Integral Transforms and Special Functions Vol. 21, No. 3, March 2010, 197–220
Integral geometry on discrete Grassmannians G (d, n) associated to Z
nAhmed Abouelaz* and Abdallah Ihsane
Department of Mathematics and Computer Science, Faculty of Sciences Aïn Chock, University Hassan II of Casablanca, Route d’El Jadida, Km 8, B.P. 5366 Maârif, 20100 Casablanca, Morocco
(Received 27 December 2008 )
The purpose of this paper is to extend carefully the discrete Radon transform, studied in [A. Abouelaz and A. Ihsane, Diophantine integral geometry, Mediterr. J. Math. 5(1) (2008), pp. 77–99], to the Radon trans- formRon the discrete GrassmannianG(d, n)(withn≥3 and 1≤d < n−1) consisting of all discrete d-planes in the latticeZndefined by systems of linear diophantine equations. By analogy with the integral geometry on Grassmann manifolds and projective spaces, which was developed by many authors, this study deals with various natural questions in this context: specific properties of the discrete Radond-plane transformRand its dualR∗,inversion formula forR(see Theorem 5.1) and also an appropriate support theorem for this Radon transform (see Theorem 6.3).
Keywords: system of linear diophantine equations; discreted-plane inZn; discrete GrassmannianG(d, n) associated to Zn; discrete Radond-plane transform; Poisson summation formula; inversion formula;
support theorem
Mathematics Subject Classification 2000: Primary: 44A12, 44A53; Secondary: 05C25, 05C65, 68R10
1. Introduction
Many authors contributed to the integral geometry on Grassmann manifolds and projective spaces, particularly Gelfand [4], Gonzalez [5,6], Grinberg [7,8], Helgason [9,10], and also Rubin [14].
We briefly recall the definition of the classical Radond-plane transformRc on the Euclidean spaceRnas well as its dualRc∗, dbeing an integer such that 0< d < n, withn≥2.We denote by G(d, n)the Grassmann manifold consisting of all affined-dimensional planes inRn.The Radon d-plane transformRcis defined by
Rcf (ξ )=
ξ
f (x)dm(x), for all(f, ξ )∈D(Rn)×G(d, n), (1) wheredmis the Euclidean measure on thed-planeξ,andD(Rn)denotes the space of all complex- valued C∞-functions onRnwith compact support. On the other hand, the dual Radond-plane
*Corresponding author. Email: a.abouelaz@fsac.ac.ma
ISSN 1065-2469 print/ISSN 1476-8291 online
© 2010 Taylor & Francis
DOI: 10.1080/10652460903092969 http://www.informaworld.com
transformRc∗is given by Rc∗ϕ(x)=
x∈ξ
ϕ(ξ )dμ(ξ ), for all(ϕ, x)∈E(G(d, n))×Rn, (2) whereE(G(d, n))denotes the space of all complex-valuedC∞-functions onG(d, n)anddμis the measure on the set of alld-planes throughx which is invariant under the group of rotations aroundxand such thatμ{ξ ∈G(d, n)|x ∈ξ} =1.
Our paper is organized as follows:
In Section 2, we fix some useful notations and also provide certain essential results on systems of linear diophantine equations. In particular, we recall the fundamental theorem which associates to each integer matrixA∈Mm,n(Z)(1≤m≤n) its Smith normal formD∈Mm,n(Z)uniquely defined byA(see Theorem 2.1). We also give an important criterion for the solvability inZn of a system of linear diophantine equations, which is essential for our study in this paper (see Proposition 2.3).
In Section 3, we study in detail the parametrization of the discreted-planes inZnas well as their properties.
In Section 4, we introduce various interesting function spaces, including the Schwartz spaces S(Zn)andS(G(d, n))(G(d, n)being the discrete Grassmannian inZn), and define the families of semi-norms on them. Afterwards, we give appropriate definitions of the discrete Radond-plane transformRand its dualR∗, and also study the specific properties of the two operatorsRandR∗. In Section 5, we establish an inversion formula for the discrete Radond-plane transform by extending that given in the authors’ paper [2]. Here, we note that the proof of the inversion theorem (see Theorem 5.1) requires some techniques that are relatively difficult compared to the case of the Radon transform on the discrete hyperplanes inZnstudied in [2].
