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The Radon Transform on Euclidean Spaces, Compact Two-Point Homogeneous Spaces and Grassmann Manifolds

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The Radon Transform on Euclidean Spaces, Compact Two-Point Homogeneous Spaces and Grassmann Manifolds THE RADON TRANSFORM ON EUCLIDEAN SPACES, COMPACT TWO-POINT HOMOGENEOUS SPACES AND GRASSMANN MANIFOLDS BY SIGURDUR HELGASON The Institute for Advanced Study, Princeton, N. J., U.S.A.(1) w 1. Introduction As proved by Radon [16] and John [13], a differentiable function / of compact support on a Euclidean space R n can be determined explicitly by means of its integrals over the hyperplanes in the space. Let J(~o, p) denote the integral of / over the hyperplane <x, co> =p where r is a unit vector and <,> the inner product in R n. If A denotes the La- placian on R n, do) the area element on the unit sphere S n-1 then (John [14], p. 13) J(x) =1(2~i)i-n (Ax)~(n-1)fs.J(~, <eo, x))dw, (n odd); (1) /(x)=(2:~i)-n(Ax) I do~f~ dJ(~'p) (neven), (2) j sn-, p - <co, x>' where, in the last formula, the Cauchy principal value is taken. Considering now the simpler formula (1) we observe that it contains two dual integra- tions: the first over the set of points in a given hyperplane, the second over the set of hyperplanes passing through a given point. Generalizing this situation we consider the following setup: (i) Let X be a manifold and G a transitive Lie transformation group of X. Let ~ be a family of subsets of X permuted transitively by the action of G on X, whence ~ acquires a G-invariant differentiable structure. Here ~ will be called the dual space of X. (ii) Given x E X, let ~ denote the set of ~ e 7~ passing through x. It is assumed that each and each ~ carry measures ju and v, respectively, such that the action of G on X and permutes the measures # and permutes the measures v. Q) Work supported in part by the :National Science Foundation, NSF GP 2600, U.S.A. 11- 652923. Acta mathematica. 113. Imprim6 le 11 mai 1965. 154 S. HELGASON (iii) If / and g are suitably restricted functions on X and ~, respectively, we can define functions ] on ~, ~ on X by ](~)= f~/(x)d/~(x), ~(x)= fi g(~)d~,(~). These three assumptions have not been made completely specific because they are not intended as axioms for a general theory but rather as framework for special examples. In this spirit we shall consider the following problems. A. Relate/unction spaces on X and ~ by means o/the trans/orms /--->] and g-->~. B. Let D(X) and D(~), respectively, denote the algebras o/ G-invariant di//erential operators on X and ~. Does there exist a map D --->b o/D(X) into D(~) and a map E -->/~ o/ D(.~.) into D(X) such that (Dl)^=bt, (Eg)"=l~ /or all / and g above ? C. In case the transforms [--> [ and g --~ ~ are one.to-one find explicit inversion/ormulas. In particular, find the relationships between f and (f)v and between q and (~)". In this article we consider three examples within this framework: (1) The already mentioned example of points and hyperplanes (w 2-w 4); (2) points and antipodal manifolds in compact two-point homogeneous spaces (w 5-w 6); p-planes and q-planes in R "+q+l (w w8). Other examples are discussed in [11] which also contains a bibliography on the Radon transform and its generalizations. See also [5]. The following notation will be used throughout. The set of integers, real and complex numbers, respectively, is denoted by Z, tt and C. If xER n, Ixl denotes the length of the vector x; A denotes the Laplacian on R n. If M is a manifold, C~(M) (respectively O(M)) denotes the space of differentiable functions (respectively, differentiable functions with compact support) on M. If L(M) is a space of functions on M, D and endomorphism of L(M) and p e M, ] eL(M) then [1)/] (p) (and sometimes D~(/(p))) denotes the value of D/at p. The tangent space to M at p is denoted My. If v is a diffeomorphism of a manifold M onto a manifold N and if ]EC~176 then F stands for the function/or -1 in C~r If D is a differential operator on M then the linear transformation of C~(N) given by Dr: J f--> (DF-1) ~ is a differential operator on N. For M=N, D is called invariant under ~ if D ~ = D. The