Diophantine Integral Geometry

1660-5446/010077-23, DOI 10.1007/s00009-008-0137-2 c 2008 Birkh¨auser Verlag Basel/Switzerland

Mediterranean Journal of Mathematics

Diophantine Integral Geometry

Ahmed Abouelaz and Abdallah Ihsane

Abstract.

In his works [1], [2] and [3], the author succeeded in establishing several inversion formulas for Radon transform on Euclidean space, Damek- Ricci space and also on a finite set. The present paper deals with Radon transform

on discrete hyperplanes in the lattice

(

2) defined by lin- ear diophantine equations. More precisely, we study carefully various natural questions in this context: specific properties of the discrete Radon transform

and its dual

inversion formula for

(see Theorem 4.1) and also an appropriate support theorem in the discrete case (see Theorem 5.3).

Mathematics Subject Classification (2000).

Primary 44A12, 44A53; Secondary 05C25, 05C65, 68R10.

Keywords.

Linear diophantine equations, discrete Radon transform, inversion formula, support theorem, discrete Fourier transform, Poisson summation for- mula, discrete moments condition.

1. Introduction

The Radon transform was firstly defined on R

by Johann Radon [13] in 1917 and was afterwards generalized on the Euclidean space R

by many authors, particu- larly I.M. Gelfand (see [6]) and S. Helgason (see [8], [9], [10]).

Let f : R

−→ C be a function integrable on each hyperplane in R

. Let P

denote the differentiable manifold R × S

/ ± 1 of all hyperplanes H ( t, ω ) in R

( H ( t, ω ) = {x ∈ R

/x.ω = t} ), with ( t, ω ) ∈ R × S

. The Radon transform of f is defined as the function R

f : P

−→ C given by

R

f ( t, ω ) =

H(t,ω)

f ( x )d µ ( x ) , for all ( t, ω ) ∈ R × S

, (1.1) where d µ is the Euclidean measure on the hyperplane H ( t, ω ) . In the case of a finite set E, the analogue of this definition consists in making the average of a function f : E −→ C over the non-empty subsets of E (see [2], [14], [16]).

In the case of the lattice Z

, we give in this paper an analogue of the definition

(1.1), which consists in making the average of a suitable complex-valued function

f on Z

over discrete hyperplanes H ( a, k ) = {x ∈ Z

/a.x = k} defined by linear diophantine equations, with ( a, k ) ∈ P × Z, where P designates the set of all elements a = ( a

, . . . , a

) ∈ Z

\ 0 such that d ( a ) = 1 , d ( a ) being the greatest common divisor of the integers a

, . . . , a

, and a.x denotes the usual inner product of a and x regarded as two vectors of the Euclidean space R

.

Now, we note that one of the difficulties of the diophantine integral geometry, relatively to its Euclidean homologue, is the problem of the existence of solutions in Z

for the linear diophantine equation a.x = c (with a = ( a

, . . . , a

) ∈ Z

\ 0 and c ∈ Z ), therefore the arithmetic of the integers a

, . . . , a

and c is essential for our study. This can be interpreted from the criterion for the solvability in Z

of a linear diophantine equation, stated in the proposition below (see Proposition 2.1).

Proposition 1.1. Let a = ( a

, . . . , a

) ∈ Z

\ 0 and c ∈ Z. Then the following two conditions are equivalent.

1) The linear diophantine equation a

x

+ · · · + a

x

= c has an infinity of solutions in Z

.

2) There exists an integer k ∈ Z such that c = kd ( a ) , where d ( a ) is the greatest common divisor of the integers a

, . . . , a

.

Remark 1.2.

(i) If the condition 2) of the proposition above is not satisfied then the linear diophantine equation a

x

+ · · · + a

x

= c has not any solution in Z

. (ii) The reader can also see [11] for further details in the simplest case n = 2 .

The proposition above allows us to give a parametrization of the discrete hyperplanes H ( a, k ) (with ( a, k ) ∈ P × Z ). More precisely, the discrete hyperplane H ( a, k ) is defined for each ( a, k ) ∈ P × Z as the set of all solutions in Z

of the linear diophantine equation a.x = k. We will show that the function ( a, k ) −→

H ( a, k ) = {x ∈ Z

/a.x = k} is a bijection of P × Z/ ± 1 onto G, where G = {H ( a, k ) / ( a, k ) ∈ P × Z} (see Proposition 2.2). In the sequel, the set of all discrete hyperplanes containing m ∈ Z

is denoted by G

.

In the second section, we fix, once and for all, some notations which shall be useful in the sequel of this paper, and also give the properties of the discrete hyperplanes H ( a, k ) , (a,k) belonging to the set P ×Z. Moreover, various interesting functional spaces are introduced. In particular, we denote by l

( Z

) the space of all complex-valued functions f defined on Z

such that

m∈Zⁿ

|f ( m ) | < + ∞.