In Section 6, we state and prove an appropriate support theorem for the discrete Radond-plane transform (see Theorem 6.3), which is analogous to the classical support theorem [9,10].
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, and also give some essential results concerning the solvability inZnof a system of linear diophantine equations under certain specific conditions. Throughout this paper, we denote bys the integern−d >1,withn≥3,andd is an integer satisfying the condition 1≤d < n−1.
Form∈Nsuch that 1≤m≤n,the set consisting of all integerm×nmatrices is designated byMm,n(Z). We denote bySL∗(k,Z) (1≤k≤n)the group of all integerk×kmatrices whose determinant is equal to±1.For simplicity, we denote a matrixA=(aij)1≤i≤m
1≤j≤n∈Mm,n(Z)by (aij). The matrixA=(aij)is called to be diagonal if aij =0,whenever i=j.The norm of A=(aij)is denoted byAand defined by
A = m
i=1
n j=1
|aij|. (3)
It is noted that throughout this paper, every row vector x=(x1, . . . , xn)∈Zn, resp. b= (b1, . . . , bs)∈Zs,is assumed to be identified with its transpose column vectortxgiven by
tx =
⎛
⎜⎝ x1
... xn
⎞
⎟⎠,
resp. with its transpose column vectortbgiven by
tb=
⎛
⎜⎝ b1
... bs
⎞
⎟⎠,
so that one can write a system ofslinear diophantine equations inZnin the reduced formAx =b, withA∈Ms,n(Z)andb∈Zs.
Before giving some essential results on systems of linear diophantine equations, it is important to note that our study in this paper is essentially based on these results. Indeed, we shall define a discreted-plane inZnas the set of all solutions inZnof a system of linear diophantine equations which is solvable in Zn. The reader can find some history, with references, on the theory of a system of linear diophantine equations in Lazebnik [13]. We just note here that many authors were interested in the conditions of the solvability inZnof this system, particularly Smith [15]
and Frobenius [3].
In the sequel, we state these results [13]. We begin with the following fundamental theorem which associates to each integer matrixA∈Mm,n(Z)of rankra diagonal matrixD∈Mm,n(Z) uniquely defined byAand called the Smith normal form ofA.
Theorem2.1 Let A be a matrix inMm,n(Z)of rankr, with 1≤m≤n.Then there exist two matricesL∈SL∗(m,Z)andR∈SL∗(n,Z)such that
LAR=D=diag(d1, d2, . . . , dr,0, . . . ,0), (4) wheredi>0,fori=1, . . . , r anddi|di+1,fori=1, . . . , r−1.
Proof The reader can find the proof of this classical theorem in [12] or [16].
Remark 2.2 In the special case whenm=r,the matrixDof Theorem 2.1 becomes
D=
⎛
⎜⎜
⎜⎝
d1 0 · · · 0 0 · · · 0 0 d2 · · · 0 0 · · · 0 ... . .. ... ... ... 0 0 · · · dr 0 · · · 0
⎞
⎟⎟
⎟⎠.
In the sequel, we will only consider the case whenr =m=s,where we recall thats=n−d >1, withn≥3 and 1≤d < n−1.
The following proposition gives us an important criterion for the solvability inZnof a system of linear diophantine equations.
Proposition2.3 LetA, L, R, Dbe as in Theorem 2.1, b∈Zmandc=Lb.Then the following two statements are equivalent:
(1) The system of linear diophantine equationsAx=bhas an infinity of integer solutions.
(2) The system of linear diophantine equationsDy=chas an infinity of integer solutions.
Proof By applying Theorem 2.1, one can writeAx=(L−1DR−1)x=b.Then for everyx∈Zn, we have the following equivalences:
Ax =b⇐⇒(L−1DR−1)x=b⇐⇒D(R−1x)=c⇐⇒(Dy=c, wherey=R−1x).
SinceR∈SL∗(n,Z),we haveR−1∈SL∗(n,Z),and thereforex ∈Znif and only ify=R−1x ∈
Zn.This completes the proof of the proposition.
