Unitarization of the Horocyclic Radon Transform on Homogeneous Trees

Following previous work in the continuous setup, we construct the unitarization of the horocyclic Radon transform on a homogeneous tree X and we show that it intertwines the quasi regular representations of the group of isometries of X on the tree itself and on the space of horocycles.


Introduction
The horocyclic Radon transform on homogeneous trees was introducted by P. Cartier [5] and studied by A. Figà-Talamanca and M.A. Picardello [11], W. Betori, J. Faraut and M. Pagliacci [3], M. Cowling, S. Meda and A.G. Setti [8], E. Casadio Tarabusi, J. Cohen and F. Colonna [6], and A. Veca [17], to name a few. Some of the typical issues considered are inversion formulae and range problems. In this paper, we treat the unitarization problem, that is the determination of some kind of pseudo-differential operator such that the pre-composition with the Radon transform yields a unitary operator. This is a classical aspect of Radon theory, addressed first by Helgason [15] in the case of the polar Radon transform. In [1], the authors consider a general setup that may be recast as a variation of the setup of dual pairs pX, Ξq à la Helgason [13] or [15]. They prove a general result concerning the unitarization of the Radon transform R from L 2 pX, dxq to L 2 pΞ, dξq and then show that the resulting unitary operator intertwines the quasi regular representations of G on L 2 pX, dxq and L 2 pΞ, dξq. As already mentioned, this unitarization really means first composing (the closure of) R with a suitable pseudo-differential operator and then extending this composition to a unitary map, as it is done in the existing and well known precedecessors of this result [15], [16]. The construction of the unitary map is the crucial step in finding the explicit and new inversion formula for the Radon transform which holds under the hypotheses assumed in [1], primarily the facts that the quasi regular representations of G on X and Ξ are both irreducible and both square integrable. This kind of consequence was one of our main motivations for adressing the issue of building the unitary "extension" in the context of homogeneous trees. The techniques used in [1] cannot be transferred directly to the case of homogeneous trees primarily because the quasi regular representation is not irreducible, much less square integrable, a fact that we explicitly recall in Appendix B. Hence, we adopt here a combination of the classical approach followed by Helgason in the symmetric space case [14] and the techniques that have been exploited in [2]. The paper is organized in three sections. In Section 1, we present the main notions and the relevant results in the theory of homogeneous trees. Then, we give a brief overview of the Helgason-Fourier transform. In Section 2, we recall the horocyclic Radon transform on homogeneous trees, we present its link with the Helgason-Fourier transform and we show its intertwining properties with quasi regular representations. Finally, in Section 3, we prove the unitarization theorem for the horocyclic Radon transform. For the reader's convenience, we add two short appendices. In the first, we briefly recall the notion of dual pair and in the second we indicate why the quasi regular representation on L 2 pXq is not irreducible.

Preliminaries
In Subsections 1.1 through 1.6 we recall the basic definitions and facts that will be used throughout, focusing on the space of horocycles. In particular, we describe the various group actions that are relevant in order to apply the machinery of dual pairs that was devised by Helgason (see Appendix A). Subsection 1.7 is devoted to a brief overview of the Helgason-Fourier transform. Standard references for these are [3], [8] and [10].

Homogeneous trees
A graph is a pair pX, Eq, where X is the set of vertices and E is the family of edges, where an edge is a two-element subset of X. We often think of an edge as a segment joining two vertices. If two vertices are joint by a segment, they are called adjacent. A tree is an undirected, connected, loop-free graph. In this paper we are interested in homogeneous trees. A q-homogeneous tree is a tree in which each vertex has exactly q`1 adjacent vertices. If q ě 1, a q-homogeneous tree is infinite. From now on, we suppose q ě 2 in order to exclude trivial cases, that is, segments and lines.
Given u, v P X with u ‰ v, we denote by ru, vs the unique ordered t-uple px 0 " u, x 1 , . . . , x t´1 " vq P X t , where tx i , x i`1 u P E and all the x i are distinct. We call ru, vs a (finite) t-chain and we think of it as a path starting at u and ending at v or, equivalently, as the finite sequence of consecutive 2-chains ru, x 1 s, rx 1 , x 2 s, . . . , rx t´2 , vs. With slight abuse of notation, if ru, vs " px 0 , . . . , x t´1 q we write u, v, x i P ru, vs and ru, x i s Y rx i , vs, i P t1, . . . , t´2u. In particular, if u and v are adjacent, both ru, vs, rv, us P X 2 are oriented, unlike the edge tu, vu P E which is not. A homogeneous tree X carries a natural distance d : XˆX Ñ N, where for every u, v P X the distance dpu, vq is the number of 2-chains in the path ru, vs.

