Integral Geometry on Discrete Grassmannians in Z n
Ahmed Abouelaz
∗, Enrico Casadio Tarabusi and Abdallah Ihsane
Abstract. We study the Radon transform R on the discrete Grassmannian of rank-d affine sublattices of Zn for 0 < d < n, extending and building on previous work of the first- and third-named authors in codimension 1.
By analogy with the integral geometry on Grassmannians in Rn, various natural questions are treated, such as definition and properties ofR and its dual transform R∗, function space setting, support theorems and inversion formulas.
Mathematics Subject Classification (2000).Primary 44A12; Secondary 05B35, 11D04.
Keywords. Radon transform, dual Radon transform, group actions, Fourier transform, systems of linear Diophantine equations, sublattices, Smith normal form.
1. Introduction
Integral geometry and its main object, the Radon transform, have been widely studied in the last few decades on several continuous or discrete settings; see, e.g., [11],[12] for extensive surveys. Considerable attention has been devoted to the setting of Grassmannians in Rn in any codimension, especially codimension 1 (one of the first instances considered) or dimension 1 (with the so-called X-ray transform). For 0< d < n, thed-plane Radon transformRc (a superscriptc will denote objects in this continuousRn setting) maps a complex-valued functionf, for instance C∞ with compact support, onRn to the functionRcf on the affine GrassmannianGcd,n of d-dimensional affine subspaces (d-planes for short) in Rn given by
Rcf(ξ) =
ξ
f(x)dm(x) for allξ∈Gcd,n,
∗Corresponding author.
© 2009 Birkhäuser Verlag Basel/Switzerland
1660-5446/09/030303-14, published online September 17, 2009 DOI 10.1007/s00009-009-0010-y
241
of Mathematics
where dm is the d-dimensional Lebesgue measure onξ. The d-plane dual Radon transform (Rc)∗, in stead, maps a function F, for instance C∞, on Gcd,n to the function onRn given by
(Rc)∗F(x) =
ξx
F(ξ)dµ(ξ) for allx∈Rn,
where dµ is the unit-mass measure on the set of d-planes through x which is invariant under the group of rigid motions that fixx. Among the contributions let us only quote [1],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13],[15]; for a detailed survey and extensive bibliography see [10].
Here we investigate the natural discrete counterpart of this situation, gener- alizing and extending some results of [2] to higher codimension. For 0< d < nwe consider the discrete Grassmannian Gd,n of rank-daffine sublattices—(discrete) d-planes for short—of the lattice Zn, and study the corresponding Radon trans- form on its elements endowed with the counting measure, as well as the dual Radon transform, their properties, function space settings, support theorems and inversion formulas.
In Section 2 we defined-planes inZn as solution sets of rank-slinear systems Ax=bof Diophantine equations, withs=n−d, as well as the affine Grassmannian Gd,nof alld-planes inZn, and the linear GrassmannianG0d,nofd-planes through 0 (or of parallelism classes ofd-planes). We study the relation betweenA, b and the correspondingd-plane. We describe the geometry ofd-planes and show thatAcan be taken in the setPs,nofs×nmatrices whose Smith normal form is a projection matrix. We study some properties of Ps,n andGd,n and prove that everyd-plane determines A ∈ Ps,n and b ∈ Zs up to the full left diagonal action of the group SLs. We show that the full group Aut(Zn) of lattice automorphisms ofZn acts on Gd,n preserving parallelism ofd-planes.
In Section 3 we define the Radon transform R on d-planes and study its properties as an operator on1(Zn) and on the Schwartz spaceS(Zn) into suitable spaces. In particular, using a special sequence ofd-planes defined in the previous section, we obtain a support theorem for R analogous to the classical support theorem inRn, although the proof is qualitatively different because no convexity is assumed on the bounded setK; as customary, injectivity ofR follows. We then invert R with a limit formula that evaluates the function at the same sequence.
We also show the expected relation of R with the Fourier transform. We define the dualR∗ and study its properties. In particular, we establish, by analogy with the Euclidean case, a duality formula withR and discuss the compositionR∗R.