The Schwartz space of Z

, denoted by S ( Z

), is consisting of all complex-valued rapidly decreasing functions f defined on Z

. We define a family ( p

)

of semi-norms on S ( Z

) by

p

( f ) = sup

m∈Zⁿ

(1 + m

)

|f ( m ) | , for all ( f, N ) ∈ S ( Z

) × N. (1.2) where m

= m

+ · · · + m

for all m = ( m

, . . . , m

) ∈ Z

.

We denote by l

( G ) the space of all complex-valued functions F defined on G such that

k∈Z

|F ( H ( a, k )) | < + ∞ for all a ∈ P . The Schwartz space of G,

denoted by S ( G ) , is the subspace of l

( G ) consisting of all functions F ∈ l

( G ) such that q

( F ) < + ∞ for all N ∈ N, where ( q

)

is defined by

q

( F ) = sup

a∈P,k∈Z

1 + k

1 + a

|F ( H ( a, k )) | , for all ( F, N ) ∈ l

( G ) × N. (1.3) We note that ( q

)

is a family of semi-norms on the Schwartz space S ( G ) . We also define a subspace S

( G ) of S ( G ) by

S

( G ) = {F ∈ S ( G ) /T

( F ) < + ∞, for all N ∈ N} , (1.4) where ( T

)

is given by

T

( F ) = sup

a∈P,k∈Z

(1 + a

+ k

)

|F ( H ( a, k )) | , for all ( F, N ) ∈ S ( G ) × N. (1.5) It is clear that ( T

)

is a family of semi-norms on the space S

( G ) such that T

( F ) ≥ q

( F ) , for all ( F, N ) ∈ S

( G ) × N.

In the third section, we give an appropriate definition of the discrete Radon transform R as well as its dual R

. The discrete Radon transform R is defined as follows

Rf ( H ( a, k )) =

m∈H(a,k)

f ( m ) , for all f ∈ l

( Z

) and all ( a, k ) ∈ P × Z. (1.6) We define the discrete dual Radon transform R

by

R

F ( m ) =

H∈G,m∈H

F ( H ) = 1 2

a∈P,k∈Z,a.m=k

F ( H ( a, k )) , (1.7) for all m ∈ Z

and all complex-valued function F defined on G satisfying the condition sup

a∈P

|F ( H ( a, k )) | < ∞, where the sum of the right-hand side of (1.7) is taken over all discrete hyperplanes H ( a, k ) ∈ G containing m, with ( a, k ) ∈ P × Z.

In this section, we also study in detail the specific properties of the dis- crete Radon transform R and its dual R

. In particular, we state and prove the following continuity theorem (see Theorem 3.7), which allows us to deduce that R ( S ( Z

)) → S

( G ) , where S

( G ) is the space of all functions F ∈ S ( G ) sat- isfying the discrete moments condition, that is, for each p ∈ N\ 0 , the function

a −→

k∈Z

F ( H ( a, k )) k

is a homogeneous polynomial of degree p in a

, . . . , a

, where a = ( a

, . . . , a

) ∈ P .

Theorem 1.3. The discrete Radon transform R is a continuous linear mapping of S ( Z

) into S ( G ) . More precisely, we have for all f ∈ S ( Z

) and all ( N, r ) ∈ N

such that r > n

2 , the following inequality

q

( Rf ) ≤ C

p

( f ) , with C

=

m∈Zⁿ

(1 + m

)

< + ∞.

We assume of course that the Schwartz space S ( Z

) (resp. S ( G )) is equipped with the topology defined by the family of semi-norms ( p

)

(resp. ( q

)

).

As in the Euclidean case, we show that the discrete Radon transform R satisfies the discrete moments condition. More precisely, we state and prove the following formula for all f ∈ S ( Z

) and ( p, a ) ∈ ( N\ 0) × P (see Corollary 3.10, (3.14) ).

k∈Z

Rf ( H ( a, k )) k

=

m∈Zⁿ

f ( m )( a.m )

. (1.8) We also give an analogue of Poisson summation formula for the discrete Radon transform R (see Corollary 3.10, (3.15) ). More precisely, we state and prove the following formula for all f ∈ S ( Z

) and a ∈ P

k∈Z

Rf ( H ( a, k )) =

m∈Zⁿ

f ( m ) . (1.9)

In the same section, we establish, by analogy with the Euclidean case, a relation- ship between the discrete Fourier transform F and the discrete Radon transform R (see Corollary 3.10, (3.16) ). We note that the discrete Fourier transform of a function f ∈ l

( Z

) is defined as the function Ff : T

−→ C given by

Ff ( x ) =

m∈Zⁿ

f ( m ) exp( −ixm ) , for all x ∈ T

, (1.10) where T

= R

/Z

is the n -dimensional torus. We recall that T

is the dual of the discrete group Z

(see [5]).

In the fourth section, we extend Strichartz’ limit formula (see [14], page 421, Problem B) to invert the discrete Radon transform R . More precisely, we state and prove the following inversion theorem for R (see Theorem 4.1).

Theorem 1.4. Let f ∈ l

( Z

) and m

∈ Z

. Then there exists a sequence ( H

)

of discrete hyperplanes in Z

containing m

such that

f ( m

) = lim

j→∞

Rf ( H

) .

