The d-plane radon transform on the torus tn

THE d-PLANE RADON TRANSFORM ON THE TORUS Tⁿ

Ahmed Abouelaz

Abstract

We define and study thed-plane Radon transform, namelyR,on then- dimensional (flat) torus. The transformationRis obtained by integrating a suitable functionf over alld-dimensional geodesics (d-planes in the torus).

We specially establish an explicit inversion formula of R and we give a characterization of the image, under the d-plane Radon transform, of the space of smooth functions on the torus.

MSC 2010: Primary 44A12; Secondary 05B35, 11D04

Key Words and Phrases: d-plane Radon transform on the torus, Fourier transform, inversion formula, characterization of the range.

1. Introduction

The Radon transform is a fundamental topic in the integral geometry.

It plays a preponderant role in many various fields of mathematics, physics and medecine.

This transform was studied and generalized in the case of Grassman manifolds and projective spaces, by many authors for example: I.M. Gelfand, M.I. Graev, and S.I. Shapiro[6],S. Helgason [8],B. Rubin[11],F. Rouvi´ere [10].

c 2011 Diogenes Co., Sofia

pp. 233–246 , DOI: 110.2478/s13540-011-0014-8

We briefly recall the definition of the classicald-plane Radon transform R_c on the Euclidean space Rⁿ, d being an integer such that 1≤d≤n−1, with n≥2.The classicald-plane Radon transformR_c is defined by

R_cf(ξ) =

ξ

f(x)d_m(x), (1.1)

where f is a complex-valued C^∞-function with compact support, d_m(x) is the Euclidean measure on the d-plane ξ. The essential problem of the integral geometry is to recover f from R_cf.

It is noted that the inversion formula for the Radon transform is closely related to fractional calculus, see [12], pages 29 and 30, formula (2.4). See also the papers [9], [13], [14], [2], [1].

The goal of this paper is to define an analogue of (1.1) and to give a inversion formula, that is to recover a continuous function f on the n- dimensional torus from the d-plane Radon transformRf.

The integral geometry in the torus was studied by I.M. Gelfand, S.G.

Gindikin and M.I. Graev in their book [7]. These authors have given the inversion formula based on the Poisson formula (see formulas (5.8), (5.13) of [7]). In addition, the inversion formula of Radon transform on the torus given in the book [7] is different from ours (see [5], see also formula 4.1 in Section4). In this work, we mainly use algebraic-arithmetic techniques and also the techniques of Fourier analysis (see Theorem4.1and its demonstra- tion). The origin of this problem was introduced by Strichartz [15], who gave a solution for n= 2. But, his method does not extend in an obvious way to n-dimensional torus (see Remark 3.2 in the paper [5]).

We consider here the n-dimensional (flat) torus Tⁿ = (R/Z)ⁿ and the d-plane Radon transform defined by integrating f along all d-dimensional geodesics (d-planes in the torus) of Tⁿ. Arithmetic techniques are used in this paper, as in the case of the Radon transform on Zⁿ, already studied by the author and his collaborators (see [3], [4]). Our purpose of this paper is to establish an explicit inversion formula for the d-plane Radon transform on the torus Tⁿ.

Our paper is organized as follows:

In Section 2, we shall fix some notations and also recall the properties of the parametrization of the discrete d-planes in Zⁿ,see [3] and [4], that will be useful in the sequel of this paper.

In Section 3, we describe a suitable set of parameters for the closed d-dimensional geodesics (d-planes in the torus). The main result of this section is the parametrization’s lemma of the d-planes of the torus Tⁿ (see Lemma 3.2). This lemma allows us to define the d-plane Radon dual of the operator R on the torus Tⁿ.

In Section 4, we establish an inversion formula for the d-plane Radon transform (see Theorem 4.1) by extending the inversion theorem proved in [5]. We note that several classical inversion formulas for the Euclidean d-plane Radon transform involveR^∗_cR_cf.However, in the case of the torus, the sum defining R^∗Rf is not convergent in general. We shall therefore introduce a weight function (see Theorem 4.1). This result is the main theorem in this paper.

