Radon Transform on the Torus

1660-5446/11/040463-9, published online December 1, 2010

© 2010 Springer Basel AG

Mediterranean Journal of Mathematics

Radon Transform on the Torus

Ahmed Abouelaz and Fran¸cois Rouvi`ere^∗

Abstract. We consider the Radon transform on the (flat) torus Tⁿ = Rⁿ/Zⁿ defined by integrating a function over all closed geodesics. We prove an inversion formula for this transform and we give a characteri- zation of the image of the space of smooth functions onTⁿ.

Mathematics Subject Classification (2010).Primary 53C65, 44A12.

Keywords.Torus, geodesic, Radon transform.

1. Introduction

Trying to reconstruct a function on a manifold knowing its integrals over a certain family of submanifolds is one of the main problems of integral geometry. In the framework of Riemannian manifolds a natural choice is the family of all geodesics. The simple example of lines in Euclidean space has suggested namingX-ray transform the corresponding integral operator, associating to a function f its integrals Rf() along all geodesics of the manifold.

Few explicit formulas are known to invert the X-ray transform. With no attempt to give an exhaustive list, let us quote Helgason [5] for Euclidean spaces, hyperbolic spaces and spheres, Berenstein and Casadio Tarabusi [4]

for hyperbolic spaces, Helgason [6] or the second author [7] for more general symmetric spaces, [8] for Damek-Ricci spaces etc.

We consider here the n-dimensional (flat) torusTⁿ =Rⁿ/Zⁿ and the Radon transform defined by integrating f along all closed geodesics of Tⁿ, that is all lines with rational slopes. Arithmetic properties will thus enter the picture, as in the case of Radon transforms onZⁿ already studied by the first author and collaborators (see [1, 2, 3]). Our present problem was introduced by Strichartz [9], who gave a solution for n = 2. But, relying on a special property of the two-dimensional case (see Remark 3.2 at the end of Section 3 below), his method does not extend in an obvious way to then-dimensional torus. The inversion formula proved here for Tⁿ (Theorem 3.1) makes use

∗Corresponding author.

464 A. Abouelaz and F. Rouvière Mediterr. J. Math.

of a weighted dual Radon transformR^∗_ϕ, with a weight functionϕto ensure convergence.

In Section 2 we describe a suitable set of parameters for the closed geodesics on the torus. Our main result (Theorem 3.1) is proved in Section 3. Section 4 is devoted to a range theorem (Theorem 4.1), characterizing the space of Radon transforms of all functions inC^∞(Tⁿ).

2. Closed geodesics of the torus

The following notation will be used throughout.

Notation. Let x·y = x₁y₁+· · ·+x_ny_n be the canonical scalar product of x, y∈Rⁿ.

Forp= (p₁, . . . , p_n)∈Zⁿ\0 the setI(p) ={k·p , k∈Zⁿ} is an ideal ofZ, not{0}, and we shall denote byd(p) =d(p₁, . . . , p_n) its smallest strictly positive element. Thusd(p) is the highest common divisor ofp₁, . . . , p_n and I(p) =d(p)Z. Let

P ={p= (p₁, . . . , p_n)∈Zⁿ\ {0} | d(p₁, . . . , p_n) = 1} and, fork∈Zⁿ,

Pk ={p∈ P |k·p= 0}.

For n≥2 the n-dimensional torus Tⁿ =Rⁿ/Zⁿ is equipped with the (flat) Riemannian metric induced by the canonical Euclidean structure ofRⁿ and the corresponding (translation invariant) measure dx; thus

Tⁿdx = 1. We denote by pr :Rⁿ→Tⁿ the canonical projection.

All functions considered here are complex-valued.

InTⁿ the geodesic from x= (x₁, . . . , x_n)∈ Tⁿ with (non zero) initial speedv= (v₁, . . . , v_n)∈Rⁿ is the line

={x+ pr(tv), t∈R}.

Since v = 0 the set of all t such that tv belongs to Zⁿ is a discrete subgroup of R, that is TZfor some T ≥0. Thus the geodesic is closed if and only ifT >0. In this case we set T v =p= (p₁, . . . , p_n)∈Zⁿ\ {0}, and the p_j’s are relatively prime: a nontrivial common divisor would contradict the definition ofT. Thus pbelongs toP.