In the fifth section, we study the support of a function f ∈ C

as well as the support of Rf (see Lemma 5.1, Remark 5.2 and Theorem 5.3), where C

is the subspace of the Schwartz space S ( Z

) consisting of all complex-valued functions f defined on Z

with finite support. We show that Supp Rf is infinite even if Supp f is finite. Consequently Supp Rf is not compact even if f ∈ C

. We also prove the support theorem in the discrete case, which is stated as follows (see Theorem 5.3).

Theorem 1.5. Let f ∈ C

and K = {x

, x

, . . . , x

} ⊂ Z

(with l ∈ N\ 0). Then Supp f ⊂ K ⇐⇒ Supp Rf ⊂

i=1

G

.

Now, we want to note that the study of the diophantine integral geometry presents many difficulties, relatively to its homologue in the Euclidean case, for several reasons. In particular, we can enunciate the following two reasons:

1) In the discrete case, the Radon transform R is defined by a series (see (1.6)), then the inversion of R is more difficult than that of its homologue R

( R

being the classical Radon transform defined by (1.1)). On the other hand, the inversion of the finite Radon transform is easy relatively to that of the discrete Radon transform. Indeed, in the case of a finite set X, Card ( G

) < + ∞ for all x ∈ X, where G

denotes the set of all lines in X containing x, while in the case of the lattice Z

( n ≥ 2), Card ( G

) = + ∞ for all m ∈ Z

.

2) The study of the diophantine integral geometry is strictly related to linear diophantine equations. Indeed, there exist many Euclidean hyperplanes which are disjoint to the lattice Z

( n ≥ 2), therefore the arithmetic of the integers defining these algebraic equations plays a fundamental role in our study of this geometry which is then more complicated than its homologue in the Euclidean case.

We end the present section by making clear that the main purpose of this paper is to establish an inversion formula for the discrete Radon transform R, and also to study an appropriate support theorem in the discrete case. The inversion theorem (see Theorem 4.1) has many applications in the fields of Tomography, Cryptography, Coding, Number Theory and Representations Theory.

2. Notations and Preliminaries

In this section, we fix some notations which will be useful in the sequel of this paper, and give the properties of the discrete hyperplanes in the lattice Z

( n ≥ 2).

We also introduce various functional spaces. For 1 ≤ p < + ∞, let l

( Z

) (resp.

l

( Z

)) be the space of all complex-valued functions f defined on Z

such that

m∈Zⁿ

|f ( m ) |

< + ∞ (resp. sup

|f ( m ) | < + ∞ ). Denote by C

( Z

) the space of all complex-valued functions f defined on Z

such that f ( m ) −→ 0 as m −→ + ∞ (with m

= m

+ · · · + m

for all m = ( m

, . . . , m

) ∈ Z

). It is clear that for 1 ≤ p < q < + ∞, we have the following inclusions

l

( Z

) ⊂ l

( Z

) ⊂ · · · ⊂ l

( Z

) ⊂ · · · ⊂ l

( Z

) ⊂ C

( Z

) ⊂ l

( Z

) . For 1 ≤ p < + ∞, we denote by .

the discrete norm on the space l

( Z

) defined by

f

=

m∈Zⁿ

|f ( m ) |

p

, for all f ∈ l

( Z

) . (2.1) We recall that l

( Z

) is a Hilbert space for the inner product < ., . > given by

< ϕ, ψ > =

m∈Zⁿ

ϕ ( m ) ψ ( m ) , for all ϕ, ψ ∈ l

( Z

) . (2.2)

For m ∈ Z

, let χ

be the characteristic function of the set reduced to the point m. It is easy to check that the system ( χ

)

is an orthonormal basis of the Hilbert space l

( Z

) . Moreover, the space l

( Z

) contains a subspace consisting of all complex-valued rapidly decreasing functions f defined on Z

, denoted by S ( Z

) and called the Schwartz space of Z

. We recall that a function f : Z

−→ C is called to be rapidly decreasing if for each N ∈ N, the family ( m

|f ( m ) | )

converges to zero as m −→ + ∞. We note that S ( Z

) is also a remarkable sub- space of l

( Z

) . We equip the Schwartz space S ( Z

) with a structure of metrisable locally convex topological vector space induced by the family ( p

)

of semi- norms on S ( Z

) given by

p

( f ) = sup

m∈Zⁿ

(1 + m

)

|f ( m ) | , for all ( f, N ) ∈ S ( Z

) × N. (2.3) Let C

be the subspace of S ( Z

) consisting of all complex-valued functions f defined on Z

with finite support. It is well known that C

is dense in S ( Z

) (see [5]). Let F be the discrete Fourier transform defined by (1.10). We can check that F is an isomorphism of the Schwartz space S ( Z

) onto the Fr´ echet space E ( T

) of the complex-valued C

-functions on T

(see [5]), where T

= R

/Z

is the n -dimensional torus. It is well known that T

is exactly the dual of the discrete group Z

(see [5]).