Section 5 is devoted to a range theorem characterizing the space R(C^∞(Tⁿ), whereC^∞(Tⁿ) is the space of all complex-valuedC^∞-functions on Tⁿ. See Theorem5.1for more details.

2. Notations and preliminaries

In this section, we shall fix once and for all some notations that will be useful in the sequel of this paper. For m ∈ N such that 1 ≤ m ≤n, the set consisting of all integer m×nmatrices is designated by M_m,n(Z).We denote by SL_∗(k,Z) (1 ≤k ≤ n) the group of all integer k×k matrices whose determinant is equal to ±1.

It is noted that throughout this paper, every row vectorx= (x₁, . . . , x_n)

∈Zⁿ is assumed to be identified with its transpose column vector ^tx given by

tx=

⎛

⎜⎝ x₁

... x_n

⎞

⎟⎠.

A matrix D= (d_ij)1≤i≤m,1≤j≤n∈M_m,n (Z) of rankr (1≤r≤m≤n) is called to be diagonal if d_ii =d_i for i= 1, . . . , r, and d_ij = 0 otherwise.

In the sequel we note this matrix by D = diag(d₁, . . . , d_r,0, . . . ,0), (see [3],[4]).

We begin with the following fundamental theorem which associates to each integer matrixA∈M_m,n(Z) of rankr a diagonal matrixD∈M_m,n(Z) uniquely defined by A and called the Smith normal form ofA.

Theorem 2.1. (See [3],[4]) Let A be a matrix inM_m,n (Z) of rank d, with 1 ≤ m ≤ n. Then there exist two matrices L ∈ SL_∗(m,Z) and V ∈SL_∗(n,Z) such that

LAV =D=diag(l₁, . . . , l_d,0, . . . ,0),

where l_i>0for all i= 1, . . . , dand l_i|l_i+1 fori= 1, . . . , d−1.

In the special case when m=d(1≤d≤n) the matrixD of the above theorem becomes

D=

⎛

⎜⎜

⎜⎝

l₁ 0 . . . 0 0 . . . 0 0 l₂ . . . 0 0 . . . 0 ... ... . .. ... ... . . . ... 0 0 . . . l_d 0 . . . 0

⎞

⎟⎟

⎟⎠. (2.1)

Let D₀ be the matrix obtained by replacing, in the above matrix, l_i by 1 for i= 1, . . . , d. In the following, we denote by M_d,n(Z) the set of all matrix of rank d, and Pd,n the subset of all elements A of M_d,n(Z) such that A = QD₀V, where Q ∈ SL_∗(d,Z) and V ∈SL_∗(n,Z), see [3], [4] for more details.

Remark that ifA= (a_ij)_1≤i≤d

1≤j≤n ∈ P_d,n,then for all k= 1, . . . , d the row v_k = (a_k1, . . . , a_kn) of the matrix A belongs to P, where

P ={a∈Zⁿ\ {0} |d(a) = 1},

with d(a) is the greatest common divisor ofa= (a₁, . . . , a_n). In addition the vectors system (v_k)_1≤k≤d is linearly independent (see [3],[4]). In the sequel, we will need the following result.

Theorem 2.2. (See [3],[4]) Let (A, b),(A, b)∈ P_d,n×Z^d.Then the following two assertions are equivalent:

(1) The two systems of linear diophantine equationsAx=b andAx= b have the same solutions inZⁿ.

(2) There exists a matrix Q∈ SL_∗(d,Z) such that A =QA and b = Qb.

For the proof, see [[3], Theorem 3.7, page 205] and also [4].

