Characterizations of some function spaces by the discrete Radon transform on Z n.

Characterizations of some function spaces by the discrete Radon transform on Z n. A. Abouelaz and T. Kawazoe Abstract Let Z n be the lattice in R n and G the set of all discrete hyperplanes in Z n. Similarly as in the Euclidean case, for a function f on Z n, the discrete Radon transform Rf is defined by the integral of f over hyperplanes, and R maps functions on Z n to functions on G. In this paper we determine the Radon transform images of the Schwartz space S(Z n ), the space of compactly supported functions on Z n, and a discrete Hardy space H 1 (Z n ). 1 Introduction Let Z n be the lattice in R n. For a = (a 1, a 2,, a n ) Z n \{0} and k Z, the linear diophantine equation a x = k has an infinity of solutions in Z n if and only if k is an integral multiple of the greatest common divisor d(a) of a 1, a 2,, a n. Therefore, for a P = {z Z n d(z) = 1}, the set of solutions H(a, k) = {x Z n a x = k} forms a discrete hyperplane in Z n. Let G be the set of all discrete hyperplanes in Z n. Then G can be parametrized as P Z/{±1} (see 2 in [1]). As an analogue of the Euclidean case, the discrete Radon transform R, which maps functions on Z n to functions on G, is given by Rf(H(a, k)) = f(m) m H(a,k) for a suitable function f on Z n. In their previous paper [1] the first author and A. Ihsane investigated the basic properties of R. Especially, the Strichartz type inversion formula for R and the support theorem were obtained. Moreover, they showed that the discrete Radon transform R is a continuous linear mapping of S(Z n ) into S(G) (see 2 for the definitions of the Schwartz spaces Z n and G). Our natural question, that is a starting point of this paper, is whether this map is bijective or not. Let us suppose that f belongs to S(Z n ) and has a suitable decay. Then it follows from Corollary 3.10 in [1] that for θ T, F 1 Rf(H(a, ))(θ) = Ff(θa), (1) wehre F and F 1 are the classical Fourier (inverse) transforms on Z n and Z respectively. We note that Ff is a C function on T n. Therefore, in order to 1

show the surjectivity of the above mapping, we have to construct a C function on T n from θa variables with θ T and a P. However, it is impossible, because θa varies in a dence subset of T n (see Remark 8). Hence, the map R from S(Z n ) into S(G) is not bijective and, to characterize the image of S(Z n ), a condition corresponding to (1) is required. When we fix an a P and we restrict our attention to a local area Z n a in Z n and the set of hyperplanes with direction a: G a = {H(a, k) k Z} in G, we use a terminology local ; for example, we call S(Z n a) and S(G a ) the local Shcwartz spaces on Z n a and G a respectively (see 3). This paper is organized as follows. A characterization of the local Shcwartz space S(Z n a) is given in 3 and a Paley-Wiener type theorem is obtained in 4. We determine the Radon transform image of the global Schwartz space S(Z n ) in 5. We introduce discrete Hardy spaces on Z n and G, locally and globally, in 6 and then characterize their Radon transform images. 2 Notations Let Z n be the vector space of all a = (a 1, a 2,, a n ) Z n equipped with the norm a 2 = a 2 1+a 2 2+ +a 2 n and the inner product a b = a 1 b 1 +a 2 b 2 + +a n b n. For 1 p < let l p (Z n ) denote the space of all complex valued functions f on Z n with finite norm: ( ) 1/p f p = <. m Z n f(m) p We introduce a space G of hyperplanes in Z n. For a = (a 1, a 2,, a n ) Z n let d(a) = d(a 1, a 2,, a n ) denote the greatest common divisor of a 1, a 2,, a n and put P = {a Z n \{0} d(a) = 1}. For each (a, k) P Z we define a discrete hyperplane H(a, k) by H(a, k) = {m Z n a m = k}. Let G be the set of all hyperplanes H (a, k) with (a, k) P Z, which is parameterized as P Z/{±1}. For 1 p < let l p (G) denote the space of all complex valued functions F on G with finite norm: ( F a,p = F (H(a, k)) p) 1/p < for all a P. For f l 1 (Z n ), the Radon transform Rf on G is given by Rf(H(a, k)) = f(m). a m=k Then R(l 1 (Z n )) l 1 (G) (see Remark 3.8 in [1]) and the Strichartz type inversion of R is given as follows: For each m Z n, f(m) = lim j Rf(H(a j, a j m)), 2