Now forp∈ P andt∈Rwe have

tp∈Zⁿ⇐⇒t∈Z.

Indeed the set of such t is a discrete subgroup τZ of R (for some τ ≥ 0), obviously containing 1. Thusτ = 1/mfor some integer m >0, so that m is a common divisor to allp_j’s, which impliesm= 1 andτ= 1.

Forx∈Tⁿ andp∈ P we denote by

(x, p) ={x+ pr(tp), t∈R}

the corresponding closed geodesic from x. By the above remark the map t→x+pr(tp) induces a bijection fromR/Zonto(x, p), since pr(tp) = pr(tp)

if and only if (t−t)p ∈Zⁿ that is t−t ∈Z. We shall therefore lett run over [0,1[ only in the sequel. The length of(x, p) is p =_n

j=1p²_j 1/2

. Lemma 2.1. Letx, y∈Tⁿ andp, q∈ P. The following are equivalent:

(i) (x, p) =(y, q)

(ii) q=±p and there existss∈Rsuch thaty=x+ pr(sp).

The set of closed geodesics fromxis therefore in one-to-one correspon- dence withP/{±1}.

Proof. (ii) implies (i) since, for anys,

(x, p) =(x,−p), (x+ pr(sp), p) =(x, p). (2.1) (i) implies (ii). Conversely, assume(x, p) =(y, q). Sinceybelongs to(x, p) there existss₀∈Rsuch thaty=x+ pr(s₀p). More generally, for anyt∈R there existss∈Rsuch thaty+ pr(tq) =x+ pr(sp). Replacingsby s+s₀ we have: for anyt∈Rthere exists∈Rand z∈Zⁿsuch that

tq=sp+z.

Let us fixj∈ {1,2, . . . , n}so thatp_j= 0. Fixingl=j let

k= (k₁, . . . , k_n) withk_j=p_l,k_l=−p_j and k_m= 0 ifm=j, l.

Thenk·p= 0, thereforet(k·q) =k·z is an integer for anyt∈R. It follows thatk·q= 0, that is

q_jp_m=p_jq_m (2.2)

for any m= 1, . . . , n(includingm=j). Applying dwe get d(q_jp) =d(p_jq) whence|q_j|d(p) =|p_j|d(q), that isp_j=±q_j sincep, q∈ P. In view ofp_j = 0,

(2.2) givesp=±qas claimed.

3. An inversion formula

3.1. Letf be a continuous function onTⁿ. We define itsX-ray transformRf as the integral off over closed geodesics ofTⁿ, namely

Rf((x, p)) = 1

0

f(x+ pr(tp))dt, (3.1)

withx∈Tⁿandp∈ P. As noted in the previous sectionx+ pr(tp) runs over the whole geodesic(x, p) whentvaries from 0 to 1.

The natural dual transformR^∗is obtained by summing over all closed geodesics through a given point, that is

R^∗F(x) = 1 2

p∈P

F((x, p)),

whereF is a function on the set of all closed geodesics and the factor 1/2 is introduced because of (2.1). However such an operator is not even defined on

466 A. Abouelaz and F. Rouvière Mediterr. J. Math.

constant functions and we shall rather replace it by a weighted dual transform as follows. Byweight function onP we mean

ϕ:P →]0,∞[ such thatϕ(−p) =ϕ(p) and

p∈P

ϕ(p)<∞, (3.2) for instance the restriction toP of any strictly positive even function inl¹(Zⁿ) such asϕ(p) =e^−p orϕ(p) = (1 + p )⁻ⁿ⁻¹. Theweighted dual transform R^∗_ϕ is then defined as

R^∗_ϕF(x) = 1 2

p∈P

ϕ(p)F((x, p)) (3.3)

and the series converges absolutely wheneverF is a bounded function on the set of all closed geodesics. The transform R^∗_ϕ is dual to R in the following sense

Tⁿ

R^∗_ϕF(x)f(x)dx= 1 2

p∈P

ϕ(p)

Tⁿ

F((x, p))Rf((x, p))dx, (3.4) valid iff is continuous onTⁿ,F is bounded andx→F((x, p)) is continuous on Tⁿ for any p. Indeed, by (2.1) and (3.3), R^∗_ϕF(x) = ¹₂

pϕ(p)F((x− pr(tp), p)) for any t∈Rand the left-hand side of (3.4) is

Tⁿ

R^∗_ϕF(x)f(x)dx = 1 2

p

ϕ(p)

Tⁿ

F((x−pr(tp), p))f(x)dx

= 1

2

p

ϕ(p)

Tⁿ

F((x, p))f(x+ pr(tp))dx.