2. The Grassmannians of discrete d -planes in Z
nDefinition 2.1. Let 0< d < nbe integers and sets=n−d. LetMm,ns be the set ofm×nmatrices of rankswith integer entries. A(discrete)d-plane inZn is the
set, if non-empty,
h=H(A, b) ={x∈Zn|Ax=b} where A∈Mm,ns ,b∈Zm, and m≥s.
An (n−1)-plane is also called a(discrete) hyperplane (cf. [2]).
The set Gd,n of all d-planes in Zn is the affine Grassmannian, while the subset G0d,n ⊂Gd,n of d-planes containing 0 is thelinear Grassmannian. (Use of the terms affine andlinear in this discrete context is an abuse of terminology.)
Ad-plane is therefore the non-empty solution set inZnof a rank-(n−d) linear system of Diophantine equations. Some history and references on such systems can be found in [14]. Several results for the hyperplane case s= 1 are proved in [2].
Remark 2.2. Different pairs (A, b),(A, b) may give rise to the samed-plane. Nev- ertheless, if H(A, b) =H(A, b) and A=A, then for anyxin either d-plane we have b=Ax=Ax=b. On the other hand, observe that in general the system Ax=bmay have solutions inRn(which is always the case ifm=s) while having none in Zn.
Thed-planesh, h areparallel if there existsz∈Zn such thath=h+z. In this case we write hh.
Proposition 2.3.
1. Parallelism is an equivalence relation on Gd,n.
2. If A=A then thed-planesH(A, b), H(A, b) are parallel.
3. For every x0 ∈Zn, to each d-plane h there exists a unique paralleld-plane h containingx0. If h=H(A, b), thenh =H(A, Ax0).
Proof. (1.) Immediate.
(2.) Takez=x0−x0 ifx0∈H(A, b) andx0∈H(A, b).
(3.) By (2), H(A, Ax0) is parallel to H(A, b). It obviously contains x0. If H(A, b) is ad-plane with the same properties, then b =Ax0. By (1) we have H(A, Ax0) = H(A, Ax0) +z for a suitable z ∈ Zn, whence A(x0+z) =Ax0, so that Az = 0. Finally, if Ax = Ax0 then Ax = A(x+z) = Ax0, so that H(A, Ax0)⊆H(A, Ax0). The reverse inclusion is proved similarly.
Applying Proposition 2.3(3) withx= 0 we obtain a natural projection γ:Gd,n→G0d,n given byγ(H(A, b)) =H(A,0),
a left inverse of the inclusion map of G0d,n into Gd,n. But γ is also the quotient map of Gd,n by the parallelism equivalence relation, since every element ofG0d,n can be regarded either as a d-plane through 0 or as the class of d-planes parallel to it.
Denote by diagm,n(λ1, . . . , λs) the matrixD= (dij)∈Mm,ns given by dij =
λi ifi=j≤s, 0 otherwise.
For k a positive integer, let SLk be the group of integer k×k matrices whose determinant is equal to ±1. In particular, SL1={±1}.
Lemma 2.4. For every A∈Mm,ns there existU ∈SLm andV ∈SLn such that U AV =DA= diagm,n(λ1, . . . , λs),
where λi >0 for i= 1, . . . , s, andλi divides λi+1 for i= 1, . . . , s−1. The matrix DA is uniquely determined by A and is called its Smith normal form. (In stead, U, V are not unique in general.)
The Smith normal form of A is preserved by multiplication of A on the left by any element of SLm or on the right by any element ofSLn.
Proof. The last statement is trivial. For the others see [14, Theorem 1].
Denote byPs,nthe set of matricesA∈Ms,ns whose Smith normal form is the projection matrix
Is,n= diags,n(1, . . . ,1),
i.e., thes×nmatrix whose left-hands×sblock is the identity, and the right-hand s×dblock is 0.
Lemma 2.5. IfT is ann×ninteger matrix whose upper rights×dblock vanishes, then T is inSLn if and only if its upper left s×sblock B is inSLs and its lower right d×dblock is inSLd, in which case
BIs,n=Is,nT .
Proof. A straightforward verification.
Proposition 2.6. IfA∈ Ps,n, then a possible choice of U in Lemma2.4is thes×s identity Is,s, that is, there exists V ∈SLn such that
AV =Is,n.
In other words, an s×n matrix A belongs to Ps,n if and only if it is the upper block (in fact, any block) of a suitable V ∈SLn.