The boundary of a homogeneous tree
An infinite chain is an infinite sequence px i q iPN of vertices of X such that, for every i P N, dpx i , x i`1 q " 1 and x i ‰ x i`2 . We denote by cpXq the set of infinite chains on X. We say that two chains px i q iPN and py i q iPN are equivalent if there exist m P Z and N P N such that x i " y i`m for every i ě N and, in such case, we write px i q iPN " py i q iPN . The boundary of X is the space Ω of equivalence classes cpXq{ ". Observe that an infinite chain identifies uniquely a point of the boundary, which may be thought of as a point at infinity. In fact, it is well known [7] that a homogeneous tree of even order q`1 can be isometrically embedded in the unit disc, the latter endowed with its hyperbolic metric, in such a way that the limit points of infinite chains correspond precisely to the points of the unit circle, the topological boundary of the unit disc.
We denote by p the canonical projection of cpXq onto Ω. For v P X and ω P Ω we write rv, ωq for the unique chain px i q iPN starting at v, i.e. x 0 " v, and "pointing at" the boundary point ω, i.e. pppx i q iPN q " ω. Furthermore, given ω 1 , ω 2 P Ω with ω 1 ‰ ω 2 , we denote by pω 1 , ω 2 q the unique infinite sequence of vertices px i q iPZ such that px´iq iPN P ω 1 and px i q iPN P ω 2 , with x 1 ‰ x´1, and we call it a doubly infinite chain. We fix an arbitrary reference point o P X. The boundary Ω is endowed with the topology (independent of the reference point) generated by the open sets Ωpuq " tω P Ω : u P ro, ωqu, u P X.
With this topology, Ω is a compact topological space. For later use, we remark that in every class ω P Ω, there is a unique infinite chain ro, ωq starting at o, and we denote by the set of all infinite chains starting at o. Clearly Γ o and Ω may be identified.

Horocycles
A 2-chain rv, us is said to be positively oriented with respect to ω P Ω if u P rv, ωq, otherwise we say that rv, us is negatively oriented.
For ω P Ω and v, u P X, we denote by κ ω pv, uq P Z the so-called horocyclic index of v and u w.r.t. ω, namely the number of positively oriented 2-chains (w.r.t. ω) in rv, us minus the number of negatively oriented 2-chains (w.r.t. ω) in rv, us. Clearly, |κ ω pv, uq| ď dpv, uq. It is easy to verify that, for every v, u, x P X and for every ω P Ω, κ ω pv, xq " κ ω pv, uq`κ ω pu, xq.
(1) Furthermore, we have the following result.
Proof. We fix x P X and we observe that Since rv, ωq " rx, ωq, then there exists N P N such that x N P rx, ωq and x N´1 R rx, ωq, with the understanding that if v P rx, ωq then N " 0. Thus, for all i ě N dpv, Furthermore, rv, xs " rv, x N s Y rx N , xs, where rv, x N s is the union of positively oriented 2-chains and rx N , xs is the union of negatively oriented 2-chains. Hence, pi´dpx, x i qq and this concludes the proof.
We are now in a position to introduce the horocycles.
h v ω,1 v ω Figure 2: A part of a 2-homogeneous tree containing portions of horocycles (unions of vertices lying on dashed lines) which are tangent to ω.