Remark 2.4 Let c=Lb=(c1, . . . , cm). First, we observe that if cl=0 for some l∈ {r+ 1, . . . , m},then the systemDy =chas no integer solution. For this, assume thatcr+1= · · · = cm=0.Under this assumption, we have for eachy=(y1, . . . , yn)∈Zn
Dy =c⇐⇒(diyi=ci, for alli∈ {1, . . . , r}).
Therefore the systemDy =c has an infinity of integer solutions if and only if cr+1= · · · = cm=0 anddi|ci,for alli∈ {1, . . . , r}(see [13]).
We state and prove the following proposition which will be useful in the sequel. It gives us the general solution inZnof a solvable system of linear diophantine equationsAx =b.
Proposition2.5 LetA, L, R, Dbe as in Theorem 2.1, b∈Zm andc=Lb.Assume that the system of linear diophantine equations Ax =b has an infinity of integer solutions. Then the general solution inZnof the systemAx =bis of the form
x =
⎛
⎜⎝ x1
... xn
⎞
⎟⎠=R
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
d1
... cr
dr
yr+1
... yn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
, (5)
whereci is the i-th component ofc=Lb,for i=1, . . . , r,andyr+1, . . . , yn are free integer parameters.
Proof Letx∈Zn.From the proof of Proposition 2.3, we have
Ax =b⇐⇒(L−1DR−1)x=b⇐⇒D(R−1x)=c⇐⇒(Dy=c, wherey=R−1x).
Then the general integer solution of the systemAx=bis given by the equalityx=Ry,where yis defined by the formula
diyi=ci, for alli∈ {1, . . . , r}, withyr+1, . . . , ynare free integer parameters. Hence
x=
⎛
⎜⎝ x1
... xn
⎞
⎟⎠=R
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
d1
... cr dr
yr+1 ... yn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠ .
The proposition is proved.
3. Parametrization of the discreted-planes inZn
In this section, we study the parametrization of the discrete d-planes inZn and also give the properties of them.
LetAbe a matrix inMs,n(Z)of rankrandb∈Zs,withs=n−d >1 (n≥3 and 1≤d <
n−1). From Theorem 2.1, there exist two matrices L∈SL∗(s,Z) andR∈SL∗(n,Z) such that LAR=D=diag(d1, . . . , dr,0, . . . ,0),wheredi >0,for i=1, . . . , r,anddi|di+1,for i=1, . . . , r−1. By using Proposition 2.3 and also Remark 2.4, we deduce that the system Ax =bhas an infinity of solutions inZn if and only ifcr+1= · · · =cs=0 anddi|ci,for all i∈ {1, . . . , r}, where(c1, . . . , cs)=c=Lb.For this reason, we consider in the sequel the integer diagonal matrixD0∈Ms,n(Z)given by
D0=
⎛
⎜⎜
⎜⎝
1 0 · · · 0 0 · · · 0 0 1 · · · 0 0 · · · 0 ... . .. ... ... ... 0 0 · · · 1 0 · · · 0
⎞
⎟⎟
⎟⎠. (6)
The choice ofD0in this form is due to the following arguments:
(1) To guarantee the existence of an infinity of integer solutions of the systemAx=b,whenever D0is the Smith normal form ofA.
(2) To make the parameterbfree inZsthroughout this paper, i.e. the systemAx=bis solvable in Znalthoughbvaries inZs,for every matrixA∈Ms,n(Z)havingD0as its Smith normal form.
This leads us to define a subset Pd,n of Ms,n(Z)such that the system of linear diophantine equationsAx=bhas an infinity of solutions inZn,for every element(A, b)ofPd,n×Zs.We definePd,nas the subset ofMs,n(Z)consisting of all matricesA∈Ms,n(Z),which can be written in the following form:
A=W D0V ,
whereW∈SL∗(s,Z)andV ∈SL∗(n,Z).It is clear that the diagonal matrixD0given by (6) is the Smith normal form of every matrixA∈Pd,n.