By analogy with the definition of the space l

( Z

), we define the space l

( G ) as the space of all complex-valued functions F defined on G such that

k∈Z

|F ( H ( a, k )) | is finite for all a ∈ P. Let us now give the definition of the Schwartz space of G , denoted by S ( G ), which is a subspace of l

( G ) . We define the Schwartz space S ( G ) by

S ( G ) =

F ∈ l

( G ) /q

( F ) < + ∞, for all N ∈ N

, (2.4)

where ( q

)

is given by q

( F ) = sup

a∈P,k∈Z

1 + k

1 + a

|F ( H ( a, k )) | , for all ( F, N ) ∈ l

( G ) × N. (2.5) We note that ( q

)

is a family of semi-norms on S ( G ) . We also define a subspace S

( G ) of the space S ( G ) by

S

( G ) = {F ∈ S ( G ) /T

( F ) < + ∞, for all N ∈ N} , (2.6) where ( T

)

is defined by

T

( F ) = sup

a∈P,k∈Z

(1 + a

+ k

)

|F ( H ( a, k )) | , for all ( F, N ) ∈ S ( G ) × N. (2.7) It is easy to check that ( T

)

is a family of semi-norms on S

( G ) such that T

( F ) ≥ q

( F ) , for all ( F, N ) ∈ S

( G ) × N.

Now, we fix, once and for all, some notations which will be useful in the

sequel of this paper. Given a = ( a

, . . . , a

) ∈ Z

\ 0 , we denote by d ( a ) the great-

est common divisor of the integers a

, . . . , a

. The set {a ∈ ( Z

\ 0) /d ( a ) = 1 } is

designated by P. For each ( a, k ) ∈ P × Z, the set {x ∈ Z

/a.x = k} is denoted

by H ( a, k ) . Moreover, we designate by G the set {H ( a, k ) / ( a, k ) ∈ P × Z} . In the sequel, the elements of G will be called discrete hyperplanes in Z

. Finally, the set of all elements of G containing m ∈ Z

is simply denoted by G

.

Let us show the following proposition which is fundamental for our study in the present paper.

Proposition 2.1. Let a = ( a

, . . . , a

) ∈ Z

\ 0 and c ∈ Z. Then the following two conditions are equivalent

1) The linear diophantine equation a

x

+ · · · + a

x

= c has an infinity of solutions in Z

.

2) There exists an integer k ∈ Z such that c = kd ( a ) , where d ( a ) is the greatest common divisor of the integers a

, . . . , a

.

Proof. The necessary condition of Proposition 2.1 is obvious. We just prove that 2) ⇒ 1). Suppose that c = kd ( a ) , with k ∈ Z, and (for instance) a

= 0 . From B´ ezout theorem, there exists ( β

, . . . , β

) ∈ Z

such that d ( a ) = β

a

+ · · · + β

a

. Then c = k ( β

a

+ · · · + β

a

) = a

( kβ

) + · · · + a

( kβ

) , and therefore γ = ( kβ

, . . . , kβ

) is a solution of the linear diophantine equation a

x

+ · · · + a

x

= c.

Let S be the set of all solutions in Z

of the linear diophantine equation a

x

+

· · · + a

x

= c. We have γ + q ( −a

, a

, 0 , . . . , 0) ∈ S, for all q ∈ Z. Hence S is

infinite. The proposition is proved.

We now state and prove the following proposition which gives us an appro- priate parametrization of the discrete hyperplanes in Z

.

Proposition 2.2. Let ψ : P × Z/ ± 1 −→ G be the function defined by ψ ( a, k )

= H ( a, k ) , for all ( a, k ) ∈ P × Z,

where ( a, k ) denotes the set { ( a, k ) , ( −a, −k ) } ∈ P × Z/ ± 1 . Then ψ is a bijection of P × Z/ ± 1 onto G.

Proof. It is obvious that ψ is well-defined and surjective. We just prove that ψ is injective. Let ( a, k ) and ( a

, k

) be two elements of P×Z. Suppose that ψ ( a, k )

= ψ ( a

, k

)

. Then we have H ( a, k ) = H ( a

, k

) . Thus the following two diophantine equations (E

) and (E

) have the same solutions in Z

(E

) a

x

+ · · · + a

x

= k,

(E

) a

x

+ · · · + a

x

= k

. (2.8) Suppose (for instance) a

= 0 and put ( x

, x

, . . . , x

) = ( k

a

, 0 , . . . , 0) as a solution of (E

). Then ( k

a

, 0 , . . . , 0) is also a solution of (E

), and therefore a

k a

= k

. Hence

ka

= k

a

. (2.9)

By the same techniques, we obtain

 

 



 

 

ka

= k

a

, ka

= k

a

,

.. . ka

= k

a

.

(2.10)

Since the integers a

, . . . , a

(resp. a

, . . . , a

) are relatively prime, there exist β

, . . . , β

∈ Z (resp. α

, . . . , α

∈ Z ) such that β

a

+ · · · + β

a

= 1 (resp.

α

a

+ · · · + α

a

= 1). From (2.10), we have

k ( α

a

+ · · · + α

a

) = k

( α

a

+ · · · + α

a

) , k ( β

a

+ · · · + β

a

) = k

( β

a

+ · · · + β

a

) .