Definition 2.
For ω P Ω, v P X and n P Z, the horocycle tangent to ω of index n with respect to the vertex v is the subset of X defined as h v ω,n " tx P X : κ ω pv, xq " nu.
We denote by Ξ the set of horocycles.
It follows immediately from (1) that for every v, u P X, n P Z and ω P Ω Hence the mapping pv, ω, nq Þ Ñ h v ω,n is not injective and so Ξ is not well parametrized by XˆΩˆZ. However, for fixed v P X, the map pω, nq Þ Ñ h v ω,n is actually bijective, so that Ξ may be identified with ΩˆZ. Formally, for every v P X, there is a bijection and, for every fixed ω P Ω, X can be covered disjointly as By equality (2), for each pair of vertices u, v P X Ψ´1 u˝Ψv pω, nq " pω, n`κ ω pu, vqq.
The topology that Ξ inherits as product of Ω and Z is proved to be independent of the choice of v P X.

Group actions
Let G be the group of isometries on X, that is the group of bijections g : X Ñ X which preserve the distance d. The group G is unimodular and locally compact, and acts transitively on X by the action pg, xq Þ ÝÑ grxs :" gpxq, g P G.
We fix an arbitrary reference point o P X and we denote by K o the corresponding stability subgroup. It turns out that K o is a maximal compact subgroup of G and under the canonical bijection gK o Þ Ñ gros we have the identification X » G{K o . The group G acts on the boundary as well. Indeed, it is easy to see that if g P G and px i q iPN " py i q iPN , then pgrx i sq iPN " pgry i sq iPN , so that the transitive action of G on X induces a transitive action of G on Ω. Indeed, if px i q iPN P cpXq, then pgrx i sq iPN P cpXq as well, because dpgrx i s, grx i`1 sq " dpx i , x i`1 q " 1 and grx i s ‰ grx i`2 s since g P G. Furthermore, px i q iPN " py i q iPN implies that there exist m P Z and N P N such that dpgrx i s, gry i`m sq " dpx i , y i`m q " 0 for every i ě N and then pgrx i sq iPN " pgry i sq iPN . Precisely, the group G acts on Ω by the action This, in turn, induces a transitive action of K o on the set Γ o of infinite chains starting at o by means of pk, ro, ωqq Þ ÝÑ ro, k¨ωq, ω P Ω.
We fix ω 0 P Ω and we denote by K o,ω0 the stabilizer of ro, The group G of isometries of X acts transitively also on the space Ξ of horocycles through the action on vertices because the G-action maps horocycles in themselves. Indeed, if ξ P Ξ, ξ " h v ω,n , with v P X, ω P Ω, n P N and rv, ωq " px i q iPN , then for every g P G grξs " tgrxs : x P X, κ ω pv, xq " nu " tgrxs : by Proposition 1. Therefore G acts transitively on Ξ by Consider the horocycle If ro, ω 0 q " px i q iPN , then g.ξ 0 " tx P X : lim iÑ8 pi´dpx, grx i sqq " 0u.
Hence, the isotropy subgroup at ξ 0 is H " where H j is the subgroup of isometries fixing the sub-path rx j , ω 0 q P cpXq, see also [3]. Therefore, Ξ » G{H. Observe that H is the isotropy subgroup of G at h o ω0,n for every n P Z. Thus, by (2), H is the isotropy subgroup of G at every horocycle tangent to ω 0 , namely at h v ω0,n for every n P Z and v P X. Let τ P G be a one-step translation along pω 1 , ω 0 q, with ω 1 P Ωztω 0 u, where ω 0 is as in the definition of H (see [10] for further details on the one-step translations in G). Assume that if v P pω 0 , ω 1 q, then τ pvq P rv, ω 0 q. Furthermore, denote by A the subgroup of G generated by the powers of τ . It is easy to see that the group A acts on H by conjugation. Indeed, for every m P Z and g P H, we have where we use that τ m¨ω 0 " τ´m¨ω 0 " ω 0 . It has been proved in [17] that the resulting semidirect product H¸A has modular function ∆ph, τ m q " q m .
With slight abuse of notation, we write ∆   and in what follows, the same notation is used for its trivial extension to ΩˆZ. The function ∆ 1 2 is the analogue of the function e ρ in the theory of symmetric spaces (see [14]).