Remark 3.1 The setPn,d coincide with the optimal subset ofMs,n(Z)consisting of all matrices A∈Ms,n(Z)such that the systemAx=b has infinitely many solutions inZn,for allb∈Zs. Indeed, letAbe a matrix ofMs,n(Z)satisfying the condition: the systemAx=bhas infinitely many integer solutions, for allb∈Zs.Prove thatA∈Pd,n.By Smith’s theorem, we can write LAR=D,whereD=diag(d1, d2, . . . , ds,0, . . . ,0)is the Smith normal form of A(di >0 fori=1, . . . , s anddi|di+1 fori=1, . . . , s−1), and(L, R)∈SL∗(s,Z)×SL∗(n,Z).Since L=(lij)1≤i,j≤s ∈SL∗(s,Z),then g.c.d(ls1, . . . , lss)=1. Thus, from Bezout theorem, there exist sintergsα1, . . . , αssuch thatα1ls1+. . .+αslss=1.On the other hand, our assumption implies that the diagonal systemDy=chas infinitely many solutions inZn(withc=(c1, . . . , cs)=Lb), thendi|ci,fori=1, . . . , s.By taking in paricularb=(α1, . . . , αs),we obtain
c=Lb=
⎛
⎜⎜
⎜⎝ c1
... cs−1
1
⎞
⎟⎟
⎟⎠.
Butds|cs,withds >0 andcs =1,thends=1.Sincedi|di+1 anddi >0 fori=1, . . . , s−1, it follows thatds−1=ds−2= · · · =d1 =1.We conclude thatD=D0,and thereforeA∈Pd,n. This proves thatPd,nis optimal.
From Proposition 2.3 and Remark 2.4, we deduce the following proposition which is essential for our study in this paper.
Proposition3.2 LetA=W D0V ∈Pd,nandb∈Zs,withW ∈SL∗(s,Z)andV ∈SL∗(n,Z).
Then we have the following two assertions:
(1) The system of linear diophantine equationsAx =bhas an infinity of integer solutions.
(2) The system of linear diophantine equationsD0y =chas an infinity of integer solutions, where c=W−1b.
Remark 3.3 Let(A, b)∈Pd,n×Zs.Throughout this paper, we denote byH (A, b)the infinite set of all solutions inZnof the system of linear diophantine equationsAx=b.The setH (A, b) will be called a discreted-plane inZn.
In the sequel, we shall apply the following proposition (see Proposition 2.5 for the proof).
Proposition3.4 LetA=W D0V ∈Pd,n,withW ∈SL∗(s,Z)andV ∈SL∗(n,Z),and letb be an arbitrary element ofZs.Then the general solution inZnof the system of linear diophantine equationsAx=bis of the following form:
x=
⎛
⎜⎝ x1
... xn
⎞
⎟⎠=V−1
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎜⎝ c1
... cs
ys+1
... yn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎟⎠
, (7)
where ci is the i-th component of c=W−1b, for i=1, . . . , s, and ys+1, . . . , yn are free integer parameters.
For simplicity, we fix some notations which will be useful in the sequel. LetV =(αij)1≤i,j≤n∈ SL∗(n,Z).We denote byV˜ the squares×ssubmatrix ofV given by
V˜ =
⎛
⎜⎝
α11 · · · α1s
... ... αs1 · · · αss
⎞
⎟⎠. (8)
The upper right-hands×d block of the matrixV is denoted byV∗and of course given by
V∗ =
⎛
⎜⎝
α1,s+1 · · · α1n
... ... αs,s+1 · · · αsn
⎞
⎟⎠. (9)
Moreover, we denote byVˆ the squared×dsubmatrix ofV defined by Vˆ =
⎛
⎜⎝
αs+1,s+1 · · · αs+1,n
... ...
αn,s+1 · · · αnn
⎞
⎟⎠. (10)
Now, we state and prove the following important theorem.
Theorem3.5 Let (A, b), (A, b)∈Pd,n×Zs. Set A=W D0V and A=WD0V, with W, W∈SL∗(s,Z)andV , V∈SL∗(n,Z).Assume that the two systems of linear diophantine equationsAx=bandAx=bhave the same solutions inZn.Then(V V−1)∗=Os,d,where Os,d denotes the zero matrix of the setMs,d(Z).