where a j = (1, j, j 2,, j n 1 ) (see Theorem 4.1 in [1]). We define the Fourier (inverse) transform Ff(t) of f as Ff(t) = m Z n f(m)e im t, t T n. Similarly, F 1 denotes the one-dimensional Fourier (inverse) transform on Z. Then Corollary 3.6 in [1] asserts that Rf satisfies that for all a P and θ T, F 1 Rf(H(a, ))(θ) = Ff(θa). (2) We introduce Schwartz spaces on Z n and G as follows. Let S(Z n ) denote the space of all complex valued functions f on Z n such that for all N N, p N (f) = sup m Z n (1 + m 2 ) N f(m) < and S(G) the space of all complex valued functions F on G such that for all N N, ( 1 + k 2 ) N q N (f) = sup F (H(a, k)) <. a P, 1 + a 2 Then R(S(Z n )) S(G) and F (H(a, k))k p = m Z n f(m)(a m) p. (see Corollary 3.10 in [1]). In particular, as a function of a P, F (H(a, k))kp is a homogeneous polynomial of degree p (3) for all p = 0, 1, 2,. The dual space S (Z n ) of S(Z n ), the space of distributions on Z n, is defined as the space of all complex valued functions f on Z n for which there exists N N such that sup m Z n(1 + m ) N f(m) <. 3 Characterization of local Schwartz spaces As in the classical case (cf. Theorem 2.4 in [3]), we shall consider local Schwartz spaces on Z n and G respectively. In what follows we fix an a P. For each hyperplane H(a, k) let p a,k Z n be the point in H(a, k) that is nearest from the origin. We set Z n a = {p a,k k Z}. We define a local Schwartz space S(Z n a) on Z n by S(Z n a) = {f S(Z n ) supp(f) Z n a}. We denote by G a the set of all hyperplanes with direction a, that is, G a = {H(a, k) k Z}. 3

Since H(a, k) H(a, k ) = if k k and H(a, k) = Z n, it is clear that G a G a = if a a and a P G a = G (see [1] for more details). We define a local Schwartz space S(G a ) on G by S(G a ) = {F S(G) supp(f ) G a }. For a function F on G we denote by P Ga (F ) the function on G such that P Ga (F )(H) = F (H) if H G a and 0 otherwise. Theorem 1. For all a P, P Ga R(S(Z n a)) = S(G a ). Proof. Since The argument used in the proof that R(S(Z n )) S(G) (see Theorem 3.7 in [1]) yields that P Ga R(S(Z n a)) S(G a ). We shall prove the converse. For F S(G a ) we put f(m) = F (H(a, k))χ pa,k (m). Then it follows Proposition 3.4 in [1] that Rf(H) = F (a, k)χ G p a,k (H). If H = H(a, k) G a, then Rf(H(a, k)) = F (H(a, k)) and Rf(H) = 0 otherwise. Therefore, it follows that P Ga R(f) = F. Hence, to complete the proof of the surjectivity, it is enough to prove that f S(Z n ). Since f is supported on Z n a and p a,k k, it follows that for all m = p a,k, (1 + m 2 ) N f(m) = (1 + p a,k 2 ) N f(p a,k ) Since F S(G), it follows that p N (f) <. (1 + k 2 ) N F (H(a, k)). 4 A Paley-Wiener type theorem We shall consider a discrete Paley-Wiener theorem relatively at the discrete Radon transform, which characterizes the image of functions on Z n with finite support. Let K = {x 1, x 2,..., x l } be a finite set in Z n. We denote by D K (Z n ) the subspace of S(Z n ) consisting of all complex-valued functions on Z n such that suppf K. Let G K = {H G H K }. We denote by D K (G) the subspace of S(G) consisting of all complex-valued functions on G such that suppf G K. Each function f D K (Z n ) is of the form f = z K f(z)χ z. 4