Measures
We endow X with the counting measure dx which is trivially G-invariant, and we denote by L 2 pXq the Hilbert space of square-integrable functions with respect to dx.
As far as Ω is concerned, recall that Ω is identified with Γ o on which K o acts transitively. Therefore, Γ o admits a unique K o -invariant probability measure µ o . We denote by ν o the measure on Ω obtained as the push-forward of µ o by means of the canonical projection p |Γ o : Γ o Ñ Ω. It has been shown in [10] that The measure ν o is G-quasi-invariant and, by definition, the Poisson kernel p o pg, ωq is the associated It is possible to prove [10] that for every F P L 1 pΩ, ν o q. Therefore, we can endow the boundary Ω with infinitely many measures which are absolutely continuous with respect to each other. In order to adequately describe the measure on Ξ relative to which we form the Lebesgue spaces L 1 pΞq and L 2 pΞq, we need the parametrization (3). The idea is to define compatible measures on ΩˆZ and Ξ in the sense that the natural pull-back of functions induced by the mapping Ψ v : ΩˆZ Ñ Ξ induces a unitary operator Ψv of the corresponding L 2 spaces. To this end, we consider the measure on Z with density q n with respect to the counting measure dn. We fix v P X and endow Ξ with the measure λ obtained as the push-forward of the measure ν v b q n dn on ΩˆZ by means of the map Ψ v , i.e.
which is independent of the choice of the vertex v (see [3]). We denote by L 1 pΞq and L 2 pΞq the spaces of absolutely integrable functions and square-integrable functions with respect to λ, respectively. Thus, by definition of λ, for every F P L 1 pΞq ż It is easy to verify that λ is G-invariant.
F˝Ψ v qqpω, nq, for almost every pω, nq P ΩˆZ. Clearly, Ψv is a unitary operator from L 2 pΞq into L 2 v pΩˆZq. Indeed, for every F P L 2 pΞq we have that ż and then Ψv is an isometry from L 2 pΞq into L 2 v pΩˆZq. Surjectivity is also clear.

Representations
Recall that X is endowed with the counting measure dx which is trivially G-invariant. Thus, the group G acts on L 2 pXq by the quasi regular representation π : G ÝÑ U pL 2 pXqq defined by πpgqf pxq :" f pg´1rxsq, f P L 2 pXq, g P G, where U pL 2 pXqq denotes the group of unitary operators of L 2 pXq. In Appendix B it is shown that π is not irreducible. Similarly, since λ is G-invariant, the group G acts on L 2 pΞq by the quasi regular unitary representationπ : G ÝÑ U pL 2 pΞqq defined bŷ πpgqF pξq :" F pg´1.ξq, F P L 2 pΞq, g P G.
These are the two representations in which we are interested.

The Helgason-Fourier transform on homogeneous trees
The Helgason-Fourier transform can be defined on homogeneous trees (see [8], [10], [11]) in analogy with the setup of symmetric spaces [14]. We briefly recall its definition and its main features. We put T " 2π{ logpqq, T " R{T Z » r0, T q and we denote by dt the normalized Lebesgue measure on T. Let C c pXq be the space of compactly supported functions on X.
As the Euclidean Fourier transform, the Helgason-Fourier transform extends to a unitary operator on L 2 pXq (see [10], [11]). The Plancherel measure involves a version of the Harish-Chandra c-function inspired by the symmetric space construction [12], namely the meromorphic function We put c q " q 2pq`1q (8) and we denote by L 2 v,c pΩˆTq the space of square-integrable functions on ΩˆT w.r.t. the measure c q |cp1{2`itq|´2 dν v dt.
Property 7. We say that f P L 2 v,c pΩˆTq satisfies Property 7 if the symmetry condition ż holds for every x P X and for almost every t P T. We denote by L 2 v,c pΩˆTq 7 the space of functions in L 2 v,c pΩˆTq satisfying Property 7.