Proof Letxbe an integer solution of the systemAx=b,thenxis also an integer solution of the systemAx =b.We have
(S1)
Ax =b Ax=b, which is equivalent to
(S2)
D0V x=W−1b D0Vx =W−1b,
whereA=W D0V andA=WD0V,withW, W∈SL∗(s,Z)andV , V∈SL∗(n,Z).From Proposition 3.4, the general solution of the systemAx=bis of the form
x=V−1
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
... cs ys+1
... yn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠ ,
whereci is thei-th component ofW−1b,for i=1, . . . , s,andys+1, . . . , yn are free integer parameters. By assumption, it follows from (S2) that
D0V V−1
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
... cs ys+1
... yn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
=D0T
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
... cs ys+1
... yn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
=W−1b,
whereT is the matrix given by
T =V V−1=
⎛
⎜⎜
⎜⎜
⎜⎜
⎝
β11 · · · β1s · · · β1n
... ... ...
βs1 · · · βss · · · βsn
... ... ...
βn1 · · · βns · · · βnn
⎞
⎟⎟
⎟⎟
⎟⎟
⎠ ,
with the coefficientsβij (1≤i, j ≤n) are some integers of course. It is clear that
D0T =
⎛
⎜⎝
β11 · · · β1s β1,s+1 · · · β1n
... ... ... ...
βs1 · · · βss βs,s+1 · · · βsn
⎞
⎟⎠,
which implies
D0T
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
... cs ys+1
... yn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
=
⎛
⎜⎝
β11 · · · β1s
... ... βs1 · · · βss
⎞
⎟⎠
⎛
⎜⎝ c1
... cs
⎞
⎟⎠+
⎛
⎜⎝
β1,s+1 · · · β1n
... ... βs,s+1 · · · βsn
⎞
⎟⎠
⎛
⎜⎝ ys+1
... yn
⎞
⎟⎠. (11)
In particular, by takingys+1= · · · =yn =0,one has
D0T
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
... cs 0 ... 0
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
=
⎛
⎜⎝
β11 · · · β1s
... ... βs1 · · · βss
⎞
⎟⎠
⎛
⎜⎝ c1
... cs
⎞
⎟⎠=W−1b. (12)
From the equalities (11) and (12), we obtain
D0T
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
... cs ys+1
... yn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
=W−1b+
⎛
⎜⎝
β1,s+1 · · · β1n
... ... βs,s+1 · · · βsn
⎞
⎟⎠
⎛
⎜⎝ ys+1
... yn
⎞
⎟⎠
=W−1b.
Thus ⎛
⎜⎝
β1,s+1 · · · β1n ... ... βs,s+1 · · · βsn
⎞
⎟⎠
⎛
⎜⎝ ys+1
... yn
⎞
⎟⎠=
⎛
⎜⎝ 0 ... 0
⎞
⎟⎠.
Sinceys+1, . . . , yn are free integer parameters, it follows that
⎛
⎜⎝
β1,s+1 · · · β1n ... ... βs,s+1 · · · βsn
⎞
⎟⎠=(V V−1)∗=Os,d.
The theorem is proved.
As consequence of Theorem 3.5, we deduce the following lemma.
Lemma3.6 LetV , Vbe as in Theorem 3.5.Under the assumption of this theorem, we have
det(V V−1)= ±1. (13)
Proof Under the assumption of Theorem 3.5, the upper right-hands×d block of the matrix V V−1vanishes, and thereforeV V−1can be written in the following form:
V V−1=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
β11 · · · β1s 0 · · · 0
... ... ... ...
βs1 · · · βss 0 · · · 0 βs+1,1 · · · βs+1,s βs+1,s+1 · · · βs+1,n
... ... ... ...
βn1 · · · βns βn,s+1 · · · βnn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
, (14)
where all the coefficientsβij are some integers. By applying Theorem 4 stated in [11, p. 113], it follows that
det(V V−1)=det(V V−1)det(V V−1).
But det(V V−1)= ±1,since the matricesV andVbelong to the groupSL∗(n,Z).Then det(V V−1)det(V V−1)= ±1,
which implies
det(V V−1)= ±1.
The lemma is proved.
Now, we will study the essential problem of the parametrization of the discreted-planes inZn. This problem consists of finding a necessary and sufficient condition for two elements(A, b)and (A, b)of the setPd,n×Zs so that the discreted-planesH (A, b)andH (A, b)coincide, i.e.
the two systems of linear diophantine equationsAx=bandAx=bhave the same solutions inZn.More precisely, we state and prove the following theorem which is fundamental for the parametrization of the discreted-planes inZn.