In the following, we denote by (e_k)_1≤k≤d an orthonormal basis ofR^d. 3. Closed d-dimensional geodesics (d-planes) of the torus For n≥2 let Tⁿ = (R/Z)ⁿ be then-dimensional torus equipped with the flat Riemannian metric induced by the canonical Euclidean structure of Rⁿ. In Tⁿ thed-dimensional geodesic from x = (x₁, . . . , x_n)∈ Tⁿ with (non zero) linearly independent vectors of Rⁿ is

D=

⎧⎨

⎩x+ pr

⎛

⎝^d

j=1

t_jv_j

⎞

⎠|t_j ∈Rand 1≤j≤d

⎫⎬

⎭,

where pr :Rⁿ→Tⁿis the natural projection. Ifd= 1,the setDcoincides with the line l

l={x+ pr (tv)|t∈R},

where the initial speed v ∈ Rⁿ, see [5] for more details. Since v₁, . . . , v_d are non zero vectors, the setGof all (t₁, . . . , t_d)∈R^dsuch that _d

j=1t_jv_j belongs toZⁿ is a discrete subgroup of R^dthat is of the form

G=Zv₁+· · ·+Zv_d,

butGcontains the subgroup of all (0, . . .0, t_i,0, . . . ,0) such thatt_iv_i ∈Zⁿ. By [5] we have τ_iv_i = (a_i1, . . . , a_in)∈ P fori= 1, . . . , d, andτ_i >0. That

is, ⎧

⎪⎨

⎪⎩

τ₁v₁ = (a₁₁, . . . , a_1n) ...

τ_dv_d= (a_d1, . . . , a_dn) .

For d= 1, we find again the results of the paper [5]. It is noted that τ_iv_i ∈ P for all i= 1, . . . , d. Let A= (a_ij)_1≤i≤d

1≤j≤n

be the matrix which is of rank d, since the non zero vectors v₁, . . . , v_d are linearly independent and τ_i>0 for alli= 1,2, . . . , d.

Now, we state and prove some results which will be useful in the sequel of this paper. We begin with the following lemma.

Lemma 3.1. Let A= (a_ij)_1≤i≤d

1≤j≤n ∈ P_d,n and (t₁, . . . , t_d)∈R^d.Then (t₁, . . . , t_d)A∈Zⁿ if and only if(t₁, . . . , t_d)∈Z^d. (3.1) P r o o f. Ifd= 1,the equivalence (3.1) can be reduces to

t₁(a₁₁, . . . , a_1n)∈Zⁿ if and only ift₁ ∈Z.

The above equivalence has been proved in [5]. Let x = (t₁, . . . , t_d), taking A = (a_ij)_1≤i≤d

1≤j≤n ∈ P_d,n,by Theorem 2.1, there exist Q∈SL_∗(d,Z) and V ∈SL_∗(n,Z) such that A=QD₀V,therefore ^tA^tx= ^tV ^tD₀^tQ^tx∈ Zⁿ, and this implies that ^tD₀^tQ^tx ∈ Zⁿ because ^tV ∈ SL_∗(n,Z). That is, ^tQ^tx ∈ Z^d since ^tD₀ =

I_d 0

, and finally ^tx ∈ Z^d because ^tQ ∈

SL_∗(d,Z).This proves the lemma. 2

For x ∈ Tⁿ and A ∈ P_d,n, we denote by D(x, A) the d-plane in the torusTⁿ given by

D(x, A) =

x+ pr ((t₁, . . . , t_d)A)|(t₁, . . . , t_d)∈R^d .

The map (t₁, . . . , t_d)∈R^d→x+ pr ((t₁, . . . , t_d)A) induces a bijection from T^d onto D(x, A), since we have

pr ((t₁, ..., t_d)A) = pr

t₁, ..., t_d A

if and only if

t₁−t₁, ..., t_d−t_d

∈Z^d, by the above lemma. We shall therefore lett_j (j= 1, . . . , d) run over [0,1[

only in the sequel.

Lemma 3.2. (Parametrization) Let x, y ∈Tⁿ and A, A ∈ P_d,n.We have D(x, A) =D(y, A) if and only if there exist a matrixQ∈SL_∗(d,Z) and (s₁, . . . , s_d)∈R^d such thaty=x+ pr ((s₁, . . . , s_d)A) and A =QA.

The set of closed d-dimensional geodesics fromx is therefore in one-to- one correspondence with P_d,n/SL_∗(d,Z).