Proof. Assume UAV = Is,n with U ∈ SLs and V ∈ SLn. If T ∈ SLn is as in Lemma 2.5 with B = (U)−1, then (U)−1Is,n =Is,nT. ThereforeAV =Is,n,
having setV =VT−1∈SLn.
Corollary 2.7. Anys×n matrixA, for 1 ≤s < s, obtained by deleting s−s rows from A∈ Ps,n, belongs toPs,n.
Proof. Up to reordering the rows of A, possible by multiplying it by an s×s permutation matrix, which is in SLs, we can assume that A is formed by the upper s lines of A, i.e., A = Is,sA. If V is as in Proposition 2.6, then AV =
Is,sAV =Is,sIs,n=Is,n.
Proposition 2.8. Ad-plane in Zn is a rank-daffine sublattice of Zn, that is, it is obtained by translating a rank-dsublattice ofZn by an integer vector.
Furthermore, in Definition 2.1 for a d-plane it is not restrictive to assume thatm=sandA∈ Ps,n. In other words, anyd-plane can be expressed asH(A, b), with A ans×nmatrix with Smith normal form Is,n, andb∈Zs.
Proof. Assume m, Aare as general as in Definition 2.1. The d-plane H(A, b) can be easily expressed asV H(DA, U b), withU, V as in Lemma 2.4. Since the bottom (m−s)×nblock ofDA is 0, the lastm−sentries ofU bmust all vanish for the d-plane to be non-empty, hence equations s+ 1 throughm of the linear system DAy=U bin the unknowny= (y1, . . . , yn)∈Znread 0 = 0 and can be eliminated, so thatH(DA, U b) =H(Is,mDA, Is,mU b).
For each i = 1, . . . , s the ith equation of the system reads λiyi = (U b)i, solvable only ifλi divides (U b)i, and in this case both sides of the equation can be divided byλi. Thus we can rewrite thed-plane asH(Is,n, c) wherec∈Zsis given byci= (U b)i/λi fori= 1, . . . , s. Explicitly we haveH(Is,n, c) ={c} ×Zd. Finally H(A, b) =V H(Is,n, c) =V({c} ×Zd) (2.1) proves the first claim; the equality V H(Is,n, c) = H(Is,nV−1, c) and Lemma 2.4
yield the other.
Proposition 2.9. AssumeA∈Ms,ns . The systemAx=b is solvable (i.e., H(A, b) is defined) for every b∈Zsif and only if A∈ Ps,n.
Proof. With notation as in Lemma 2.4, the system Ax=bis solvable if and only if so is DAy =U b (with x=V y). Ifλ1 =· · · =λs = 1 then the solution set of the latter system is {U b} ×Zd; this gives the “if” implication. Else, ifλi >1 for somei, then, if theith entry ofc∈Zsis 1, forb=U−1cthe latter system has no solution, because itsith equation isλiyi= 1; this proves the “only if” part.
Corollary 2.10. For s= 1, a row n-vector A is in P1,n if and only if the greatest common divisor of its entries is 1.
Proof. AssumeA∈ P1,n. The equalityAx0= 1, for which Proposition 2.9 ensures the existence ofx0∈Zn, expresses 1 as a linear integer combination of the entries ofA, whence the “only if” implication. Conversely, ifx0 is such a vector, then for everyb∈Zthe equationAx=bis solved byx=bx0. ThereforeA∈ P1,n. In view of these results we shall henceforth assume thatd-planes are assigned by matrices in Ps,nin stead ofMm,ns .
Proposition 2.11 (see[2, Proposition 2.6]fors= 1). For everyh∈Gd,n, the lattice Zn is the disjoint union of the distinctd-planes parallel toh. More precisely, for every A∈ Ps,n
Zn=
hh
h =
b∈Zs
H(A, b) (disjoint union).
Proof. It follows from Propositions 2.3(3) and 2.9.
Remark 2.12. In the same notation,Zsmay be regarded as a parameter space for the d-planes of each parallelism class, although the parametrization depends on the choice of A.
Remark 2.13. Every d-plane H(A, b) can be obtained as the intersection of s linearly independent hyperplanes, given by the rows A1, . . . , As of A ∈ Ps,n as matrices and the entries ofb∈Zs as right-hand terms. That is,
H(A, b) = s i=1
H(Ai, bi).