As shown in [1], Proposition 3.4 and Theorem 3.5, Rf is of the form Rf = z K f(z)χ G z (4) and supprf z K G z. Hence Rf belongs to D K (G). In what follows we shall characterize functions of the form (4). We define D,K (G) as the subspace of D K (G) consisting of all functions F with moment condition (3) and for each m Z n, there exists j m N for which F (H(a j, a j m)) = F (H(a jm, a jm m)) for all j j m, (5) where a j = (1, j, j 2,..., j n 1 ). Lemma 2. Let F D K (G) and a P. Assume that F (H(a, k))k p = 0 for all p = 0, 1, 2,. Then F (H(a, k)) = 0 for all k Z. Proof. We note that for a fixed a P, F (H(a, k)) = 0 except finite k and p=0 F (H(a, k)) (iθk)p p! = F (H(a, k))e iθk = 0 for θ T. Therefore F 1 F (H(a, ))(θ) = 0 and thus, F (H(a, k)) = 0 for all k Z. Hence F = 0. Lemma 3. Let m Z n. (i) The case of m / K: Let j m,k = z K z m 2. If j > j m,k, then H(a j, a j m) K =. (ii) The case of m K: Let j K = z K z 2. If j > 2j K, then H(a j, a j m) K = {m}. Proof. (i) Let m / K and j > j m,k. We assume that H(a j, a j m) K and z H(a j, a j m) K. This implies that (z 1 m 1 ) + (z 2 m 2 )j + + (z n m n )j n 1 = 0. Hence j (z 1 m 1 ) and thus, there exists α Z such that z 1 m 1 = αj. Then it follows that (αj) 2 = (z 1 m 1 ) 2 z m 2 j m,k. Therefore, α must be 0 and z 1 = m 1. We repeat the same argument for z 2 m 2 and so on. Then z = m. This contradicts to m / K and z K. Hence H(a j, a j m) K = for all j > j m,k. (ii) Let m K and j > j K. We assume 5

that z H(a j, a j m) K. Similarly as above, z 1 m 1 = αj. Then it follows that j K z 2 z 2 1 = m 2 1 + (αj) 2 + 2αjm 1 (αj) 2 2 α jj K = α j( α j 2j K ). Therefore, α must be 0 and z 1 = m 1. We repeat the same argument for z 2 m 2 and so on. Then z = m. Theorem 4. R is a bijection of D K (Z n ) onto D,K (G). Proof. Let f D K (Z n ). As said above, Rf D K (G). Moreover, it follows from Corollary 3.10 in [1] that Rf satisfies (3) and from the proof of Theorem 4.1 in [1] that Rf satisfies (5). Therefore, Rf belongs to D,K (G). Since the inversion formula of R exists, it is sufficient to prove the surjectivity of R. Let F D,K (G). We define a function g on Z n by g(m) = lim j F (H(a j, a j m)). Then if follows from Lemma 3 that g D K (Z n ). Moreover, since F satisfies (5) and K is finite, there exists J K N such that g(m) = F (H(a j, a j m)) for all m K and j J K. Then it follows from Corollary 3.10 in [1] and Lemma 3 that for all j > j 0 = max{2j K, J K } and p = 0, 1, 2,, Rg(H(a j, k))k p = g(m)(a j m) p m Z n = F (H(a j, a j m))(a j m) p m K = ( ) F (H(a j, k)) k p m H(a j,k) K = F (H(a j, k))k p. Hence (Rg F )(H(a j, k))k p = 0 for all j j 0. Since Rf and F satisfies the moment condition (3), as a function of a P, (Rg F )(H(a, k))kp is a homogeneous polynomial of degree p, which is equal to 0 at a = a j for all j j 0. Therefore, (Rg F )(H(a, k))kp = 0 for all a P. Then by Lemma 2, we see that F = Rg. This completes the proof of the theorem. 5 Characterization of global spaces We shall obtain a global characterization of the Radon transform images of subspaces of l 1 (Z n ). For a subspace X of L 1 (T n ), we denote by ˆX the space on Z n consisting of all Fourier coefficients of G X, that is, ˆX = {f : Z n C there exists G X such that f(m) = Ĝ(m)}, where Ĝ(m) = G(t)e im t dt. T n 6