P r o o f. It is clear that

D(x, A) = D(x, QA) =D(x+ pr ((s₁, . . . , s_d)QA), QA), (3.2) for any (s₁, . . . , s_d)∈R^dand for allQ∈SL_∗(d,Z).Indeed, lety∈D(x, A), y can be written in the form

y = x+ pr ((t₁, . . . , t_d)A)

= x+ pr

(t₁, . . . , t_d)Q⁻¹QA

= x+ pr ((s₁, . . . , s_d)QA),

where (s₁, . . . , s_d) = (t₁, . . . , t_d)Q⁻¹. If y ∈ D(x, QA) the element y have the form x+ pr((r₁, . . . , r_d)QA) which is equal to x+ pr ((t₁, . . . , t_d)A), where (t₁, . . . , t_d) = (r₁, . . . , r_d)Q. Conversely, assume that D(x, A) = D(y, A) whereA= (a_ij)_1≤i≤d

1≤j≤n

and A= (a_ij)_1≤i≤d

1≤j≤n

.The equality D(x, A)

=D(y, A) is equivalent toD(x, A) =D(x, A).IndeedD(x, A) =D(y, A) shows that x ∈ D(y, A), it follows that y = x −pr (sA) where s = (s₁, . . . , s_d) ∈ R^d but D(x−pr (sA), A) = D(x, A). Now, assume that D(x, A) =D(x, A). This equality shows that for all t= (t₁, . . . , t_d)∈ R^d there exists (s₁, . . . , s_d) =s∈R^d such that

tA−sA ∈Zⁿ,

for allt= (t₁, . . . , t_d)∈R^d.But by [3], Proposition 3.10, page 209; see also [4], Proposition 2.11, page 307, we have

tA−sA ∈

b∈Z^d

H(A, b) =

b∈Z^d

H A, b

(disjoint union), where H(A, b) is ad-plane ofZⁿ.Thus

tA−sA ∈H(A, b(t, s)), and

tA−sA∈H

A, b(t, s) ,

where b(t, s) and b(t, s) are vectors of Z^d which depend on t and s. This gives the two systems of linear diophantine equations

⎧⎨

⎩

A(tA−sA) =b(t, s) A(tA−sA) =b(t, s)

tA−sA∈Zⁿ, for all t= (t₁, . . . , t_d)∈R^d.

These two systems have the same solutions, therefore by Theorem 2.2, Section 2, there exists Q ∈ SL_∗(d,Z) such that A = QA and b(t, s) =

Qb(t, s),and this proves the lemma. 2

4. An inversion formula

Let f be a continuous function on Tⁿ. We define its d-plane Radon transform Rf as the integral of f over closed d-dimensional geodesics namely (d-planes in the torus) onTⁿ by

Rf(D(x, A)) = ₁

0 · · · ₁

0 f(x+ pr((t₁, . . . , t_d)A)dt₁. . . dt_d, (4.1) withx∈Tⁿ,A∈ P_d,n.As noted in the previous section,x+pr((t₁, . . . , t_d)A) run over the wholed-plane whent_j varies from 0 to 1,withj∈ {1,2, . . . , d}. Remark 4.1. Let f be a continuous function on Tⁿ and (x, x₀) ∈ (Tⁿ)².Then for all A∈ P_d,n,we have

Rf_x0(D(x, A)) =Rf(D(x+x₀, A)),

where f_x0(x) =f(x+x₀).It suffices to replace f by f_x0 in formula 4.1.

Let x∈Tⁿ and G^(x) be the set of all d-planes ofTⁿ fromx. LetG₀ be the set G₀=

x∈TⁿG^(x) and H_x∈G^(x).Denote by Λ_Hx the set given by Λ_Hx ={A∈ P_d,n|D(x, A) =H_x}.

Let Γ be the function defined on G0 by Γ (H_x) = inf

A∈Λ_Hx||A||², where

||A||=

⎛

⎜⎜

⎝

1≤i≤d

1≤j≤n

|a_ij|²

⎞

⎟⎟

⎠

12

,

with A = (a_ij)_1≤i≤d

1≤j≤n

and for a= (a₁, . . . , a_n) ∈ P, ||a|| = _n

j=1|a_j|²¹₂ . Let S(G0) be the Schwartz space defined by a family (T_N,x)_{N∈N, x∈T}ⁿ of semi-norms as follows

S(G₀) =

F :G₀−→C| sup

x∈TⁿT_N,x(F)<∞, for allN ∈N

, with

T_N,x(F) = sup

Hx∈G^(x)

(1 + Γ (H_x))^N|F(H_x)|, for all function F :G0−→C.