For everyi= 1, . . . , s, by Corollary 2.7 the row vectorAi is inP1,n.
More generally, assume given a partitionQof the index set{1, . . . , s}of the rows ofA. For each cosetQ∈ Q, the (#Q)×nmatrixAQ made of the rows ofA with index inQis in P#Q,n by Corollary 2.7, soH(AQ, bQ) is a (n−#Q)-plane (wherebQis the vector made by the entries ofbwith index inQ), and thed-plane H(A, b) can be expressed as
H(A, b) =
Q∈Q
H(AQ, bQ).
For any n, endowZn with the norm
|x|= max
j=1,...,n|xj| for everyx= (x1, . . . , xn)∈Zn. Lemma 2.14. We have
#{x∈Zn| |x|=r} ≤2n(2r+ 1)n−1 for everyr≥0, so that
C=
x∈Zn
1
(1 +|x|)n+1 = ∞ r=0
#{x∈Zn| |x|=r} (1 +r)n+1 <∞
Proof. Subtract the equality #{x∈Zn| |x| ≤r} = (2r+ 1)n from itself for two
consecutive values ofr.
Proposition 2.15. For everyh∈Gd,n there exist constants M, m >0 such that mkd ≤#{x∈h| |x|< k} ≤M kd for large enough k.
Proof. Follows easily from Proposition 2.8 and expression (2.1).
Proposition 2.16. There exists a sequence A(k) ∈ Ps,n such that, denotingh(k)= H(A(k),0), for everyk∈Nand everyx∈h(k)\{0}we have|x| ≥k. Thus for any b ∈Zs the minimum distance between distinct elements of the d-plane H(A(k), b) is not less than k.
Proof. Fork∈Nlet thes×nmatrixA(k)= (A(ijk)) be given by
A(ijk)=
⎧⎪
⎨
⎪⎩
1 ifi=j, kj−i ifj > i=s, 0 otherwise.
If then×nmatrixV(k)is defined by the same equalities (althoughiruns through n rather than s), thenV(k)∈SLn and A(k) =Is,nV(k), so thatA(k) ∈ Ps,n. For k >0, ifx∈h(k)then
x1=· · ·=xs−1= d t=0
ktxs+t= 0.
If x = 0 let xs+u, for 0 ≤ u ≤ d, be the first non-zero component of x; i.e., xs+u= 0, whilexs+t= 0 whenever 0≤t < u. Thenu < d and
0=xs+u=−k d t=u+1
kt−u−1xs+t, whence kdividesxs+u, so that|x| ≥ |xs+u| ≥k.
Ify, z∈H(A(k), b) are different, thenA(k)(y−z) = 0, so|y−z| ≥k.
Define a pseudonorm onGd,n by
|h|= min
x∈h|x| for everyh∈Gd,n, which satisfies|kh|=|k||h|fork∈Z.
We shall henceforth regard H as a map onPs,n×ZstoGd,n. We have that H mapsPs,n× {0}, identified withPs,n, ontoG0d,n. Ifπ:Ps,n×Zs→ Ps,nis the projection onto the first factor, then γ◦H =H◦π, that is, the diagram
Ps,n×Zs −−−−→ Pπ s,n
⏐⏐
H ⏐⏐H|Ps,n Gd,n −−−−→γ G0d,n
(2.2)
commutes, and all maps are onto.
By Lemma 2.4 the group SLsacts onPs,n, as well as diagonally onPs,n×Zs, by left multiplication.
Theorem 2.17 (see[2, Proposition 2.2]for the cases= 1of the second statement).
For A, A ∈ Ps,n, thed-planesH(A,0), H(A,0) through 0 coincide (equivalently:
for b, b ∈Zs the d-planes H(A, b), H(A, b) are parallel) if and only if A, A are in the same SLs-orbit (i.e., there existsU ∈SLssuch that U A=A). That is,
G0d,n=Ps,n/SLs, and H|Ps,n is the quotient map.