Theorem3.7 Let(A, b), (A, b)∈Pd,n×Zs.Then the following two assertions are equiva- lent:
(1) The two systems of linear diophantine equationsAx=bandAx=bhave the same solutions inZn.
(2) There exists a matrixQ∈SL∗(s,Z)such thatA=QAandb=Qb.
Proof It is clear that the second assertion implies the first one. We just prove the implication (1)⇒(2).SetA=W D0VandA=WD0V,withW, W∈SL∗(s,Z)andV , V∈SL∗(n,Z).
Letc1, . . . , csbe thescomponents ofW−1b.From Proposition 3.4, the general solution of the systemAx =bis of the form
x=V−1
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
... cs ys+1
... yn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
, (15)
whereys+1, . . . , yn are free integer parameters. Then, under the assumption of our theorem, the general solution of the systemAx =bis also of the form (15).
By takingys+1 = · · · =yn =0,we obtain (see (12))
D0V V−1
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
... cs 0 ... 0
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
=V V−1
⎛
⎜⎝ c1
... cs
⎞
⎟⎠
=W−1b.
Since det(V V−1)= ±1 (see Lemma 3.6), it follows that
⎛
⎜⎝ c1
... cs
⎞
⎟⎠=(V V−1)−1W−1b. (16)
By applying the same techniques as above to the systemAx =b,we get
D0VV−1
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
... cs 0 ... 0
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
=D0
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
... cs 0 ... 0
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
=W−1b. But it is clear that
D0
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝ c1
... cs
0 ... 0
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
=
⎛
⎜⎝ c1
... cs
⎞
⎟⎠.
Then ⎛
⎜⎝ c1
... cs
⎞
⎟⎠=W−1b. (17)
From (16) and (17), we deduce that
(V V−1)−1W−1b=W−1b. Thus
b=Qb,
whereQis the square matrix belonging toSL∗(s,Z)given by
Q=W(V V−1)−1W−1. (18)
To complete the proof of this theorem, it suffices to prove that A=QA.For this, prove that A−Q−1A=Os,n,whereOs,ndenotes the matrix of the setMs,n(Z)whose coefficients are all equal to zero. By settingB=A−Q−1A,we have
B=W D0V −WV V−1W−1WD0V
=W (D0V −V V−1D0V)
=W B,
whereB=D0V −V V−1D0V.SinceW∈SL∗(s,Z),it is clear thatB=Os,nis equivalent to B=Os,n.Then, to complete the proof of Theorem 3.7, it suffices to prove thatB=Os,n.Let x =(x1, . . . , xs, xs+1, . . . , xn)∈Zn. Show thatBV−1x=Os,1,whereOs,1 denotes the zero column vector of the setMs,1(Z).SinceB=D0V −V V−1D0V,we have
BV−1x=D0V V−1x−V V−1D0VV−1x
=D0V V−1x−V V−1D0x.
Then
BV−1x=D0V V−1x−V V−1x (19) wherex=(x1, . . . , xs).But from (14), the matrixV V−1can be written as
V V−1=
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
β11 · · · β1s 0 · · · 0
... ... ... ...
βs1 · · · βss 0 · · · 0 βs+1,1 · · · βs+1,s βs+1,s+1 · · · βs+1,n
... ... ... ...
βn1 · · · βns βn,s+1 · · · βnn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠ ,
which implies
D0V V−1=
⎛
⎜⎜
⎜⎝
1 0 · · · 0 0 · · · 0 0 1 · · · 0 0 · · · 0 ... . .. ... ... ... 0 0 · · · 1 0 · · · 0
⎞
⎟⎟
⎟⎠
⎛
⎜⎜
⎜⎜
⎜⎜
⎜⎜
⎝
β11 · · · β1s 0 · · · 0
... ... ... ...
βs1 · · · βss 0 · · · 0 βs+1,1 · · · βs+1,s βs+1,s+1 · · · βs+1,n
... ... ... ...
βn1 · · · βns βn,s+1 · · · βnn
⎞
⎟⎟
⎟⎟
⎟⎟
⎟⎟
⎠
=
⎛
⎜⎝
β11 · · · β1s 0 · · · 0 ... ... ... ... βs1 · · · βss 0 · · · 0
⎞
⎟⎠,
and therefore
D0V V−1x =
⎛
⎜⎝
β11 · · · β1s ... ... βs1 · · · βss
⎞
⎟⎠
⎛
⎜⎝ x1
... xs
⎞
⎟⎠
=V V−1x.