Thus

k = k

( α

a

+ · · · + α

a

) , k ( β

a

+ · · · + β

a

) = k

. It follows that

k = k ( β

a

+ · · · + β

a

)( α

a

+ · · · + α

a

) . (2.11) We distinguish two cases.

1

) The case when k = 0 .

In this case, the equality (2.11) becomes

( β

a

+ · · · + β

a

)( α

a

+ · · · + α

a

) = 1 .

Thus β

a

+ · · · + β

a

= α

a

+ · · · + α

a

= λ, with λ = ± 1 . If λ = 1 then k = k

, which implies a = a

(from (2.10)). If λ = − 1 then k = −k

, and therefore a = −a

(from (2.10)). We conclude that ( a, k ) = ( a

, k

) .

2

) The case when k = 0 .

In this case, we have also k

= 0 . Then (2.8) becomes a

x

+ · · · + a

x

= 0 ,

a

x

+ · · · + a

x

= 0 .

(2.12) There exists b = ( b

, . . . , b

) ∈ Z

\ 0 such that a

b

+ · · · + a

b

= 0 . From (2.12), b is a solution of the diophantine equation a

x

+ · · · + a

x

= 0 . Since a.b = a

.b = 0 , there exists α ∈ Z such that a

= αa. Then d ( a

) = |α| d ( a ) , which implies α = ± 1 . Consequently a

= ±a. Hence ( a, k ) = ( a

, k

) . This completes the

proof of the proposition.

Remark 2.3. The set G plays a role as discrete Grassmannian in the present paper.

On the other hand, the set P × Z/± 1 plays a role as R×S

/ ± 1 in the Euclidean

case.

Definition 2.4. A subset H of Z

is called a discrete hyperplane if it can be written in the form

H = {x ∈ Z

/a.x = k} , where a ∈ P and k ∈ Z.

Remark 2.5. Let a = ( a

, . . . , a

) ∈ P and k ∈ Z. We have d ( a ) = 1 , then d ( a ) |k . It follows, from Proposition 2.1, that the discrete hyperplane H ( a, k ) is infinite. Moreover, H ( a, k ) is contained in the Euclidean hyperplane H ( t, ω ) = {x∈ R

/x.ω = t} , where ( t, ω ) =

k a , a

a

∈ R×S

, with a =

i=1

a

. We note that the set G of all discrete hyperplanes in Z

is contained in P

= R×S

/ ± 1 , P

being the set of all hyperplanes in the Euclidean space R

. Proposition 2.6. Let a be a fixed element of P. Then the lattice Z

can be written in the form

Z

=

k∈Z

H ( a, k ) . (2.13)

Proof. Let a be a fixed element of P and m ∈ Z

. By taking k = m.a ∈ Z, it is clear that m belongs to the discrete hyperplane H ( a, k ) . Then Z

is contained in

k∈Z

H ( a, k ) , and therefore Z

=

k∈Z

H ( a, k ) . The proposition is proved.

Let m ∈ Z

. It is easy to check that the characteristic function χ

of the set G

belongs to the Schwartz space S ( G ) . Moreover, we can define an incidence relation between the discrete hyperplane H ( a, k ) (with ( a, k ) ∈ P × Z ) and the set G

by

H ( a, k ) ∈ G

⇐⇒ m ∈ H ( a, k ) , (2.14) which can be written

χ

( H ( a, k )) = χ

( m ) . (2.15) We end this section by noting that in the case of a finite set X, the cardinal of G

( x ∈ X ) is finite (see [2], [4]), where G

denotes the set of all lines in X containing x , but the cardinal of G

( m ∈ Z

) is infinite, which makes the establishment of an explicit inversion formula for the discrete Radon transform R very difficult relatively to its homologue in the finite case.

3. Properties of the discrete Radon transform and its dual

We begin this section with appropriate definitions of the discrete Radon transform R and its dual R

. We shall afterwards study the specific properties of R and R

. Definition 3.1. Let f ∈ l

( Z

) . The discrete Radon transform of f is the complex- valued function Rf defined on G by

Rf ( H ( a, k )) =

m∈H(a,k)

f ( m ) , for all ( a, k ) ∈ P × Z. (3.1)

Definition 3.2. Let F be a complex-valued function defined on G satisfying the condition sup

a∈P

|F ( H ( a, k )) | < ∞ . The discrete dual Radon transform of F is the complex-valued function R

F defined on Z

by

R

F ( m ) =

H∈G,m∈H

F ( H ) = 1 2

a∈P,k∈Z,a.m=k

F ( H ( a, k )) , for all m ∈ Z

, (3.2) (the sum of the right-hand side of (3.2) is taken over all discrete hyperplanes H ( a, k ) ∈ G containing m, with ( a, k ) ∈ P × Z ).

