t. h = h_{ω,0}^{u}, v ∈ [u, ω)}. The topology generated by the sets S(u, v) makes H totally disconnected. Then H is homeomorphic to ℤ × Ω, where Ω is endowed with the compact topology generated by I_{v}^{u} = {ω ∈ Ω : v ∈ [u, ω)}. For any u ∈ T, there is a measure μ^{u} on Ω: μ^{u}(I_{v}^{u}) = 1/c_{d(u,v)}.

The family of horocycles through a fixed ω does not depend on the choice of the reference vertex u, but indices do: h_{ω,n}^{v} = h_{ω,n+κ_{ω}(u,v)}^{u}.

For simplicity of notation, we fix a root e throughout, and set h_{ω,n} = h_{ω,n}^{e}, μ = μ^{e}, dω = dμ^{e}(ω), k(v, ω) = κ_{ω}(e, v), and I_{v} = I_{v}^{e}. Notice that dμ^{v}(ω) = q^{k(v,ω)} dω.

For ω ∈ Ω, let ω_{n} ∈ [e, ω) be the vertex of length n. For v ∈ T, and 0 ≤ n ≤ |v|, let v_{n} ∈ [e, v] be the vertex of length n. For v ∈ T and n ≥ |v|, the set D_{n}(v) = {u : |u| = n and u_{|v|} = v} is the set of descendants of v of length n.

Definition 2. For a function φ on H, we define the Radon conditions as follows: (R_{1}) ∑_{n} φ(h_{ω,n}^{v}) is independent of v and ω.

(R_{2}) For any v ∈ T, n ∈ ℤ,

∫_{Ω} φ(h_{ω,n}^{v})dμ^{v}ω = q^{−n} ∫_{Ω} φ(h_{ω,−n}^{v})dμ^{v}ω.

Proposition 1. If f ∈ L^{1}T, then Rf is a continuous function satisfying the Radon conditions.

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

Proposition 1 is proved by showing first that the Radon conditions are satisfied for the function φ = Rχ_{u}, where χ_{u} is the characteristic function of {u}, and then extending linearly.

Fix v ∈ T. For 0 ≤ t ≤ |v|, let I_{v}^{t} = {ω ∈ Ω : k(v, ω) = 2t−|v|}. Then for t ≠ |v|, I_{v}^{t} = I_{v_{t}} − I_{v_{t+1}}, I_{v}^{|v|} = I_{v}, and Ω = ∐_{t=0}^{|v|} I_{v}^{t}. Using the relations h_{ω,n}^{v} = h_{ω,n+k(v,ω)} and dμ^{v}ω = q^{k(v,ω)} dω, condition (R_{2}) may be rewritten as

(R′_{2}) ∑_{t=0}^{|v|} q^{2t−|v|} ∫_{I_{v}^{t}} φ(h_{ω,n+2t−|v|}) = q^{−n} ∑_{t=0}^{|v|} q^{2t−|v|} ∫_{I_{v}^{t}} φ(h_{ω,−n+2t−|v|}) dω.

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

3. Functions of compact support

Theorem 1. The image of R on the space of functions on T of finite (i.e. compact) support is the space of functions on H of compact support satisfying the Radon conditions.

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

Definition 3. Let N be a non-negative integer.

(1) A function f on T is N-radial if for all v ∈ T with |v| ≥ N, f(v) depends only on v_{N} and |v|.

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

(3) A function φ on H is N-radial if φ(h_{ω,n}) depends only on ω_{N} and n.

(4) φ has radius N if [−N, …, N] × Ω is the smallest such set containing the support of φ (so φ(h_{ω,n}) = 0 for |n| > N).

In particular, a 0-radial function on T is what is generally called radial.

We actually prove a more precise version of Theorem 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 H of radius less than or equal to N satisfying the Radon conditions. This result is established by means of Propositions 2 and 3, whose proofs are outlined below.

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

Proposition 2. E^{N} = E^{N−1} ⊕ ⨁_{|v|=N} ℂ Rχ_{v}.

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.

Proposition 3. If φ is a function on H of compact support satisfying the Radon conditions, then there exists N such that φ ∈ E^{N}.