Then the equality (19) becomes
BV−1x=V V−1x−V V−1x
=Os,1.
ThusBV−1=Os,n,and thereforeB=Os,n.It follows thatB=W B=Os,n.This completes
the proof of the theorem.
The theorem above shows that H (A, b)=H (A, b) if and only if there exists a matrix Q∈SL∗(s,Z)such that A=QAandb=Qb.This might suggest how to define correctly an equivalence relationRon the setPd,n×Zsso as to ensure the existence of a bijection between the quotient setPd,n×Zs/RandG(d, n)= {H (A, b)|(A, b)∈Pd,n×Zs},which associates to everyR-equivalence class(A, b)the discreted-planeH (A, b),with(A, b)∈Pd,n×Zs.More precisely, we defineRas follows:
(A, b)R(A, b)⇐⇒(∃Q∈SL∗(s,Z)such thatA=QAandb=Qb), (20) for all(A, b), (A, b)∈Pd,n×Zs.
As in the case of the discrete hyperplanes in Zn(see Proposition 2.2 of [2]), we now state and prove the following essential theorem which gives us an appropriate parametrization of the discreted-planesH (A, b),with(A, b)∈Pd,n×Zs.
Theorem3.8 Let :Pd,n×Zs/R→G(d, n)be the function defined by ((A, b))=H (A, b),for all(A, b)∈Pd,n×Zs,
where(A, b)denotes theR-equivalence class of(A, b). Thenis a bijection of Pd,n×Zs/R ontoG(d, n).
Proof It is obvious thatis well-defined and surjective. We just prove the injectivity of the func- tion.Let(A, b)and(A, b)be two elements ofPd,n×Zs.Suppose((A, b))=((A, b)) and prove that(A, b)=(A, b).By hypothesis, the discreted-planesH (A, b)andH (A, b)coin- cide, then there exists a matrixQ∈SL∗(s,Z)such thatA=QAandb=Qb(see Theorem 3.7).
Therefore(A, b)R(A, b). Hence(A, b)=(A, b).And the theorem is proved.
Definition3.9 A subset H of Zn is called a discrete d-plane if it can be written in the following form:
H = {x ∈Zn|Ax=b}, where(A, b)∈Pd,n×Zs.
We now state and prove the following proposition which will be useful in the sequel.
Proposition3.10 LetAbe a fixed element of Pd,n. Then the latticeZncan be written as Zn=
b∈Zs
H (A, b). (21)
Proof Letm∈Zn. Putb=Am∈Zs. Thenm∈H (A, b). Thus the latticeZnis contained in
b∈ZsH (A, b), which implies the equality (21).
Remark 3.11 LetAbe a fixed element ofPd,nand(b, b)∈Zs×Zssuch thatb=b.Then the two discreted-planesH (A, b)andH (A, b)are disjoint. Indeed, if there existsm∈H (A, b)∩ H (A, b),thenb=b=Am,which is absurd.
4. Properties of the discrete Radond-plane transform and its dual
In this section, we first define various interesting function spaces, and afterwards, we give appro- priate definitions of the discrete Radond-plane transformRas well as its dualR∗.We will also study the specific properties ofRandR∗.
Here, we fix some notations which will be useful in the sequel of this paper. LetH ∈G(d, n).
We always denote byH the set{(A, b)∈Pd,n×Zs|H (A, b)=H}.We set α(H )= inf
(A,b)∈H
(A2), (22)
β(H )= inf
(A,b)∈H(b2), (23)
and
(H )= inf
(A,b)∈H
(A2+ b2). (24)
It is clear thatα(H ), β(H )and(H )are positive integers, for allH ∈G(d, n).