Remark 3.3. Let F be a complex-valued function defined on G satisfying the condition sup

a∈P

|F ( H ( a, k )) | < ∞ . For all m ∈ Z

, the equalities of (3.2) can be reduced as follows

R

F ( m ) = 1 2

a∈P

F ( H ( a, a.m )) . (3.3) The equality (3.3) has a sense. Indeed, it is clear that

a∈P

|F ( H ( a, a.m )) | ≤ sup

k∈Z

a∈P

|F ( H ( a, k )) |

< ∞, for all m ∈ Z

. In particular, if F is a complex-valued function defined on G satisfying the con- dition sup

(1 + a

+ k

)

|F ( H ( a, k )) | < ∞, with N ∈ N such that 2 N > n, then R

F is well-defined. Indeed, we have under this condition

sup

k∈Z

a∈P

|F ( H ( a, k )) |

= sup

k∈Z

a∈P

(1 + a

+ k

)

|F ( H ( a, k )) | (1 + a

+ k

)

≤ C ( F, N ) sup

k∈Z

a∈P

1

(1 + a

+ k

)

,

where C ( F, N ) = sup

(1 + a

+ k

)

|F ( H ( a, k )) | < ∞.

But for all ( a, k ) ∈ P × Z

(1 + a

+ k

)

≥ (1 + a

)(1 + k

) . Thus

sup

k∈Z

a∈P

|F ( H ( a, k )) |

≤ C ( F, N ) sup

k∈Z

1 (1 + k

)

a∈P

1 (1 + a

)

≤ C

C ( F, N ) < ∞, where C

is a positive constant.

We now state and prove the following proposition which is useful in the sequel of this paper.

Proposition 3.4. Let m

∈ Z

. Then Rχ

= χ

, where G

is the set of all dis-

crete hyperplanes in Z

containing m

, and χ

(resp. χ

) is the characteristic

function of the set {m

} (resp. of G

).

Proof. Let m

∈ Z

and H ∈ G. We have Rχ

( H ) =

m∈H

χ

( m ) =

1 , if m

∈ H, 0 , otherwise.

Then

Rχ

( H ) = χ

( H ) . Thus

Rχ

= χ

,

and the proposition is proved.

Now, let us give an example of the set G

( m ∈ Z

) which proves that Card ( G

) is infinite. We note that there exists an infinity of discrete lines in Z

containing a fixed point ( m

, k

) of Z

. Indeed, R. Strichartz has given in [14]

an interesting example of a sequence ( y

)

of discrete lines in Z

containing ( m

, k

) , defined by

y

=

( k, m ) ∈ Z

/k − k

= j ( m − m

) ,

for all j ∈ N. It is clear that G

contains the set of all discrete lines y

(with j ∈ N ), then Card ( G

) = + ∞. Moreover, R. Strichartz has also constructed with this sequence ( y

)

an example of a complex-valued function f on Z

, which is not absolutely summable, such that Rf ≡ 0 and f = 0 . This example can be generalized in the case of Z

.

By analogy with the Euclidean case, we now state and prove the duality formula between the discrete Radon transform R and its dual R

.

Proposition 3.5. Let f ∈ l

( Z

) and let F be a complex-valued function defined on G satisfying the condition sup

a∈P

|F ( H ( a, k )) | < ∞. Then we have the duality formula

m∈Zⁿ

f ( m ) R

F ( m ) =

H∈G

Rf ( H ) F ( H ) . (3.4) In particular, the duality formula (3.4) holds if f is a complex-valued function defined on Z

such that sup

(1 + m

)

|f ( m ) | < ∞ and F is a complex- valued function defined on G such that sup

(1 + a

+ k

)

|F ( H ( a, k )) | is finite, with 2 N > n ( N ∈ N ) .

Proof. The assumptions f ∈ l

( Z

) and F = sup

a∈P

|F ( H ( a, k )) | < ∞ imply

m∈Zⁿ

|f ( m ) |

a∈P

|F ( H ( a, a.m )) | ≤

m∈Zⁿ

|f ( m ) | F = f

F < ∞.

Thus, by Fubini’s theorem for series, all series in the following lines converge absolutely and the summation signs can be permuted :

m∈Zⁿ

f ( m ) R

F ( m ) = 1 2

m∈Zⁿ

f ( m )

a∈P

F ( H ( a, a.m ))

= 1

2

a∈P





k∈Z

m∈Zⁿ,a.m=k

f ( m ) F ( H ( a, a.m ))





= 1

2

a∈P





k∈Z

F ( H ( a, k ))

m∈H(a,k)

f ( m )





= 1

2

a∈P,k∈Z

F ( H ( a, k )) Rf ( H ( a, k )) . We deduce the duality formula

m∈Zⁿ

f ( m ) R

F ( m ) =

H∈G

Rf ( H ) F ( H ) .

In particular, the duality formula (3.4) holds if sup

(1 + m

)

|f ( m ) | < ∞ and sup

(1+ a

+ k

)

|F ( H ( a, k )) | < ∞, with N ∈ N such that 2 N > n.

Indeed, these conditions imply f ∈ l

( Z

) and (from Remark 3.3) F ≤ C

sup

a∈P,k∈Z

(1 + a

+ k

)

|F ( H ( a, k )) | < ∞,

where C

is a positive constant. This completes the proof of the proposition.