Let X G denote a subspace of l 1 (G) consisting of all F l 1 (G) such that there exists a G X for which F 1 F (H(a, ))(θ) = G(θa). (6) Theorem 5. Suppose that ˆX l 1 (Z n ). Then R is a bijection of ˆX onto XG. Proof. Since ˆX l 1 (Z n ) and R(l 1 (Z n )) l 1 (G), R is defined on ˆX and each Rf, f ˆX, belongs to l 1 (G). Moreover, it follows from (2) that Rf satisfies (6) with G = Ff X. Hence R( ˆX) X G. Since the inversion formula of R exists, it is sufficient to prove the surjectivity of R. Let F X G and suppose that F 1 F (H(a, ))(θ) = G(θa) for G X. Since ˆX l 1 (Z n ), it follows that G(θa) = Ĝ(m)e im θa = Ĝ(m)e iθm a m Z n m Z n = ( ) Ĝ(m) e iθk. m H(a,k) Hence F (H(a, k)) = m H(a,k) Ĝ(m). Therefore, if we define a function g on Z n by g(m) = Ĝ(m), m Zn, it follows that F = Rg. Clearly, G X implies g ˆX. This completes the proof of the theorem. Let S (G) denote a subspace of S(G) consisting of all F S(G) such that there exists a G C (T n ) for which F 1 F (H(a, ))(θ) = G(θa). Then Corollary 6. R is a bijective continuous mapping of S(Z n ) onto S (G). Let A 1 (T n ) denote a subspace of L 1 (T n ) consisting of all G L 1 (T n ) such that m Z n G(m) < and let L 1 (G) be a subspace of L 1 (G) consisting of all F L 1 (G) such that there exists a G A 1 (T n ) for which F 1 F (H(a, ))(θ) = G(θa). Then it follows that Corollary 7. R is a bijective continuous mapping of l 1 (Z n ) onto L 1 (G). Remark 8. (i) The left hand side of (6) is a function of θ with 0 θ 2π and G in the right hand side is a function on T n. Therefore, for a fixed a P, θ in the right hand side varies in the set of 0 θ L a a, where L a is the length of the line segment with direction a between the origin and the boundary of T n. Hence, if we rewrite (6) as ( θ ) ( F 1 F (H(a, )) = G θ a ), 0 θ L a, (7) a a then a a, a P moves all rational points in Sn 1 T n. Therefore, the condition (7) implies that the left hand side defined on {rω ω S n 1 T n and rational, 0 r L ω } can be extended to a function G on T n. (ii) In Corollary 6 and Corollary 7, the continuity of R 1 is an open problem. 7