We denote byl∗1(G(d, n))the space of all complex-valued functions F defined onG(d, n) satisfying the condition
b∈Zs|F (H (A, b)|<∞,for allA∈Pd,n.The Schwartz space of the discrete GrassmannianG(d, n), denoted byS(G(d, n)),is the subspace ofl1∗(G(d, n))consist- ing of all functionsF ∈l∗1(G(d, n))such that qN(F ) <∞ for allN ∈N, where(qN)N∈N is defined by
qN(F )= sup
H∈G(d,n)
1+β(H ) 1+α(H )
N
|F (H )|, (25)
for all(F, N )∈l∗1(G(d, n))×N. We note that(qN)N∈Nis a family of semi-norms onS(G(d, n)).
We also define an useful subspace ofS(G(d, n)),denoted byS∗(G(d, n)),as follows:
S∗(G(d, n))= {F ∈S(G(d, n))|TN(F ) <∞, for allN ∈N}, (26) where(TN)N∈Nis given by
TN(F )= sup
H∈G(d,n)
(1+(H ))N|F (H (A, b))|, (27) for all (F, N )∈S(G(d, n))×N. We check that (TN)N∈N is a family of semi-norms on S∗(G(d, n)).It is clear thatTN(F )≥qN(F ),for all(F, N )∈S∗(G(d, n))×N.
We also recall some useful notations which can be found in [2]. We denote byl1(Zn)the space of all complex-valued functions defined onZnsuch that
m∈Zn|f (m)|<∞.The Schwartz space
ofZn, consisting of all complex-valued rapidly decreasing functionsf defined onZn,is denoted byS(Zn).We denote by(pN)N∈Nthe family of semi-norms onS(Zn)defined by
pN(f )= sup
m∈Zn
(1+ m2)N|f (m)|,for all(f, N )∈S(Zn)×N, (28) wherem2=m21+ · · · +m2n,for allm=(m1, . . . , mn)∈Zn.
Moreover, the subspace ofS(Zn)consisting of all complex-valued functionsf defined onZn with finite support is denoted byC(Zn).Finally, we assume that the Schwartz spaceS(Zn)(resp.
S(G(d, n))) is equipped, in the sequel of this paper, with the topology defined by the family of semi-norms(pN)N∈N(resp.(qN)N∈N).
Now, we define the discrete Radond-plane transform and its dual.
Definition4.1 Letf ∈l1(Zn).The discrete Radond-plane transform offis the complex-valued functionRfdefined onG(d, n)by
Rf (H (A, b))=
m∈H (A,b)
f (m), for all(A, b)∈Pd,n×Zs. (29)
Definition4.2 LetF ∈S∗(G(d, n)).The discrete dual Radond-plane transform ofF is the complex-valued functionR∗F defined onZnby
R∗F (m)=
H∈G(d,n),m∈H
F (H )=
H∈Gm(d,n)
F (H ),for allm∈Zn, (30) whereGm(d, n) is the subset of the discrete GrassmannianG(d, n) consisting of all discrete d-planes inZncontainingm.
Remark 4.3 LetF ∈S∗(G(d, n))andm∈Zn.Show thatR∗F (m)has a sense. For this, it suf- fices to prove that the series
H∈Gm(d,n)F (H )is absolutely convergent. SinceF ∈S∗(G(d, n)), there exist an integerN > (s×n)/2 and a positive constantC(F, N )=T2N(F ) <∞such that
H∈Gm(d,n)
|F (H )| ≤C(F, N )
H∈Gm(d,n)
1 (1+(H ))2N. But for allH ∈Gm(d, n),there exists an element(A(H ), b(H ))ofH such that
(H )= inf
(A,b)∈H(A2+ b2)= A(H )2+ b(H )2, since{A2+ b2|(A, b)∈H}is a non empty subset ofN.Then
H∈Gm(d,n)
|F (H )| ≤C(F, N )
H∈Gm(d,n)
1
(1+ A(H )2+ b(H )2)2N
≤C(F, N )
(A,b)∈Pd,n×Zs
1
(1+ A2+ b2)2N.
By using the inequality (1+ A2+ b2)2≥(1+ A2)(1+ b2), which holds for all (A, b)∈Pd,n×Zs,we get
H∈Gm(d,n)
|F (H )| ≤C(F, N )
⎛
⎝
A∈Pd,n
1 (1+ A2)N
⎞
⎠
b∈Zs
1
(1+ b2)N <∞. (31) This proves thatR∗F (m)has a sense, and therefore the functionR∗F is well-defined.
