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

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1 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

2 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 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

3 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

4 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

5 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

6 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 (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

7 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

8 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

9 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

10 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

11 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), [2] Folland, G.B. and Stein, E.M., Hardy Spaces on Homogeneous Groups, Mathematical Notes 28, Princeton University Press, New Jersey, [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 B.P Maarif, Casablanca Japan Morocco 11