Then (3.4) follows by integration with respect to t∈[0,1]. The calculations are valid since, for anyt,

p

ϕ(p)

Tⁿ|F((x, p))||f(x+ pr(tp))|dx≤

p

ϕ(p) sup|F|sup|f|<∞. 3.2. Several classical inversion formulas for Radon transforms involveR^∗Rf.

As noted before the sum defining it does not converge here in general (not even for a constant function f) and we shall use R^∗_ϕRf instead with R^∗_ϕ defined by (3.3). As usual we denote by¹

f(k) =

Tⁿ

f(x)e^−2iπk·xdx

with k ∈ Zⁿ the Fourier coefficients of f. Let us recall the notation P_k = {p∈ P |k·p= 0}.

Theorem 3.1. Let ϕbe a weight function onP satisfying (3.2) and, fork∈ Zⁿ,

ψ(k) = 1 2

p∈Pk

ϕ(p).

1The exponentiale^−2iπk·xis of course unambiguously defined, withx∈Rⁿ/Zⁿreplaced by any representative inRⁿ.

Thenψis strictly positive onZⁿ, the operatorR^∗_ϕRis a convolution operator onTⁿ(n≥2) and, for any continuous functionf onTⁿsuch thatf∈l¹(Zⁿ), the X-ray transformR is inverted by

f(x) =

k∈Zⁿ

Tⁿ

e^{2iπk·(x−y)}

ψ(k) R^∗_ϕRf

(y)dy, x∈Tⁿ.

This inversion formula applies in particular to any function f ∈Cⁿ(Tⁿ).

A convolution inverse forR^∗_ϕRis thusformally given by the series

k

e^2iπk·xψ(k)⁻¹.

This series does not converge absolutely however:

kψ(k)⁻¹ = ∞ since ψ(lk) =ψ(k)>0 for any strictly positive integerl.

Proof. (i) Our definitions imply, for any continuousf, R_ϕ^∗Rf(y) = 1

2

p∈P

ϕ(p)Rf((y, p)) = 1 2

p∈P

ϕ(p) ₁

0

f(y+ pr(tp))dt

= < S_ϕ(x), f(y−x)>, whereS_ϕ is the distribution onTⁿ defined by

< S_ϕ, f >= 1 2

p∈P

ϕ(p) ₁

0

f(−pr(tp))dt.

Indeed the estimate|< S_ϕ, f >| ≤ ¹₂

p∈Pϕ(p)

sup|f| shows thatS_ϕ is actually a measure onTⁿ. Thus

R_ϕ^∗Rf =S_ϕ∗f (3.5)

(convolution onTⁿ), and this convolution equation can be easily inverted by means of Fourier coefficients. From (3.5) we have

R_ϕ^∗Rf(k) =S_ϕ(k)f(k), with

S_ϕ(k) =< S_ϕ(x), e^−2iπk·x>= 1 2

p∈P

ϕ(p) ₁

0

e^2iπtk·pdt.

The integral vanishes wheneverk·p= 0, therefore S_ϕ(k) = 1

2

p∈Pk

ϕ(p) =ψ(k) .

(ii) Given an arbitrary k= (k₁, . . . , k_n)∈Zⁿ we claim that ψ(k)>0.

Indeedϕ >0 and the set Pk is nonempty:

• ifk_j= 0 for allj, then Pk =P.

• if k_j = 0 for some j and k_l = 0 for all l =j, then Pk is the set of all p∈ P such thatp_j= 0.

468 A. Abouelaz and F. Rouvière Mediterr. J. Math.

• if k_j = 0 and k_l = 0 for some j, l with j = l, then Pk contains p = (p₁, . . . , p_n) with

p_j = k_l

d(k_j, k_l) ,p_l=− k_j

d(k_j, k_l) ,p_m= 0 ifm=j, l .