The horocyclic Radon transform
In this section we recall the definition of the horocyclic Radon transform on homogeneous trees and its fundamental properties. As already mentioned, the horocyclic Radon transform is precisely the Radon transform à la Helgason relative to the dual pair pX, Ξq. The case of homogeneous trees is not covered by the general setup considered in [1] by the authors since the quasi regular representation π of G on L 2 pXq is not irreducible. For this reason, we can not apply the results presented in [1] in order to obtain a unitarization theorem and we therefore adopt an approach which mimics the one used in [15] and [2] in the case of the polar and the affine Radon transforms, respectively.
Definition 5. The horocyclic Radon transform Rf of a function f P C c pXq is the map Rf : Ξ Ñ C defined by Rf pξq " ÿ xPξ f pxq.
We recall that for every v P X there exists a bijection Ψ v : ΩˆZ Ñ Ξ given by pω, nq Þ Ñ h v ω,n and we shall write We need to introduce the Fourier transform on L 2 pZq. We denote by L 2 T the space of T -periodic functions f on R such that Let s P L 2 pZq, the Fourier transform F s of s is defined as the Fourier series of the T -periodic function with Fourier coefficients pspnqq nPZ . Precisely, where the series converges in L 2 T . The Parseval identity reads Furthermore, if s P L 1 pZq, for almost every t P T F sptq " ÿ nPZ spnqq int .
We are now ready to state the result which relates the Helgason-Fourier transform with the horocyclic Radon transform. For the reader's convenience, we include the proof.
Proposition 7 (Fourier Slice Theorem, version I, [4,8]). Let v P X. For every f P C c pXq and ω P Ω, A v f pω,¨q P L 1 pZq and for almost every t P T.
Proof. Let f P C c pXq and ω P Ω. By formula (4) Then, A v f pω,¨q is in L 1 pZq and applying again (4) we have that for almost every t P T and this concludes the proof.
We refer to Proposition 7 as the Fourier Slice Theorem for the horocyclic Radon transform in analogy with the polar Radon transform, see [15] as a classical reference.
In order to prove our first intertwining result, we show that Rf P L 2 pΞq for every f P C c pXq. Let f P C c pXq and v P X. By Parseval identity and Proposition 7 we have that ż Since f has finite support, then by the definition of the Helgason-Fourier transform, the inequality |κ ω pv, xq| ď dpv, xq and ν v pΩq " 1, the above leads to ż Therefore, Rf P L 2 pΞq for every f P C c pXq. The horocyclic Radon transform intertwines the regular representations of G on L 2 pXq and L 2 pΞq. This result is a direct consequence of the fact that X and Ξ carry G-invariant measures dx and dλ.
Proposition 8. For every g P G and f P C c pXq Rpπpgqf q "πpgqpRf q.
Proof. For all g P G and f P C c pXq for every ξ P Ξ.
We now introduce a closed subspace of L 2 pΞq which will play a crucial role because it is the range of the unitarization of the horocyclic Radon transform.
Let v P X. For every F P L 2 pΞq Hence, the function Ψv F pω,¨q is in L 2 pZq for almost every ω P Ω. Moreover, by Parseval identity and Fubini theorem Then, for almost every t P T the function pI b F qΨv F p¨, tq is in L 2 pΩ, ν v q and Property 5. We say that F P L 2 pΞq satisfies Property 5 if the symmetry condition ż holds for every v P X and for almost every t P T. We denote by L 2 5 pΞq the space of all such functions.
Our main results in Section 3 are based on the following characterization of L 2 5 pΞq. For every v P X, we denote by L 2 v pΩˆTq the space of square-integrable functions on ΩˆT w.r.t. the measure ν v b dt.
a.e. pω, tq P ΩˆT, is an isometry from L 2 pΞq into L 2 v pΩˆTq. Furthermore, for every other u P X Φ u F pω, tq " p u pv, ωq for almost every pω, tq P ΩˆT. Finally, a function F belongs to L 2 5 pΞq if and only if Φ v F satisfies Property 7.
By Proposition 9, F P L 2 5 pΞq implies that Φ v F satisfies (9) for every v P X. Conversely, if we want to prove that a function F P L 2 pΞq satisfies (11) it is enough to verify that (9) holds true for at least one, hence every, v P X. This last remark will prove very useful in our proofs.
Proof. By Parseval identity, for every F P L 2 pΞq we have that ż so that Φ v is an isometry from L 2 pΞq into L 2 v pΩˆTq. Now, let u P X and F P L 2 pΞq. For almost every ω P Ω we have that in L 2 T , we conclude that relation (12) holds true. Finally, let F P L 2 pΞq. For every x P X and for almost every t P T, (12) Then, for every x P X and almost every t P T ż This equality allows us to conclude that F P L 2 5 pΞq if and only if Φ v F satisfies (9) and this concludes our proof.
Corollary 10. For every f P C c pXq, in L 2 v pΩˆTq and Rf P L 2 5 pΞq. Proof. The proof follows immediately by Proposition 7 and the fact that the Helgason-Fourier transform satisfies (9). Some comments are in order. Proposition 9 with Corollary 10 show that RpC c pXqq Ď L 2 5 pΞq and it highlights the link between the range of the Radon transform with the range of the Helgason-Fourier transform, which will play a crucial role in our main result. The range RpC c pXqq has already been completely characterized in [6]. We recall the result in [6] for completeness and in order to understand the relation with L 2 5 pΞq.
Theorem 11 (Theorem 1, [6]). The range of the horocyclic Radon transform on the space of functions with finite support on X is the space of continuous compactly supported functions on Ξ satisfying the following two conditions (i) for some v P X, hence for every v P X, ř nPZ F˝Ψ v pω, nq is independent of ω P Ω; (ii) for every v P X and n P Z ż It is worth observing that condition (11) is the equivalent on the frequency side of equation (14) for continuous compactly supported functions on Ξ. As it will be made clear in the next section, condition (11) better suits our needs.