Definition 4.1. LetF ∈ S(G0).The dual Radon d-plane transform of F is the complex-valued functionR^∗F defined on Tⁿ by

R^∗F(x) =

Hx∈G^(x)

F(H_x). (4.2)

Remark 4.2. Let F ∈ S(G₀) and x ∈ Tⁿ. Show that R^∗F(x) has a sense. For this, it suffices to prove that the series

Hx∈G^(x)F(H_x) is absolutely convergent. Since F ∈ S(G0) there exist an integer N > d×n and a positive constant C(F, N, x) =T_N,x(F)<∞ such that 2

Hx∈G^(x)

|F(H_x)| ≤C(F, N, x)

Hx∈G^(x)

1

(1 + Γ (H_x))^N.

But for all H_x∈G^(x), there exists an element A(H_x) of Λ_Hx such that Γ (H_x) = inf

A∈Λ_Hx||A||²=||A(H_x)||², since

||A||² |A∈Λ_Hx

is a non empty subset of N.Then

Hx∈G^(x)

|F(H_x)| ≤ C(F, N, x)

Hx∈G^(x)

1 (1 + Γ (H_x))^N

≤ C(F, N, x)

A∈Pd,n

1

1 +||A||²_N <∞.

This proves that R^∗F(x) has a sense, and therefore the functionR^∗F is well-defined. The dual of the d-plane Radon transform is obtained by summing over all closed d-dimensional geodesics (d-planes) through a given point. Where F is a function belonging to S(G₀). The transform R^∗ is actually dual of R in the following sense

TⁿR^∗F(x)f(x)dx=

A∈Pd,n

TⁿF(D(x, A))Rf(D(x, A))dx, (4.3) where dx is the canonical invariant measure on the torus Tⁿ. Indeed, by (3.2) and (4.2)

R^∗F(x) =

A∈Pd,n

F(D(x−pr ((t₁, . . . , t_d)A), A)),

for any (t₁, . . . , t_d)∈R^d,and the left-hand side of (4.3) can be transformed as follows

TⁿR^∗F(x)f(x)dx =

A∈Pd,n

TⁿF(D(x−pr ((t₁, ..., t_d)A), A))f(x)dx

=

A∈Pd,n

TⁿF(D(x, A))f(x+pr ((t₁, ..., t_d)A))dx.

Then (4.3) follows by integration with respect tot₁, . . . , t_d.The calculations are valid whenever for any (t₁, . . . , t_d)∈R^d,

A∈Pd,n

Tⁿ|F(D(x, A))f(x+ pr ((t₁, . . . , t_d)A))|dx <∞.

Several classical inversion formulas for the Radon transform involve the function R^∗Rf. However the sum defining this function does not converge here in general. We then introduce a weight function ϕ:P_d,n → ]0,+∞[ given by

ϕ(A) = exp −

inf

||QA||² |Q∈SL_∗(d,Z)

, (4.4)

and ϕ_i(A) = exp −

inf

||e_i(QA)||²|Q∈SL_∗(d,Z)

, for all A ∈ P_d,n. It is clear that ϕ(A) = ϕ(QA) for all Q ∈ SL_∗ (d,Z). Since the subset

||QA||² |Q∈SL_∗(d,Z)

ofN is non empty, we have inf

||QA||² |Q∈SL_∗(d,Z)

=||Q(A)A||², (4.5) where Q(A) is an element of SL_∗(d,Z) which depends on the matrix A.