Similarly, for A, A ∈ Ps,n and b, b ∈ Zs, the d-planes H(A, b), H(A, b) coincide if and only if (A, b),(A, b) are in the sameSLs-orbit (i.e., there exists U ∈SLs such thatU A=A andU b=b). That is,
Gd,n= (Ps,n×Zs)/SLs, and H is the quotient map.
Proof. The “if” parts are immediate. For the equivalence of the parenthetical variant about parallels use Proposition 2.3.
For the first “only if” implication, assume H(A, b) = H(A, b), and write AV =Is,n=AVwithV, V∈SLnby Proposition 2.6. Up to changing unknowns byx=V y and replacingV withV V, we can suppose thatV =In,n.
Thus the systems Is,ny = 0 and Is,n(V)−1y = 0 are equivalent. For each j =s+ 1, . . . , n, the former is solved by the standard basis’jth vector y = ej, which thereby satisfies the latter, thus yielding that the first s entries of thejth column of (V)−1 vanish. So the whole upper rights×dblock of (V)−1vanishes, therefore Lemma 2.5 applies to T = (V)−1, and we gather thatU =B is in SLs and U A=U Is,n=Is,n(V)−1=A.
As to the second “only if”, apply γ to both d-planes to get H(A,0) = H(A,0), then takeU such thatU A=A. SoH(A, b) =H(A, b) =H(U A, U b) = H(A, U b), which implies U b=b by Remark 2.2.
Thus both vertical arrows of the commutative diagram (2.2) are the respective quotient maps of the actions of SLs.
The full group of lattice automorphisms ofZn is the semidirect product Aut(Zn) = SLnZn,
whose factor groupZn acts by translations.
Proposition 2.18. The GrassmannianGd,n inherits: the action ofSLn, that is V H(A, b) =H(AV−1, b) for everyV ∈SLn,A∈ Ps,n andb∈Zs, transitive on G0d,n (although not onZn orZn\ {0}); the action of Zn, namely
x+H(A, b) =H(A, b+Ax) for every x∈Zn,A∈ Ps,n andb∈Zs, transitive on each parallelism class; whence the action of Aut(Zn), namely
(V, x)H(A, b) =H(AV−1, b+AV−1x)
for every(V, x)∈SLnZn,A∈ Ps,n andb∈Zs, therefore transitive on the whole of Gd,n. Furthermore Aut(Zn) preserves paral- lelism: ifhh andg∈Aut(Zn) thenghgh.
Proof. Transitivity of SLn onG0d,n is by Proposition 2.6, while that ofZn on any parallelism class is by Proposition 2.3. The remaining claims are easy.
3. The Radon transform on d -planes and its dual
The actions of Aut(Zn) onZnand onGd,nnaturally induce ones on functions: for g∈Aut(Zn) andf :Zn→C, letfg:Zn→Cbe given by
fg(x) =f(gx) for everyx∈Zn; similarly, forF :Gd,n→C letFg:Gd,n→C be given by
Fg(h) =F(gh) for everyh∈Gd,n.
Definition 3.1. The d-plane Radon transform of f : Zn → C is the function Rf :Gd,n→C, wherever defined, given by
Rf(h) =
x∈h
f(x) for everyh∈Gd,n for which convergence is absolute.
Proposition 3.2 (see [2, Proposition 3.6]for the case s= 1 of translations). The linear transform Ris equivariant for Aut(Zn), i.e.,
(Rf)g=Rfg for every g∈Aut(Zn) andf :Zn →C for which Rf is defined.
Proof. A standard consequence of definitions.
Let
f=
x∈Zn
|f(x)|, for everyf :Zn→C, be the usual 1 norm with respect to the counting measure; the space
1(Zn) ={f :Zn→C| f<∞}
is the largest natural domain ofR, and is invariant for Aut(Zn).
The following support theorem, a stronger version of injectivity, is analogous to the result in the Euclidean case [11],[12], where K is assumed bounded and convex.
Theorem 3.3 (see [2, Theorem 5.3]for the cases= 1on finitely supported func- tions). Let f ∈ 1(Zn) and let K be a bounded (i.e., finite) subset of Zn. Then suppf ⊂Kif and only if Rf vanishes at everyd-plane that does not intersectK.