Let {v^{1}, …, v^{c_{N}} } be an enumeration of the vertices of length N. If v ∈ T, |v| ≤ N, let A_{v}^{t} = {j : I_{v^{j}} ⊂ I_{v}^{t}}. Thus I_{v}^{t} = ∐_{j∈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, …, c_{N}} = ∐_{t=0}^{|v|} A_{v}^{t} and recall that Ω = ∐_{t=0}^{|v|} I_{v}^{t}. Let φ ∈ E^{N}, and set a_{n,j} = φ(h_{ω,n}) for ω_{N} = v^{j}. Then (R′_{2}) becomes

(R′′_{2}) ∑_{t=0}^{M} q^{2t} ∑_{j∈A_{v}^{t}} a_{n+2t−M,j} = q^{−n} ∑_{t=0}^{M} q^{2t} ∑_{j∈A_{v}^{t}} a_{−n+2t−M,j},

for |v| = M ≤ N.

The proof of Propositions 2 and 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 ∑_{j∈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 ∑_{j∈A_{v}^{N}} a_{−N,j}, which is a_{−N,j_{0}}, where v = v^{j_{0}}. Thus ∑_{j∈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 φ ∈ E^{N}, then the function φ̃ = φ − ∑_{j=1}^{c_{N}} a_{M,j}R(χ_{v^{j}}) has the property that φ̃(h_{ω,n}) = 0 for n = N as well as for |n| > N. Hence ã_{N,j} = 0 for all j, and so, by what we just proved, ã_{−N,j} = 0 for all j. Thus φ̃ ∈ E^{N−1}, proving Proposition 2.

Now let φ be a function with compact support satisfying the Radon conditions. Since topologically H ≃ ℤ × Ω with Ω compact, there is some positive integer N such that the support of φ is contained in [−N, N]×Ω, i.e. φ(h_{ω,n}) = 0 for |n| > N. Then φ has radius less than or equal to N. Again using (R′′_{2}), it is possible to show that φ is N-radial. Thus φ ∈ E^{N}, proving Proposition 3, and hence Theorem 1.

4. Non-compact support

In this section we develop a parallel theory for distributions on H and define certain decay conditions for functions on T and distributions on H.

For r > 0, define A_{r} as the class of all functions f : T → ℂ satisfying the decay condition:

∑_{n=|v|}^{∞} t^{n} | ∑_{u∈D_{n}(v)} f(u)| < ∞ ∀t ∈ [0, r), ∀v ∈ T.

Observe that L^{1}T ⊂ A_{1}, since for f ∈ L^{1}T and 0 ≤ t < 1,

∑_{n=|v|}^{∞} t^{n} | ∑_{u∈D_{n}(v)} f(u)| ≤ ∑_{n=|v|}^{∞} ∑_{u∈D_{n}(v)} |f(u)| ≤ ∑_{u∈T} |f(u)| = ∥f∥_{1}.

The elementary measurable sets in H can be generated by all sets of the form {h_{ω,n} ∈ H : ω ∈ I_{v}}, which may be identified with {n} × I_{v}. A distribution on H is an element of the dual of the vector space generated by the characteristic functions of the elementary measurable sets of H. Thus, since I_{v} = ∐_{u^{−}=v} I_{u}, we may think of a distribution on H as a function φ on the sets {n} × I_{v} satisfying the property

φ({n} × I_{v}) = ∑_{u^{−}=v} φ({n} × I_{u}).

If f ∈ 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

Rf({n} × I_{u}) = ∫_{I_{u}} Rf(h_{ω,n})dω.

Now for a larger class of functions on T, this leads to the following definition of the Radon transform as a distribution:

Definition 4. For a function f on T, let

Rf({n} × I_{u}) = ∑_{m=0}^{∞} ∑_{|v|=m} f(v)Rχ_{v}({n} × I_{u}),

if this is defined for all u ∈ T, and all n ∈ ℤ.

This definition is consistent with the previous formula, since

f = ∑_{m=0}^{∞} ∑_{|v|=m} f(v)χ_{v}.