Consequently, ϕ(A) = exp

− ||Q(A)A||²

and ϕ_i(A) = exp

− ||e_i(Q(A)A)||²

. In the sequel, we associate to each A ∈ P_d,n an element Q(A) ∈SL_∗(d,Z) given by formula (4.5). It is clear that ϕ(A) = ^d

i=1ϕ_i(A).Consider the operatorR^∗ϕR defined onC(Tⁿ) as follows

R^∗ϕRf(y) =

A∈Pd,n

ϕ(A)Rf(D(y, A)) (4.6)

=

A∈Pd,n

ϕ(A) ₁

0 · · · ₁

0 f(y+ pr ((t₁, . . . , t_d)A))dt₁· · ·dt_d. It is noted that for all A∈ P_d,n andi∈ {1, . . . , d},the vector e_iA∈ P, because e_iA= (a_i1, . . . , a_in) is the ith row of the matrix A,see [3],[4]. As usual we denote the Fourier coefficients of f by

f!(k) =

Tⁿf(x) exp (−2iπkx)dx, withk∈Zⁿ.

In the following, we state and prove the main result of this paper, which is the inversion theorem for the d-plane Radon transform on the torusTⁿ. Theorem4.1. Letf be a continuous function onTⁿ, n≥2,such that f!∈l¹(Zⁿ).Withϕas in (4.4) the d-plane Radon transform R is inverted by

f(x) =

k∈Zⁿ

1 ψ(k)

TⁿR^∗ϕRf(y) exp (2iπk(x−y))dy, (4.7) for anyx∈Tⁿ,whereψdenotes the strictly positive function onZⁿdefined by

ψ(k) =

A∈Pd,n k(e1A)=···=k(edA)⁼⁰

" _d

#

i=1

ϕ_i(A)

$

=

A∈Pd,n, Ak=0

" _d

#

i=1

ϕ_i(A)

$

=

A∈Pd,n, Ak=0

ϕ(A). In particular, this theorem applies to any function f ∈Cⁿ(Tⁿ).

P r o o f. LetT be a distribution on Tⁿdefined by

< T, f >=

A∈Pd,n

ϕ(A) ₁

0 · · · ₁

0 f(pr (−((t₁, . . . , t_d)A)))dt₁· · ·dt_d, whereT is depending on the functionϕdefined byϕ(A) = ^d_i=1ϕ_i(A) for all A∈ P_d,n, see formula (4.4). Indeed, the estimate

|< T, f >| ≤

A∈Pd,n

" _d

#

i=1

ϕ_i(A)

$

||f||=

⎛

⎝

A∈Pd,n

ϕ(A)

⎞

⎠||f||,

where||f||= sup_x∈Tn|f(x)|,shows thatT is actually a distribution onTⁿ. Then

R^∗ϕRf =T∗f, convolution onTⁿ, (4.8) since by (4.6)

R^∗ϕRf(y) = < T(x), f(y−x)> .

The convolution equation (4.8) can be easily inverted by means of Fourier coefficients. From (4.8) we have

R^∗ϕRf(k) =T!(k)f!(k), (4.9) with

T!(k) = < T(x),exp (−2iπkx)>

=

A∈Pd,n

ϕ(A)

⎛

⎝#^d

j=1

₁

0 exp (2iπt_j(e_jA)k)dt_j

⎞

⎠

=

A∈Pd,n

#d j=1

(ϕ_j(A)) ₁

0 exp (2iπt_j(e_jA)k)dt_j

.

The right hand side of the last equality vanishes whenever there exists j₀ ∈ {1, . . . , d} such that k·(e_j0A)= 0,therefore

T!(k) =

A∈Pd,n k(e1A)=···=k(edA)⁼⁰

⎛

⎝#^d

j=1

ϕ_j(A)

⎞

⎠ (4.10)

=

A∈Pd,n, Ak=0

⎛

⎝#^d

j=1

ϕ_j(A)

⎞

⎠=

A∈Pd,n, Ak=0

ϕ(A) =ψ(k).

Now, given an arbitrary k = (k₁, . . . , k_n) ∈ Zⁿ, we claim that ψ(k) >

0. Indeed ϕ_j > 0 for all j = 1, . . . , d, see the expression of ϕ given by formula (4.4). Let P_k be the set defined by

P_k = {A∈ P_d,n|k·(e₁A) =· · ·=k·(e_dA) = 0}

= {A∈ P_d,n|Ak= 0}.