Unitarization and Intertwining
In order to obtain the unitarization for the horocyclic Radon transform that we are after, we need some technicalities. Figure 3 might help the reader to keep track of all the spaces and operators involved in our construction. Figure 3: Spaces and operators that come into play in our construction.
Let v P X. We set D v " tϕ P L 2 v pΩˆZq : pI b F qϕ P L 2 v,c pΩˆTqu and we define the operator J v : D v Ď L 2 v pΩˆZq Ñ L 2 v pΩˆZq as the Fourier multiplier where c q is given by (8). We define the set of functions and we consider the operator Λ : E Ď L 2 pΞq Ñ L 2 pΞq given by Lemma 12. The operator Λ is independent of the choice of v P X.
Proof. Take u P X and putΛ F " Ψů´1J u ΨůF.
We verify that Λ "Λ. By Proposition 9, it is sufficient to prove that Φ v pΛF q " Φ v pΛF q for every F P L 2 pΞq. For almost every pω, tq P ΩˆT (12) yields and we can conclude that Λ "Λ.
As a direct consequence of Lemma 12, for every v P X we have that for every F P E and for almost every pω, tq P ΩˆT The operator Λ intertwines the regular representationπ as shown by the next proposition.
Proposition 13. The subspace E isπ-invariant and for all F P E and g P Ĝ πpgqΛF " ΛπpgqF.
The next result follows directly by Proposition 9 and equation (15).