We extend the Radon conditions to the case of distributions as follows: (R_{1}) ∑_{n∈ℤ} φ({n} × I_{v})/μ(I_{v}) is independent of v. (R_{2}) For all v ∈ T, n ∈ ℤ,

∑_{t=0}^{|v|} q^{2t−|v|}φ({n + 2t − |v|} × I_{v}^{t}) = q^{−n} ∑_{t=0}^{|v|} q^{2t−|v|}φ({−n + 2t − |v|} × I_{v}^{t}).

For r > 0, define B_{r} as the class of all distributions φ on H satisfying the decay condition:

∑_{n=|v|}^{∞} t^{n}q^{n} |φ({n} × I_{v})| < ∞ for all t ∈ [0, r), v ∈ T.

Theorem 2. For r > 1/\sqrt{q}, R(A_{r}) is the set of all φ ∈ B_{r} satisfying the Radon conditions.

A distribution φ on H is N-radial if φ({n} × I_{v}) depends only on n and v_{N}. The proof of Theorem 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 A_{r}^{N} be the space of N-radial functions in A_{r}, and let B_{r}^{N} be the space of N-radial distributions in B_{r}. The key result in proving Theorem 2 is the following

Proposition 4. For r > 1/\sqrt{q}, the image of the Radon transform on A_{r}^{N} is the set of all φ ∈ B_{r}^{N} satisfying the Radon conditions.

The following example shows that the use of distributions is necessary:

Example. Let l_{1}, …, l_{q} be complex numbers of absolute value one, such that ∑_{j=1}^{q} l_{j} = 2/3, and set l_{q+1} = l_{1}. Label the vertices as follows: let x_{1}, …, x_{q+1} be the vertices of length 1. If v ≠ e has already been labeled, write the immediate descendants of v as vx_{1}, …, vx_{q}. Thus a typical vertex v of length N is labeled as x_{i_{1}} … 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}} … l_{i_{N}} (\frac{4}{3})^{N} , f(e) = 1. Thus

| ∑_{u∈D_{n}(v)} f(u)| = |f(v)|(8/9)^{n−N} = (\frac{8}{9})^{n} (\frac{3}{2})^{N} .

If 0 < t < 9/8, then ∑_{n} t^{n} (\frac{8}{9})^{n} (\frac{3}{2})^{N} converges, and so f ∈ A_{9/8}. By Theorem 2, Rf ∈ B_{9/8}.

On the other hand, we now show that Rf cannot be evaluated at any horocycle. A horocycle h_{ω,n} is the disjoint union of the sets D_{n+2k}(ω_{n+k}) − D_{n+2k}(ω_{n+k+1}) over the set of all non-negative integers k, for n ≥ 0. Now

| ∑_{v∈D_{n+2k}(ω_{n+k})} f(v)| = (\frac{4}{3})^{n} (\frac{32}{27})^{k}

and

| ∑_{v∈D_{n+2k}(ω_{n+k+1})} f(v)| = (\frac{3}{2}) (\frac{4}{3})^{n} (\frac{32}{27})^{k} .

Since the second sum has a larger absolute value, the absolute value of the difference is at least \frac{1}{2} (\frac{4}{3})^{n} (\frac{32}{27})^{k}. Thus the series for defining Rf(h_{ω,n}) does not converge, for n ≥ 0. For n < 0,

h_{ω,n} = ∐_{k=0}^{∞} (D_{n+2k}(ω_{k}) − D_{n+2k}(ω_{k+1})) ,

and the same conclusion holds. Since point evaluation cannot be defined, Rf cannot be induced by a function.

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Dipartimento di Matematica “G. Castelnuovo”, Università di Roma “La Sapienza”, Piazzale A. Moro 2, 00185 Roma, Italy

E-mail address: casadio@alpha.science.unitn.it

Department of Mathematics, University of Maryland, College Park, MD 20742

E-mail address: jmc@math.umd.edu

Department of Mathematical Sciences, George Mason University, 4400 University Drive, Fairfax, VA 22030

E-mail address: fcolonna@osf1.gmu.edu