We fix some notations which will be useful in the sequel. Let be the function :Z →Zdefined by

(r) =

⎧⎨

⎩

1 if r = 0 r if r= 0.

And let δ be the function defined on Z² by δ(α, β) =

⎧⎨

⎩

1 if α=β = 0 0 otherwise.

Now, let us show that P_k=∅ for allk∈Zⁿ. For allk= (k₁, . . . , k_n)∈Zⁿ, we seek a matrix A ∈ Pd,n depending on k such that Ak = 0. Let A =

(a_rl)_1≤r≤d

1≤l≤n

be the matrix defined as follows

a_rl=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

a_rr = k_r+1

d(ε(k_r+1), ε(k_r))+δ(k_r, k_r+1) ifl=r, r= 1, . . . , d a_r,r+1 =− k_r

d(ε(k_r+1), ε(k_r))+δ(k_r, k_r+1) if r= 1, . . . , d, l=r+ 1

0 otherwise.

We distinguish the following cases:

(1) If k_i= 0 for all i= 1, . . . , n, thenP_k =P_d,n.

(2) Ifk_i = 0 for alli= 1, . . . , d+ 1, we have(k_i) =k_i andδ(k_i, k_i+1) = 0.In this casea_ii= _d(k^ki+1

i,ki+1), a_ii+1 =−_d(ki^k_,kⁱ_i+1),thusa_iik_i−k_i+1a_ii+1 = 0.

(3) Ifk_i= 0 andk_i+1= 0 for all i= 1, . . . , d, then(k_i) = 1, (k_i+1) = k_i+1 and δ(k_i, k_i+1) = 0. In this case a_ii = _d(1,k^ki+1

i+1) and a_ii+1 = 0 (since k_i = 0), it follows thata_iik_i−a_ii+1k_i+1= 0.

(4) In the simple case wherek_i= 0 andk_i+1= 0 for alli= 1, . . . , d,we have a_ii =a_ii+1 = 1 and a_iik_i−a_ii+1k_i+1 = 0 (sincek_i =k_i+1 = 0). Then P_k is non empty for all k∈Zⁿ.

In particular, this theorem applies to any function f ∈ Cⁿ(Tⁿ), it suffices to use the same technics as in [5]. And this completes the proof of

the theorem. 2

Remark 4.3. (1) LetF be a function defined onTⁿ× P_d,nas follows F!(k, A) =

TⁿF(x, A) exp (−2iπkx)dx, it is clear that

Rf% (k, A) =

⎧⎨

⎩

f!(k) ifAk= 0 0 ifAk= 0.

(4.11) Indeed,

Rf% (k, A) =

Tⁿf(x) exp (−2iπkx)dx

⎛

⎝#^d

j=1

₁

0 exp (2iπk(e_jA))dt_j

⎞

⎠,

therefore the equality (4.11) follows.

(2) For k∈Zⁿ, letA(k) be an element of P_d,n such that A(k)·k= 0.

By (4.11) we have f!(k) =Rf% (k, A(k)),then

Tⁿ|f(x)|²dx=

k∈Zⁿ

&&

& !f(k)&&&² =

k∈Zⁿ

&&

& %Rf(k, A(k))&&&².

5. A range theorem

As in [5], we give a characterization of the space R(C^∞(Tⁿ)), where R is thed-plane Radon transform on the torusTⁿ.LetF be a function on Tⁿ× P_d,n,we writeF!(k, A) ='

TⁿF(x, A) exp (−2iπkx)dx.

LetEbe the space of all functionsFonTⁿ×P_d,nsatisfying the following three conditions:

(1) For any A ∈ P_d,n,the map x → F(x, A) belongs to C^∞(Tⁿ) and for any multi-index α ∈Nⁿ,the partial derivative ∂_x^αF(x, A) is uniformly bounded on Tⁿ× P_d,n.

(2) F!(k, A) = 0 whenever k∈Zⁿ, A∈ P_d,n and Ak= 0.