Proof. The “only if” implication is obvious. To prove the “if” part, assume by contradiction thatf(x0)= 0 for somex0∈Zn\K; by equivariance (see Proposi- tion 3.2), up to a translation we can assumex0= 0. Takeh(k)∈G0d,n as in Propo- sition 2.16 and assume k >maxx∈K|x|. Ifx∈h(k), then eitherx= 0 or|x| ≥k;
in either casex /∈K, whenceh(k)∩K=∅. Nevertheless
|Rf(h(k))−f(0)|=
x∈h(k)
|x|≥k
f(x) ≤
|x|≥k
|f(x)|; (3.1)
ifkis also large enough that
|x|≥k
|f(x)|<|f(0)|,
then these inequalities together are incompatible withRf(h(k)) = 0.
Theorem 3.4 (see [2, Remark 3.8(2)]for the case s= 1of norm estimates). The transform R is a one-to-one,Aut(Zn)-equivariant, unit-norm operator on 1(Zn) into the Aut(Zn)-invariant Banach space
1par(Gd,n) ={F :Gd,n→C| Fpar<∞}, where
Fpar= sup
h∈G0
d,n
hh
|F(h)| for everyF :Gd,n→C. Proof. For everyf ∈1(Zn) andh∈Gd,n we have
hh
|Rf(h)| ≤
hh x∈h
|f(x)|=f,
where the equality follows from Proposition 2.11; therefore Rfpar ≤ f. The inequality is itself an equality if f does not change sign.
IfRf ≡0, apply Theorem 3.3 withK empty and deducef ≡0.
Theorem 3.5 (see [2, Theorem 4.1] for the case s = 1). Any f ∈ 1(Zn) can be expressed in terms of its transform Rf by
f(x0) = lim
k→∞Rf(x0+h(k)) for every x0∈Zn, where the sequenceh(k)∈G0d,n is as in Proposition 2.16.
Proof. A translation reduces again tox0= 0. Then we have (3.1), whose rightmost
term tends to 0 ask→ ∞.
Let the sequence of seminormspN on functions onZn be given by pN(f) = sup
x∈Zn(1 +|x|)N|f(x)| for everyN ∈N andf :Zn →C. By analogy with the continuous case, theSchwartz space
S(Zn) ={f :Zn →C|pN(f)<∞for everyN ∈N}
of fast decreasing functions onZn is, endowed with the topology induced by the sequence of seminorms pN, another natural Aut(Zn)-invariant domain ofR.
Proposition 3.6. We have S(Zn)⊆1(Zn)with continuous embedding; explicitly, f ≤Cpn+1(f) for every f :Zn →C,
where C is given by Lemma2.14.
Proof. A straightforward computation.
Similarly, let the sequence of seminormsqN on functions onGd,nbe given by qN(F) = sup
h∈Gd,n
(1 +|h|)N|F(h)| for everyN ∈NandF :Gd,n→C, and define the Schwartz spaceonGd,n as
S(Gd,n) ={F:Gd,n →C|qN(f)<∞for everyN ∈N},
with the topology induced by the sequence of seminormsqN; this is too an Aut(Zn)- invariant space.
Theorem 3.7 (see[2, Theorem 3.7]for the cases= 1). The transformRis a one- to-one, Aut(Zn)-equivariant, bounded operator of S(Zn)intoS(Gd,n); explicitly
qN(Rf)≤CpN+n+1(f) for everyN ∈N andf ∈1(Zn), with C given by Lemma2.14.
Proof. For eachh∈Gd,n, setting w(x) = (1 +|x|)N we have (1 +|h|)N|Rf(h)| ≤min
x∈hw(x)
x∈h
|f(x)|
≤R(w|f|)(h)≤ wf ≤Cpn+1(wf) =CpN+n+1(f), where the rightmost inequality holds by Proposition 3.6. The remaining claims
follow from this and Theorem 3.4.
A functionφ:Zs→Cis ofmoderateM-growth, whereM ≥0, if
˜
p−M(φ) = sup
b∈Zs(1 +|b|)−M|φ(b)|<∞.
Proposition 3.8 (see[2, Proposition 3.9]for the cases= 1inS(Zn)). Iff ∈1(Zn) and φ:Zs→C is bounded (i.e., of moderate0-growth), or iff ∈ S(Zn)andφis of moderate M-growth for some M ≥0, then
b∈Zs
Rf(H(A, b))φ(b) =
x∈Zn
f(x)φ(Ax) for every A∈ Ps,n, both sides being absolutely converging sums.