As in the Euclidean case, we now show the invariance of the discrete Radon transform R under translations. More precisely, we state and prove the following proposition.

Proposition 3.6. Let f ∈ l

( Z

) and ( m

, k

) ∈ Z

× Z. Then we have, for all ( a, k ) belonging to P × Z such that a.m

= k

, the following equality

R (

f )( H ( a, k )) = Rf ( H ( a, k + k

)) , (3.5) where (

f )( m ) = f ( m + m

) for all m ∈ Z

.

Proof. Let f ∈ l

( Z

) and ( m

, k

) ∈ Z

× Z. Let ( a, k ) ∈ P × Z such that a.m

= k

. From (3.1), we have

R (

f )( H ( a, k )) =

m∈Zⁿ,a.m=k

f ( m + m

) ,

which gives by putting b = m + m

R (

f )( H ( a, k )) =

b∈Zⁿ,(b−m0).a=k

f ( b )

=

b∈Zⁿ,b.a=k+a.m0

f ( b )

=

b∈H(a,k+k0)

f ( b ) . Thus

R (

f )( H ( a, k )) = Rf ( H ( a, k + k

)) .

The proposition is proved.

Let ( p

)

and ( q

)

be the families of semi-norms on S ( Z

) and S ( G ) respectively, defined in Introduction (see (1.2) and (1.3)). We assume that in the sequel of this paper, the space S ( Z

) (resp. S ( G )) is equipped with the topology defined by the family ( p

)

(resp. ( q

)

). We now state and prove the fol- lowing continuity theorem for the discrete Radon transform R, which allows us to deduce that R ( S ( Z

)) → S

( G ) (see Remark 3.8).

Theorem 3.7. The discrete Radon transform R is a continuous linear mapping of S ( Z

) into S ( G ) . More precisely, we have for all f ∈ S ( Z

) and all ( N, r ) ∈ N

such that r > n

2 , the following inequality

q

( Rf ) ≤ C

p

( f ) , (3.6) with C

=

m∈Zⁿ

(1 + m

)

< + ∞.

Proof. Let f ∈ S ( Z

) and ( N, r ) ∈ N

such that r > n

2 . Let ( a, k ) ∈ P × Z. It is clear that

(1 + k

)

|Rf ( H ( a, k )) | ≤

m∈Zⁿ,a.m=k

|f ( m ) | (1 + k

)

. (3.7) Replacing k by a.m in the right-hand side of (3.7), we obtain

(1 + k

)

|Rf ( H ( a, k )) | ≤

m∈Zⁿ

|f ( m ) | (1 + ( a.m )

)

. But for all m ∈ Z

(1 + ( a.m )

)

≤ 1 + a

m

≤ 1 + a

1 + m

1 + m

. Then

(1 + k

)

|Rf ( H ( a, k )) | ≤ 1 + a

m∈Zⁿ

1 + m

|f ( m ) | 1 + m

.

Since f ∈ S ( Z

) , the expression (1 + m

)

|f ( m ) | is majorized by p

( f ) for all m ∈ Z

. Thus

(1 + k

)

|Rf ( H ( a, k )) | ≤ 1 + a

p

( f )

m∈Zⁿ

1 + m

.

It follows that sup

k∈Z

(1 + k

)

|Rf ( H ( a, k )) | ≤ C

p

( f ) 1 + a

, (3.8)

with C

=

m∈Zⁿ

(1 + m

)

. Consequently, the inequality (3.8) gives sup

a∈P,k∈Z

1 + k

1 + a

|Rf ( H ( a, k )) | ≤ C

p

( f ) , which implies

q

( Rf ) ≤ C

p

( f ) .

This completes the proof of the theorem.

Remark 3.8.

1

) Theorem 3.7 shows that R ( S ( Z

)) → S ( G ) , the symbol → signifying that R ( S ( Z

)) ⊂ S ( G ) and the identity mapping of R ( S ( Z

)) into S ( G ) is con- tinuous. Since the discrete Radon transform R satisfies the discrete moments condition (see Corollary 3.10, (3.14)), we deduce that

R ( S ( Z

)) → S

( G ) , (3.9) where S

( G ) is the space of all functions F ∈ S ( G ) satisfying the discrete mo- ments condition, that is, for all p∈ N\ 0 , the function a −→

k∈Z

F ( H ( a, k )) k

is a homogeneous polynomial of degree p in a

, . . . , a

, with a = ( a

, . . . , a

) ∈ P.

2

) We have the following inclusion

R ( l

( Z

)) ⊂ l

( G ) . (3.10) Indeed, we have from (3.1)

Rf ( H ( a, k )) =

m∈H(a,k)

f ( m ) , for all ( a, k ) ∈ P × Z, where f ∈ l

( Z

) . Then for each a ∈ P

k∈Z

|Rf ( H ( a, k )) | ≤

k∈Z

m∈H(a,k)

|f ( m ) | = f

< ∞. (3.11) Thus Rf ∈ l

( G ) , which proves the inclusion (3.10). We recall that in the Euclidean case, we have

R

( L

( R

) , d x ) ⊂ L

( P

, d t d ω ) ,

where d x (resp. d t d ω ) is the Euclidean measure on R

(resp. P

), P

being

the set of all Euclidean hyperplanes in R

.