(3) F!(k, A) =F!(k, B) whenever k∈ Zⁿ, with A, B ∈ P_d,n and Ak = Bk= 0.

Theorem 5.1. The d-plane Radon transform f → F, F(x, A) = Rf(D(x, A)),is a bijection of C^∞(Tⁿ)onto E.

P r o o f. The proof is exactly as in ([5], Theorem 4.1). 2 LetGbe the set of alld-planes of the torusTⁿand K be a compact of Tⁿ.We put

G_K ={D(x, A)∈G|D(x, A)∩K=∅}, with (x, A)∈Tⁿ× P_d,n. Remark 5.1. Letf be a continuous function onTⁿ.Then

suppf ⊂K implies that suppRf ⊂G_K.

Indeed, letD(x, A) be an element ofGwhich does not belong toG_K.Then D(x, A)∩K =∅,this implies that D(x, A)⊂Tⁿ\K.It follows that

|Rf(D(x, A))| ≤

D(x,A)|f(y)|dy ≤

Tⁿ\K|f(y)|dy= 0, since D(x, A)⊂Tⁿ\K and suppf ⊂K.

Acknowledgements

The author is grateful to the referee for his suggestions and comments.

He also thanks professor V. Kiryakova for scientific advice. Finally, he would like to express his deep gratitude to professor F. Rouvi`ere for his suggestions and encouragements.

References

[1] A. Abouelaz, Integral geometry in the sphere S^d. Research Notes in Mathematics 422 (2001), 83–125.

[2] A. Abouelaz and R. Daher, Sur la transformation de Radon de la sph`ere S^d.Bull. Soc. Math. France 121 (1993), 353–382.

[3] A. Abouelaz and A. Ihsane, Integral geometry on discrete Grassman- nians G(d, n) associated to Zⁿ. Integr. Transf. Spec. Funct. 21, No 3 (2010), 197–220.

[4] A. Abouelaz, E. Casadio Tarabusi and A. Ihsane, Integral geometry on discrete Grassmannians in Zⁿ.Mediterr. J. Math. 6, No 3 (2009), 303–316.

[5] A. Abouelaz and F. Rouvi`ere, Radon transform on the torus. To appear in: Mediterranean Journal of Mathematics 9, No 3 (2012).

[6] I.M. Gelfand, M.I. Graev, and S.I. Shapiro, Differential forms and In- tegral geometry. Funct. Anal. Appl.3 (1969), 24–40.

[7] I.M. Gelfand, S.G. Gindikin and M.I. Graev,Selected Topics in Integral Geometry. Transl. of Math. Monographs, AMS, Providence, RI, 2003.

[8] S. Helgason, The Radon Transform. Progress in Mathematics 5, Birkh¨auser, Boston, Inc. MA, Second Ed., 1999.

[9] E. Ournycheva and B. Rubin, On Y. Nievergelt’s inversion formula for the Radon transform.Fract. Calc. Appl. Anal.13, No 1 (2010), 43–56.

[10] F. Rouvi`ere, Transformation aux rayonsX sur un espace sym´etrique.

Comptes Rendus Acad. Sci. Paris, Ser. I,342(2006), 1–6.

[11] B. Rubin, Radon transforms on affine Grassmannians. Trans. Amer.

Math. Soc.356 (2004), 5045–5070.

[12] B. Rubin, Notes on Radon Transforms in integral Geometry, Fract.

Calc. Appl. Anal.6, No 1 (2003), 25–72.

[13] B. Rubin, Inversion of fractional integral related to the spherical Radon transform. In: Fractional Analysis, Vol. 157, No 2 (20 August 1998), 470–487.

[14] B. Rubin, Fractional calculus and wavelet transforms in integral ge- ometry. Fract. Calc. Appl. Anal.1, No 2 (1998), 193–219.

[15] R. Strichartz, Radon inversion - variations on a theme. Amer. Math.

Monthly 89, (June-July 1982), 377–384 and 420–423.

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

Route d’El Jadida, Km 8, B.P. 5366 Maˆarif 20100 Casablanca, MOROCCO

e-mail: a.abouelaz@fsac.ac.ma Received: July 7, 2010