Proof. FixA∈ Ps,nand set ψ(x) =φ(Ax) for everyx∈Zn. By Proposition 2.11
b∈Zs Ax=b
|f(x)φ(b)|=f ψ.
If φ is bounded, then f ψ ≤ fp˜0(φ) < ∞. Else if f ∈ S(Zn), then, since p−M(ψ)≤(1 +A)Mp˜−M(φ) ifA= supx∈Zn\{0}|Ax|/|x| is the norm ofA as operatorZn →Zs, we have
f ψ ≤Cpn+1(f ψ)≤Cpn+1+M(f)(1 +A)Mp˜−M(φ)<∞,
where we used Proposition 3.6 again. Therefore in either case the summations on b or x(under the constraint Ax =b) can be performed in any order, and both sides of the equality in the statement equal
b∈Zs
Ax=bf(x)φ(b).
For any n letTn = (R/2πZ)n be the n-dimensional torus, whose elements will be considered as row vectors. The(n-dimensional) Fourier transform off ∈ 1(Zn) is the functionFnf :Tn →Cgiven by
Fnf(θ) =
x∈Zn
f(x)e−iθx for everyθ∈Tn.
In the continuous setting, the following, especially fors= 1, is customarily referred to asFourier slice theorem.
Proposition 3.9 (see[2, formulas (3.15) and (3.16)]for the case s= 1in S(Zn)).
If f ∈1(Zn)then
FsRf(H(A,·))(θ) =Fnf(θA) for every A∈ Ps,nandθ∈Ts
(the left-hand side denoting the s-dimensional Fourier transform ofRf(H(A,·)), namely Rf(H(A, b))regarded as a function of b∈Zs alone).
Specializing for θ= 0 we have
hh
Rf(h) =
x∈Zn
f(x) for everyh∈Gd,n.
Proof. For eachθ∈Tn apply Proposition 3.8 withφ(b) =e−iθb for everyb∈Zs, a bounded function.
The caseθ= 0 follows from Proposition 2.11.
The bottom identity of the last statement is analogous to Poisson summation formula in the Rnsetting.
Definition 3.10. Thed-plane dual Radon transformofF:Gd,n→Cis the function R∗F :Zn →C, wherever defined, given by
R∗F(x) =
hx
F(h) for everyx∈Zn for which convergence is absolute.
Proposition 3.11. The linear transformR∗ is equivariant forAut(Zn), i.e., (R∗F)g =R∗Fg for everyg∈Aut(Zn)andF :Gd,n→C for which R∗F is defined.
Proof. As forR.
The Banach space
1ver(Gd,n) ={F :Gd,n→C| Fver<∞}, where
Fver= sup
x∈Zn
hx
|F(h)| for everyF:Gd,n→C, is the largest natural domain ofR∗, and is invariant for Aut(Zn).
Theorem 3.12 (see[2, Proposition 3.5] for the case s = 1, although on different function spaces). The transformsR, R∗ are indeed dual of each other, that is,
x∈Zn
f(x)R∗F(x) =
h∈Gd,n
Rf(h)F(h) for everyf ∈1(Zn), F ∈1ver(Gd,n).
Proof. As in Proposition 3.8, both sides of the equality in the statement equal
x∈Znf(x)
hxF(h), provided the summations in x ∈ Zn and in h ∈ Gd,n (under the constraintx∈h) can be exchanged. The absolute summability
x∈Zn
|f(x)|
hx
|F(h)| ≤ fFver<∞
grants this.
Nevertheless, for everyx∈Zneven the Radon transform of the characteristic functionδx of the set {x} is not in1ver(Gd,n), sinceRδx(h) = 1 for every hx.
More generally we have the following.
Proposition 3.13. The image ofR only intersects the domain ofR∗ at0, that is, R1(Zn)∩1ver(Gd,n) ={0}.
Proof. Forf ∈1(Zn) andx∈Zn, if the sequenceh(k)∈G0d,nis as in Theorem 3.5, then thed-planesx+h(k)fork∈Nare infinitely many (because they are distinct), they all containx, and Rf(x+h(k)) tends to f(x). At the same time, if Rf ∈ 1ver(Gd,n) then
hx|Rf(h)|<∞, thereforeRf(x+h(k)) tends to 0.