We now state and prove the following useful proposition.

Proposition 3.9. Let f ∈ S ( Z

) and let ϕ : Z −→ C be a function which has a moderate growth (i.e. |ϕ ( k ) | ≤ C (1+ k

)

for all k ∈ Z, where C and α are positive constants). Then for all a ∈ P

k∈Z

Rf ( H ( a, k )) ϕ ( k ) =

m∈Zⁿ

f ( m ) ϕ ( a.m ) . (3.12) Proof. Fix an element a of P . First, we check, under the assumptions of the propo- sition, that the double series

k∈Z

m∈H(a,k)

f ( m ) ϕ ( k ) is absolutely convergent.

We have

k∈Z

m∈H(a,k)

|f ( m ) | |ϕ ( k ) | ≤ C

k∈Z

m∈H(a,k)

|f ( m ) | (1 + k

)

≤ C

m∈Zⁿ

|f ( m ) |

1 + ( a.m )

.

Since f ∈ S ( Z

) and (1 + ( a.m )

)

≤ (1 + m

)

(1 + a

)

, it follows that

k∈Z

m∈H(a,k)

|f ( m ) | |ϕ ( k ) | ≤ C

(1 + a

)

m∈Zⁿ

1 + m

< ∞, where N is a positive integer chosen sufficiently great and C

is a positive constant depending on f .

Now, the absolute convergence of the double series

k∈Z

m∈H(a,k)

f ( m ) ϕ ( k ) is enough to validate all computations in the following lines :

k∈Z

Rf ( H ( a, k )) ϕ ( k ) =

k∈Z





m∈Zⁿ,a.m=k

f ( m )



 ϕ ( k )

=

k∈Z

m∈H(a,k)

f ( m ) ϕ ( a.m )

=

m∈Zⁿ

f ( m ) ϕ ( a.m ) .

The proposition is proved.

In the sequel of this paper, we denote by F

the discrete Fourier transform in one dimension. The one-dimensional torus is designated by T

. For f ∈ S ( Z

) and a ∈ P, let Rf ( H ( a, . )) denote the function defined on Z by

Rf ( H ( a, . ))( k ) = Rf ( H ( a, k )) , for all k ∈ Z. (3.13)

Corollary 3.10. Let f ∈ S ( Z

) and p ∈ N\ 0 . Then for all ( a, θ ) ∈ P×T

k∈Z

Rf ( H ( a, k )) k

=

m∈Zⁿ

f ( m )( a.m )

, (3.14)

k∈Z

Rf ( H ( a, k )) =

m∈Zⁿ

f ( m ) , (3.15)

F

Rf ( H ( a, . ))( θ ) = Ff ( θa ) . (3.16) Proof. To prove the equality (3.14), resp. (3.15), resp. (3.16), it suffices to apply the equality (3.12) of Proposition 3.9 by taking ϕ ( k ) = k

, resp. ϕ ( k ) = 1 , resp.

ϕ ( k ) = exp( −ikθ ) . The Corollary is proved.

Remark 3.11.

1

) From (3.14), the discrete Radon transform R satisfies the discrete moments condition, that is, for all f ∈ S ( Z

) and p ∈ N\ 0 , the function a −→

k∈Z

Rf ( H ( a, k )) k

is a homogeneous polynomial of degree p in a

, . . . , a

, with a = ( a

, . . . , a

) ∈ P . We recall that the Euclidean Radon transform R

satisfies the classical moments condition, that is, for all f ∈ S ( R

) and p ∈ N \ 0 , the function ω −→

R

Rf ( t, ω ) t

d t is a homogeneous polynomial of degree p in ω

, . . . , ω

, with ω = ( ω

, . . . , ω

) ∈ S

.

2

) The equality (3.15) is analogous to Poisson summation formula for the Fourier transform (see [5], [15]).

3

) The equality (3.16) gives a relationship between the discrete Fourier trans- form F and the discrete Radon transform R as in the Euclidean case (see [6], [8], [9] for more precision).

Now, let T

= R

/Z

be the n -dimensional torus ( T

is the dual of the discrete group Z

) and f ∈ S ( Z

) . We recall that the discrete Fourier transform of f is defined as the function Ff : T

−→ C given by

F f ( θ ) =

m∈Zⁿ

f ( m ) exp( −iθm ) , for all θ ∈ T

. We denote by e

, with θ ∈ T

, the function defined on Z

by

e

( m ) = exp( −iθm ) , for all m ∈ Z

. Proposition 3.12. Let f ∈ S ( Z

) . Then for all ( a, θ ) ∈ P × T

k∈Z

R ( f e

)( H ( a, k )) = Ff ( θ ) , (3.17) where Ff is the discrete Fourier transform of f .

Proof. Let f ∈ S ( Z

) and ( a, θ ) ∈ P × T

. From (3.1), we have R ( f e

)( H ( a, k )) =

m∈H(a,k)

f ( m ) exp( −iθm ) , for all k ∈ Z.