Finally, in view of the assumptions f ∈ C(Tⁿ) and f ∈ l¹(Zⁿ), the Fourier inversion applies to f whence, by(i),

f(x) =

k∈Zⁿ

f(k)e^2iπk·x=

k

1

ψ(k)R^∗_ϕRf(k)e^2iπk·x for allx∈Tⁿ and the inversion formula follows.

(iii) If f belongs to Cⁿ(Tⁿ) we have ∂_jⁿf(k) = (2iπk_j)ⁿf(k) (with

∂_j = ∂/∂x_j) and

k∈Zⁿk²ⁿ_j |f(k)|² < ∞ by Parseval’s formula applied to

∂_jⁿf ∈L²(Tⁿ). Therefore

⎛

⎝

k∈Zⁿ,k=0

|f(k)|

⎞

⎠

2

≤

k∈Zⁿ,k=0

k₁²ⁿ+· · ·+k²ⁿ_{n −1}

k∈Zⁿ

k²ⁿ₁ +· · ·+k²ⁿ_n

|f(k)|²<∞ by Cauchy-Schwarz inequality, thusfbelongs to l¹(Zⁿ).

Variant of the proof. Though natural, the distributionS_ϕ can be skipped in the first part of the proof of Theorem 3.1. Given a functionF onTⁿ× P let us write

F(k, p) =

Tⁿ

F(x, p)e^−2iπk·xdx. (3.6) We then have the following ”Fourier slice theorem”, with f continuous on Tⁿ (and a slight abuse of notation),

Rf(k, p) = f(k) ifp∈ Pk

0 otherwise, (3.7)

emphasizing the important rˆole in our problem of the set Pk of all (k, p)∈ Zⁿ× P such thatk·p= 0. Indeed

Rf(k, p) = ₁

0

dt

Tⁿ

f(x+ pr(tp))e^−2iπk·xdx

=

Tⁿ

f(x)e^−2iπk·xdx ₁

0

e^2iπtk·pdt and (3.7) follows.

Since

p∈Pϕ(p)<∞and|Rf((x, p))| ≤sup|f| the series R^∗_ϕRf(x) =1

2

p∈P

ϕ(p)Rf((x, p))

converges uniformly on Tⁿ, therefore R^∗_ϕRf(k) = 1

2

p∈P

ϕ(p)Rf(k, p) = 1 2

p∈P_k

ϕ(p)f(k) =ψ(k)f(k), and the proof ends as before.

Remark 3.2. Forn= 2 andk= 0 the setPk only has two elements:

Pk={p(k),−p(k)} withp(k) =

k₂

d(k₁, k₂),− k₁ d(k₁, k₂)

.

Indeed k₁p₁ =−k₂p₂ withd(p₁, p₂) = 1 impliesk₁ =lp₂ andk₂=−lp₁ for somel∈Z, whenced(k₁, k₂) =|l|andp=±p(k).

The finiteness of Pk in this case was the key to Strichartz’ inversion formula forn= 2 in [9] p. 422. Writinge_k(x) =e^2iπk·x he observed that, for k = 0, Re_k((x, p)) = e_k(x) if p ∈ Pk,Re_k((x, p)) = 0 ifp /∈ Pk (cf. (3.7) above), therefore

e_k(x) = 1 2

p∈P

Re_k((x, p))

where the sum only contains two (equal) terms. Multiplying by the Fourier coefficientf(k) and summing over allk= 0 he obtained

k=0

f(k)e_k(x) = 1 2

p∈P

R

⎛

⎝

k=0

f(k)e_k

⎞

⎠(x, p),

that is

f(x)−f(0) =1 2

p∈P

Rf((x, p))−f(0)

since Rc = c obviously for any constant c. This is actually an inversion formula forRbecause

f(0) = 1

0

1 0

f(x₁, x₂)dx₁dx₂= 1

0

Rf((x(s), p₀))ds withx(s) = (s,0) andp₀= (0,1) for instance.