Corollary 14.
For every F P E , ΛF P L 2 5 pΞq if and only if F P L 2 5 pΞq. Proof. By Proposition 9, ΛF P L 2 5 pΞq if and only if Φ v pΛF q satisfies (9). By (15) and since t Þ Ñ |cp1{2`itq| is even, Φ v pΛF q satisfies (9) if and only if Φ v pF q satisfies (9), which is equivalent to F P L 2 5 pΞq. This concludes the proof. We are now in a position to prove our main result. which intertwines the representations π andπ, i.e.
Theorem 15 implies thatπ is not irreducible, too.
Proof. We first show that ΛR extends to a unitary operator Q from L 2 pXq onto L 2 5 pΞq. Let f P C c pXq and v P X. By the Fourier Slice Theorem (10), the Parseval identity and the definition of Λ, we have that Hence, ΛR is an isometric operator from C c pXq into L 2 pΞq. Since C c pXq is dense in L 2 pXq, ΛR extends to a unique isometry from L 2 pXq onto the closure of RanpΛRq in L 2 pΞq. We must show that ΛR has dense image in L 2 5 pΞq. The inclusion RanpΛRq Ď L 2 5 pΞq follows immediately from Corollary 10 and Corollary 14. Let F P L 2 5 pΞq be such that xF, ΛRf y L 2 pΞq " 0 for every f P C c pXq. By the Parseval identity and the Fourier Slice Theorem (10) we have that For simplicity of notation, we denote by Θ v F the function on ΩˆT defined as a.e. pω, tq P ΩˆT.
Hence we have proved that xΘ v F, H v f y " 0 for every f P C c pXq. The following two facts follow immediately from Proposition 9. Since Φ v is an isometry from L 2 pΞq into L 2 v pΩˆTq, then Θ v F belongs to L 2 v,c pΩˆTq. Furthermore, since F P L 2 5 pΞq and since t Þ Ñ |cp1{2`itq| is even, then Therefore, RanpΛRq " L 2 5 pΞq and ΛR extends uniquely to a surjective isometry Q : L 2 pXq ÝÑ L 2 5 pΞq.
Observe that Qf " ΛRf for every f P C c pXq. Then, the intertwining property (18) follows immediately from Proposition 8 and Proposition 13.
As a byproduct, one obtains an extended Fourier Slice Theorem.
Proposition 16 (Fourier Slice Theorem, version II). Let v P X. For every f P L 2 pXq pI b F qpΨv pQf qqpω, tq " ?
Proof. Let v P X. For every f P C c pXq, by (13) and (15) we have that for almost every pω, tq P ΩˆT. Let f P L 2 pXq, since C c pXq is dense in L 2 pXq, then there exists a sequence pf m q m Ď C c pXq such that f m Ñ f in L 2 pXq. Then, since Q is a unitary operator from L 2 pXq onto L 2 5 pΞq and Φ v is an isometry from L 2 pΞq into L 2 v pΩˆTq, then Φ v pQf m q Ñ Φ v pQf q in L 2 v pΩˆTq. Since f m P C c pXq for every m P N, Definition 17. We define the Radon transform of f : X Ñ C as the map Rf : Ξ Ñ C given by for any f for which the integral converges.
Interchanging the roles of X and Ξ, we can defině x 0 " K. ξ 0 , and for every x " gK P G{K we setx " g.x 0 " gK. ξ 0 , which is independent of the choice of the representative g of gK P G{K. We may think ofx as the sheaf of manifolds in Ξ passing through x P X. By definition, G acts transitively on the orbit tg.x 0 : g P Gu and we denote by r K the stability subgroup atx 0 . F pξqdµ 0 pξq, F P L 1 px 0 , dµ 0 q, k P K.
We push-forward the measure dµ 0 tox " pgKqˇby the mapx 0 Q ξ Þ Ñ g.ξ Px. Since the measurex 0 is K-invariant, the so obtained measure dµ x does not depend on the choice of the representative of x.
Definition 18. The dual Radon transform of F is the map R # F : X Ñ C given by for any F : Ξ Ñ C for which the integral converges.
We conclude this Appendix showing that the horocyclic Radon transform on homogeneous trees recalled in Section 2 is precisely the Radon transform à la Helgason for the dual pair pX, Ξq, where X is an homogeneous tree and Ξ is the set of horocycles on X. We keep the notation of Section 1.4. Once we have fixed the origins o P X and ω 0 P Ω, and consequently the closed subgroup H of G " AutpXq, we define the root horocycle ξ 0 as ξ 0 " Hros.
By direct computation, ξ 0 " h o ω0,0 , and (5) implies that ξ " h gros g¨ω0,0 . So that, Ξ " tgrξ 0 s : g P Gu is exactly the set of horocycles in X. By the definition of H, we have that H " tg P G : grξ 0 s " ξ 0 u.
Thus, condition (19) is satisfied and Ξ » G{H. We endow the root horocycle ξ 0 with the counting measure dµ 0 , which is an H-invariant measure. Then, all the horocycles in Ξ are equipped with the counting measure by pushing forward dµ 0 to ξ " gH by the map ξ 0 Q x Þ Ñ grxs P ξ. It is therefore clear that the horocyclic Radon transform on homogeneous trees is precisely the Radon transform in Definition 17 when X is an homogeneous tree and Ξ is the family of horocycles on X.