The compositionR∗Ris thus only defined at the trivial function, then it can- not be used to obtain any inverse, unlike in theRn setting. The dual transformR∗ could be defined differently—for instance, assigning different, summable weights in the set G(d,nx) of d-planes through the same vertex x ∈ Zn—but the counting measure is essentially the only non-trivial measure onG(d,nx) that is invariant for the isotropy of Aut(Zn) atx, which is indeed transitive by Proposition 2.18. The situation inRnis different, because the group of automorphisms considered there, namely rigid motions, allows for an invariant measure of finite total mass on the set ofd-planes through a point.
Acknowledgment
The authors wish to express their gratitude to the referee for valuable suggestions and comments.
References
[1] A. Abouelaz and R. Daher,Sur la transformation de Radon de la sph`ereSd, Bull.
Soc. Math. France121(no. 3) (1993), 353–382.
[2] A. Abouelaz and A. Ihsane, Diophantine integral geometry, Mediterr. J. Math. 5 (no. 1) (2008), 77–99.
[3] I.M. Gelfand, M.I. Graev and Z.Ja. ˇSapiro, Integral geometry on k-dimensional planes, (Russian) Funkcional. Anal. i Priloˇzen.1(1967), 15–31. (Translation) Funct.
Anal. Appl.1(1967), 14–27.
[4] I.M. Gelfand, M.I. Graev and Z.Ya. ˇSapiro,Differential forms and integral geometry, (Russian) Funkcional. Anal. i Priloˇzen.3(no. 2) (1969), 24–40. (Translation) Funct.
Anal. Appl.3(no. 2) (1969), 101–114.
[5] F.B. Gonzalez,Radon transforms on Grassmann manifolds, J. Funct. Anal.71(no. 2) (1987), 339–362.
[6] F.B. Gonzalez and T. Kakehi, Pfaffian systems and Radon transforms on affine Grassmann manifolds, Math. Ann.326(no. 2) (2003), 237–273.
[7] F.B. Gonzalez and T. Kakehi, Dual Radon transforms on affine Grassmann mani- folds, Trans. Amer. Math. Soc.356(no. 10) (2004), 4161–4180.
[8] F.B. Gonzalez and T. Kakehi, Moment conditions and support theorems for Radon transforms on affine Grassmann manifolds, Adv. Math.201(no. 2) (2006), 516–548.
[9] E.L. Grinberg, Radon transforms on higher Grassmannians, J. Differential Geom.
24(no. 1) (1986), 53–68.
[10] E.L. Grinberg and B. Rubin, Radon inversion on Grassmannians via G˚arding- Gindikin fractional integrals, Ann. of Math. (2)159(no. 2) (2004), 783–817.
[11] S. Helgason, Geometric Analysis on Symmetric Spaces, Mathematical Surveys and Monographs39, American Mathematical Society, Providence, RI, 1994.
[12] S. Helgason, The Radon Transform, 2nd edition, Progress in Mathematics 5, Birkh¨auser, Boston, Inc. MA, 1999.
[13] T. Kakehi,Range theorems and inversion formulas for Radon transforms on Grass- mann manifolds, Proc. Japan Acad. Ser. A Math. Sci.73(no.5) (1997), 89–92.
[14] F. Lazebnik, On systems of Linear Diophantine Equations, Math. Mag. 69(no. 4) (1996), 261–266.
[15] B. Rubin,Radon transforms on affine Grassmannians, Trans. Amer. Math. Soc.356 (no. 12) (2004), 5045–5070.
Ahmed Abouelaz and Abdallah Ihsane
D´epartement de Math´ematiques et Informatique, Universit´e Hassan II Casablanca 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 a ihsane morocco@yahoo.fr Enrico Casadio Tarabusi
Dipartimento di Matematica “G. Castelnuovo”, Sapienza Universit`a di Roma Piazzale A. Moro 2, 00185 Roma, Italy
e-mail:casadio@mat.uniroma1.it Received: November 1, 2007.
Revised: August 20, 2008.
Accepted: December 1, 2008.