adjoint representation of a Lie group G (respectively, Lie algebra (~) will be denoted Ada (respectively ad~). These subscripts are omitted when no confusion is likely. THE RADON TRANSFORM ON EUCLIDEAN SPACES 155 w 2. The Radon transform in Euclidean space Let R ~ be a Euclidean space of arbitrary dimension n and let E denote the manifold of hyperplanes in R ". If ] is a function on R ~, integrable on each hyperplane in R ~, the Radon trans]orm of ] is the function f on ~ given by f~/(x)da(x), ~e~, (1) where da is the Euclidean measure on the hyperplane ~. In this section we shall prove the following result which shows, roughly speaking, that / has compact support if and only if ] does. THE o R~ • 2.1. Let /E C~(R n) satis/y the ]ollowing conditions: (i) For each integer k>0 {x{kl(x) is bounded. (if) There exists a constant A >0 such that ~(~) =0/or d(O,~) >A, d denoting distance. Then /(x)=O /or {x[ >A. Proo/. Suppose first that / is a radial function. Then there exists an even function E E C ~ (R) such that/(x) = E( I x { ) for x E R n. Also there exists an even function _~ E C ~ (R) such that P(d(0,~))=](~). Because of (1) we find easily ~(p)= fR._lF((p~+{y{~)+)dy=~_l j~ F((p~+t~)'tn-2dt, (2) where ~n-1 is the area of the unit sphere in R ~-1. Here we substitute s = (p~ +t2) - and then put u =p-L Formula (2) then becomes u n-3 -~'(u-1 ) = ~n--1 f (F(s -1 ) s -n) (u 2 - sS) ds. (3) This formula can be inverted (see e.g. John [14], p. 83) and we obtain ~(S--1)S-n=CS~)] X (S2--U2) (4) where c is a constant. :Now by (if), /~(u -1) = 0 for 0 < u 4 A -1 so by (4), F(s -1) = 0 for 0 <s ~< A -1, proving the theorem for the case when / is radial. :Now suppose /E Cr n) arbitrary, satisfying (i) and (if). Let K denote the orthogonal group O(n). For x,y E R ~ we consider the spherical average 156 s. H~.LGASON Y) = f /(x+k.y) dk, where dk is the Haar measure on O(n), with total measure 1. Let Rj* be the Radon trans- form of/* in the second variable. Since (f) ^ = (f)~ for each rigid motion v of R n it is clear that [R~ /*] (x, 2) = f~ [(x + k. 2) dk, x E R n, ~ E ~, (5) where x +/r ~ is the translate of k. ~ by x. Now it is clear that the distance d satisfies the inequality d(O, x+ k.~) >~d(O, 2)- Ix[ for all xEIt n, kEK. Hence we conclude from (5) [R2/*J(x, 2) =0 if d(0, 2) >A + Ix]. (6) For a fixed x, the function y-->/*(x, y) is a radial function in C~176~) satisfying (i). Since the theorem is proved for radial functions, (6) implies that K/(x+k.y)dk=O if ly[>A+ix]. The theorem is now a consequence of the following lemma. LE~ 2.2. Let / be a/unction in C~176n) such that [xik/(x) is bounded on R n/or each integer lc > O. Suppose / has sur/ace integral 0 over every sphere which encloses the unit sphere. Then/(x)--O/or Ix] >1. Proo]. The assumption about / means that fs /(x+Lw)deo=O for L>lxl+l. (7) n--i This implies that f l /(x + y) dy = 0 for L>ix[+l. (8) yI>~L Now fix L>I. Then (8) shows that l~i<L ](x + y) dy is constant for 0 ~< I x I < L- 1. The identity THE RADON TRANSFORM ON EUCLIDEAN SPACES 157 fs~_ /(x+ Lw) (x~ + Lo~)do~=X~ fs~_ /(x+ Lw)dw + Le-~ s fl~l<L/(x+y)dy then shows that the function xJ(x) has surface integral 0 over each sphere with radius L and center x (0 <~ Ix I <L-1). In other words, we can pass from/(x) to xJ(x) in the identity (7). By iteration, we find that on the sphere l Yl =/~ (L > 1)/(y) is orthogonal to all poly- nomials, hence/(y) =-0 for l Yl =L. This concludes the proof. Remark. The proof of this lemma was suggested by John's solution of the problem of determining a function on R n by means of its surface integrals over all spheres of radius 1 (John [14], p. 114). w 3. Rapidly decreasing functions on a complete Riemnnnian manifold Let M be a connected, complete Riemannian manifold, M its universal covering manifold with the Riemannian structure induced by that of M, /]l=lll I • ... • the de Rham decomposition of M into irreducible factors ([17]) and let Mi =~t(M~) (1 <~i ~<l) where ~ is the covering mapping of M onto M. Let A, A, A~, A~ denote the Laplaee-Beltrami operators on M, M, Mi, ~ri, respectively. It is clear that A~ can be regarded as a differ- ential operator on 21~/.
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