6 Hardy spaces The lattice Z n is a space of homogeneous type, because Z n is equipped with a Euclidean distance and a counting measure. Hence, we can introduce real Hardy spaces on Z n according to the process in G. Folland and E. Stein [2]. We shall prove that the Radon transform R locally maps H 1 (Z n ) onto H 1 (Z) and globally maps to an atomic Hardy space on G. We briefly overview the definition of H 1 (Z n ) and its atomic decomposition. For ϕ S(R n ) we define a discrete dilation ϕ t, t = 1, 2, 3, of ϕ by ϕ t (x) = t n ϕ(t 1 m)χ m (x), x Z n. m Z n In the Euclidean case, ϕ t is an approximate identity. Hence, to keep this propperty, we put ϕ 0 (x) = ϕ(0)χ 0 (x). For any f S (Z n ), we define a radial maximal function M ϕ f on Z n by M ϕ f(x) = sup f ϕ t (x), x Z n, t=0,1,2, where is the discrete convolution on Z n. We say that a distribution f S (Z n ) belongs to H 1 (Z n ) if there is a ϕ S(R n ) with ϕ(x)dx 0 so that M R n ϕ f L 1 (Z n ). We put f H 1 = M ϕ f 1. Since f(x) cm ϕ f(x), it follows that H 1 (Z n ) l 1 (Z n ). According to the process in [2], Chap. 3, we can obtain an atomic decomposition of H 1 (Z n ) as follows. We say that a function b on Z n is a (1,, 0)-atom if it satisfies (i) suppb B(m 0, r), (ii) b r n (iii) b(m) = 0, m Z n where B(m 0, r) is a closed ball centered at m 0 Z n and radius r N, which depends on b. Then f H 1 (Z n ) if and only if there exist a collection {b i } of (1,, 0)-atoms on Z n and a sequence {λ i } of complex numbers with i λ i < so that f = i λ ib i. Moreover, f H 1 inf i λ i, where the infimum is taken over all atomic decomposition of f. Similarly, we can define H 1 (Z) on Z. For l N, we denote by H l (Z) the subspace of H(Z) which is constructed by using (1,, 0)-atoms supported on intervals with length 2lr, r N. Now we shall characterize the image Rb on G of a (1,, 0)-atom b on Z n. As pointed in 3, Rb = z B(m 0,r) b(z)χg z. Therefore, for each fixed a P, as a function of k Z, Rb(H(a, k)) is supported on {a z z B(m 0, r)}. We denote by k 0 = a m 0 the middle point of this support. Then we can easily deduce that supprb(h(a, )) [k 0 a r, k 0 + a r], 8

because if k = a z is in the support of Rb(H(a, )), k k 0 = a (z m 0 ) a r. We note that Rb(H(a, k)) b(m) m H(a,k) r n H(a, k) B(m 0, r) (8) n crn 1 r a = c( a r) 1 and Rb(H(a, k)) = m Z n b(m) = 0. These properties imply that c 1 Rb(H(a, k) is a (1,, 0)-atom on Z with radius a r. Therefore, if we define R a by we can obtain the following. R a f(k) = Rf(H(a, k)), Proposition 9. For each a P, R a continuously maps H 1 (Z n ) into H 1 a (Z). For the surjectivity we shall prove the following lemmas. Lemma 10. Let a P and Q be a cube in R n such that it is centered at the origin, each side length is 2 a r and a face parallels to H(a, 0). Then, for each integer l [ a r, a r], Q H(a, l) contains at least (2r) n 1 elements. Proof. Since Q is cubic, we may suppose that l = 0. When n = 2, the assertion is clear. Let a = (a 1, a 2,, a n ), n 3 and b i = (0,, 0, a i, a i+1, 0,, 0) for 1 i n 1. Then it follows from the case of n = 2 that there exist at least 2r elements n i,j = (0,, 0, m i,j, m i+1,j, 0,, 0) Q for which m i,j a i + m i+1,j a i+1 = 0, 1 j 2r. Hence each n i,j belongs to Q H(a, 0). Since n i,j, 1 i n 1, are linearly independent, the desired result follows. Lemma 11. Let a P. For each (1,, 0)-atom B on Z with radius a r, r N, there exist a (1,, 0)-atom b on Z n and a constant C for which B = CR a b, where C depends only on n and a. Proof. We may suppose that B is supported on [ a r, a r]. Let Q be a cube in R n such that Q is centered at the origin, each side length 2 a rand a face parallels to H(a, 0). By Lemma 10, for each integer l [ a r, a r], Q H(a, l) contains (2r) n 1 integral points, say {m q l }, 1 q (2r)n 1. We define a function b on Z n as b(m q l ) = (2r) (n 1) B(l) 9