Remark 3.3. Strichartz’ method does not extend in an obvious way to Tⁿ for n > 2, the sets Pk being infinite. Indeed let n ≥ 3 and k ∈ Zⁿ. If (k₁, k₂) = (0,0), Pk contains (l,1,0, . . . ,0) for all l ∈Z. If (k₁, k₂)= (0,0) let k_j =k_j/d(k₁, k₂). By Bezout’s theorem there existq₁, q₂ ∈Z such that k₁q₁+k₂q₂+k₃= 0, therefore

(q₁+lk₂, q₂−lk₁, d(k₁, k₂),0, . . . ,0)

is orthogonal tok for alll∈Z. Dividing the first three components by their highest common divisor we obtain elements ofPk, easily seen to be distinct whenlruns overZ.

470 A. Abouelaz and F. Rouvière Mediterr. J. Math.

Remark 3.4. As noted above we can pick, for any k∈Zⁿ, an elementp(k) ofP such that k·p(k) = 0. By (3.7) we havef(k) =Rf(k, p(k)) therefore

Tⁿ|f(x)|²dx=

k∈Zⁿ

f(k)²=

k∈Zⁿ

Rf(k, p(k))².

This may be viewed as a Plancherel type theorem, expressing the L² norm off by means of its Radon transform.

4. A range theorem

In order to state the next theorem we shall denote by V the space of all functionsF onTⁿ× P satisfying the following three conditions:

(i) for any p ∈ P the map x → F(x, p) belongs to C^∞(Tⁿ) and, for any multi-index α∈Nⁿ, there exists a constantC_α such that |∂^α_xF(x, p)| ≤C_α for all (x, p)∈Tⁿ× P

(ii) F(k, p) = 0 whenever k∈Zⁿ,p∈ P andp /∈ Pk

(iii) F(k, p) =F(k, q) wheneverk∈Zⁿ andp, q∈ Pk.

Properties(ii) and (iii) of F defined by (3.6) are the ”moment conditions”

relevant to our problem.

Theorem 4.1. The X-ray transform f →F, with F(x, p) =Rf((x, p)), is a bijection ofC^∞(Tⁿ)ontoV.

Proof. Given f ∈ C^∞(Tⁿ) the function F(x, p) = Rf((x, p)) = ₁

0 f(x+ pr(tp))dtclearly satisfies (i). And(ii), (iii) follow from (3.7).

By Theorem 3.1 the map f → F is injective; only the surjectivity re- mains to be proved. GivenF ∈ V let

g(k) =F(k, p) for any p∈ Pk, (4.1) well defined by assumption(iii), and let us consider the Fourier series

f(x) =

k∈Zⁿ

g(k)e^2iπk·x. (4.2)

By(i) for anyl ∈Nthere exists a constant C_l such that|F(k, p) | ≤C_l(1 + k )^−lfor allk∈Zⁿ,p∈ P. The series (4.2) therefore defines aC^∞function on the torus. Finally let Gbe the function defined byG(x, p) =Rf((x, p)).

Using (3.7), (4.2) and (4.1) successively we have, fork·p= 0,

G(k, p) =f(k) =g(k) =F(k, p).

Besides, by (3.7) again and (ii), G(k, p) = 0 = F(k, p) for k·p= 0. Thus

GandFcoincide and it follows thatRf((x, p)) =F(x, p), which completes

the proof.

Acknowledgment

The authors are grateful to the referee for several helpful remarks.

References

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3) (2010), 197–220.

[4] C. Berenstein and E. Casadio Tarabusi, Inversion formulas for the k-dimen- sional Radon transform in real hyperbolic spaces, DukeMath.J.62(1991), 613–

631.

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[6] S. Helgason, The Abel, Fourier and Radon transforms on symmetric spaces, Indag.Mathem.16(2005), 531–551.

[7] F. Rouvi`ere,Transformation aux rayons X sur un espace sym´etrique, Comptes Rendus Acad. Sci. Paris, Ser. I,342(2006), 1–6.

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Ahmed Abouelaz

D´epartement deMath´ematiques et Informatique Facult´e des Sciences A¨ın Chock

Route d’ElJadida, Km8, B.P. 5366Maˆarif 20100 Casablanca

Morocco

e-mail:a.abouelaz@fsac.ac.ma Fran¸cois Rouvi`ere

LaboratoireJ.A. Dieudonn´e Universit´e de Nice, ParcValrose 06108Nice cedex 2

France

e-mail:frou@unice.fr

URL:http://math.unice.fr/~frou Received:January8, 2010.

Revised:July 5, 2010.

Accepted: October 12, 2010.