for a r l a r, 1 q (2r) n 1 and 0 otherwise. Then b is supported on Q B(0, ([ n] + 1) a r). We note that b(m) = b(m q l ) = B(l) = 0 m Z n l,q l by the moment condition of B. Moreover, b(m q l ) (2r) (n 1) B(l) 2 a 1 (2r) n. Therefore, it is easy to see that 2 n 1 ([ n] + 1) n a n+1 b is a (1,, 0)-atom on Z n. Last we note that R a b(l) = b(m q l ) = B(l). m H(a,l) b(m) = q Hence B = R a b = C n,a R a (C 1 n,ab), where C n,a = 2 n+1 ([ n] + 1) n a n 1. Let F H 1 a (Z) and F = i λ ib i an atomic decomposition of F, where each atom is supported on an interval with length 2 a r. Then it follows from Lemma 11 that there exist a collection of (1,, 0)-atoms b i on Z n so that B i = C n,a R a b i. Therefore, F = R a ( i λ ic n,a b i ) and i λ i C n,a C n,a F H 1. Hence, F belongs to the image of H 1 (Z n ). Theorem 12. For each a P, R a continuously maps H 1 (Z n ) onto H 1 a (Z). Now we shall introduce an atomic Hardy space on G. Let B(m, r) Z n be a closed ball centered at m Z n with radius r N. We use the notations in 3 and we recall Theorem 4 and (8). We say that a function B on G is a (1,, 0)-atom on G if it satisfies (i) B D,B(m,r) (G), (ii) B(H(a, k)) H(a, k) B(m, r) r n for all H(a, k) G, (9) (iii) B(H(a, k)) = 0 for all a P, where B(m, r) depends on B. Clearly, if b is a (1,, 0)-atom on Z n, then Rb is a (1,, 0)-atom on G (see the argument before Proposition 9). Let B be a (1,, 0)-atom on G. Then by Theorem 4 and its proof, R 1 B is supported on B(m, r) and, if x B(m, r), then for a sufficiently large j, H(a j, a j x) B(m, r) = {x} (see Lemma 3). Hence, it follows from (9) that R 1 B(x) = B(H(a j, a j x)) r n and thus, R 1 B r n. Moreover, R 1 B(m) = R 1 B(m) m Z n m H(a,k) = B(H(a, k)) = 0. 10

Therefore, it is easy to see that R 1 B is a (1,, 0)-atom on Z n. Finally, we define the atomic Hardy space H,0(G) 1 on G as F H,0(G) 1 if and only if there exist a collection {B i } of (1,, 0)-atoms on G and a sequence {λ i } of complex numbers with i λ i < so that F = i λ ib i. We put F H 1,0 = inf i λ i, where the infimum is taken over all atomic decomposition of F. Then the previous argument yields the following. Theorem 13. R is an isomorphism of H 1 (Z n ) onto H 1,0(G). References [1] A. Abouelaz and A. Ihsane, Diophantine Integral Geometry, Mediterr. J. Math., 5 (2008), 77-99. [2] Folland, G.B. and Stein, E.M., Hardy Spaces on Homogeneous Groups, Mathematical Notes 28, Princeton University Press, New Jersey, 1982. [3] S. Helgason, The Radon Transform, Progress in Mathematics. 5 (1980). Ahmed Abouelaz Takeshi Kawazoe Department of Mathematics Department of Mathematics Faculty of Sciences and Informatics Keio University at Fujisawa University Hassan II Endo, Fujisawa, Kanagawa 252-8520 B.P. 5366 Maarif, Casablanca Japan Morocco 11