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INVERTING RADON TRANSFORMS THE GROUP THEORETIC APPROACH

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33 pages
INVERTING RADON TRANSFORMS : THE GROUP-THEORETIC APPROACH François Rouvière Abstract. In the framework of homogeneous spaces of Lie groups, we propose a synthetic survey and several generalizations of various inversion formulas from the literature on Radon transforms, obtained by group-theoretic tools such as invariant di?erential operators and harmonic analysis. We introduce a general concept of shifted Radon transform, which also leads to simple inversion formulas and solves wave equations. Mathematics Subject Classi?cation (MSC 2000): primary 44A12, secondary 43A85, 53C35, 58J70. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 2 Geometric setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 3 Convolution on X and inversion of R . . . . . . . . . . . . . . . . . . . . . . . . 7 4 Radon transforms on isotropic spaces . . . . . . . . . . . . . . . . . . . . . . . . 12 5 Harmonic analysis on X and inversion of R .

  • radon transform

  • invariant measure

  • all ?xed

  • riemannian manifold

  • left-invariant measure

  • transitive lie

  • means ofk-invariant

  • lie group


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INVERTING RADON TRANSFORMS : THE GROUP-THEORETIC APPROACH
François Rouvière
Abstract.homogeneous spaces of Lie groups, we proposeIn the framework of a synthetic survey and several generalizations of various inversion formulas from the literature on Radon transforms, obtained by group-theoretic tools such as invariant di¤erential operators and harmonic analysis. We introduce a general concept of shifted Radon transform, which also leads to simple inversion formulas and solves wave equations.
Mathematics Subject Classication (MSC 2000): primary 44A12, secondary 43A85, 53C35, 58J70.
Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Geometric setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Convolution onXand inversion ofR. . . . . . . . . .. . . . . . . . . . . . . . 4 Radon transforms on isotropic spaces . . . . . . . . . . . . . . . . . . . . . . . . 5 Harmonic analysis onXand inversion ofR. . . . . . . .. . . . . . . . . . . . 6 Shifted Radon transforms, waves, and the amusing formula . . . . . . . . . . . 7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1. Introduction
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The Radon transform on a manifoldXassociates to a functionuon this manifold its integralsRu(y)over a given familyYof submanifoldsy(equipped with suitable measures). One of the main problems of integral geometry is to recoverufromRuby means of an explicit inversion formula. The dual Radon transformRthen enters the picture in a natural way: it maps functions onYinto functions onX, by integrating (with respect to a suitable measure) over all submanifoldsy2Ywhich contain a given pointx2X. Here we assume that a Lie groupGacts transitively on bothXandY, so that they are homogeneous spaceX=G=K,Y=G=HwhereK,Hare Lie subgroups ofG; besides Kwill be compact throughout the paper. main examples for OurXwill be Riemannian symmetric spaces of the noncompact type, often assumed to have rank one (hyperbolic spaces). ForYthey will be a family of totally geodesic submanifolds ofX, or the family of horocycles. We rst look for a left inverse ofRof the following form u(x) =DRRu(x), (*)
whereDis some operator acting on functions onX all known examples. InDis an integro-di¤erential operator, sometimes even di¤erential. The purpose of the present paper is to emphasize three simple ideas leading to such results (or related to them), sometimes hidden under long calculations dealing with some specic example. As a benet we can unify several proofs from the literature, and obtain some generalizations. a.The rst idea stems from Proposition 3 (section 3.1) :RRis always aconvolution op-eratoronX, by aK-invariant measureS. BesidesScan be easily written down explicitly on rank one examples (Propositions 4 and 5). The problem is thus to nd a convolution inverseDtoS study it in section 4 for noncompact isotropic spaces (i.e.. We all Euclidean or hyperbolic spaces), looking forDas a polynomial of the Laplace-Beltrami operator of
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François Rouvière
Xwith given fundamental solutionS. This can be done for the Radon transform on even-dimensional totally geodesic submanifolds (with an additional assumption, see Theorem 8), or on horocycles of odd-dimensional hyperbolic spaces (Theorem 9). Another natural approach is to seek the convolution inverseDby means ofK-invariant harmonic analysis onX. We discuss this in section 5 for the totally geodesic Radon transform on hyperbolic spaces. Unfortunately it seems di¢ cult to ndDexplicitly by this method, except under the assumptions of Theorem 8 (proved by simpler tools) or for the case ofX=Hn(R)(already solved by Berenstein and Tarabusi [1]). b.Radon himself, and will be developed here in fullThe second idea goes back to Johann generality. If we replaceRby theshifted dual transformRt, obtained by integrating over submanifoldsyat distancet(in some sense) from a pointx, we may prove new inversion formulas forR. More precisely forX=G=K,Y=G=Hwe consider (section 6.1) Ru(gH) =ZHu(ghK)dh,Rtv(gK) =ZKv(gktH)dk, whereg,tare elements ofG,uis a function onXandvonY. Of courseRt=Rwhen t is then quite elementary to observe (section 6.2) that an inversionis the identity. It formula ofRat the originxoforK-invariantu, sayu(xo) =< T(y); Ru(y)>, implies the following new result u(x) =< T(t); RtRu(x)>(**) for arbitraryuandx. The notationT(t)means that the operatorTnow acts on the shift variablet, instead ofxas in (*). Applying this method to the horocycle transform on Rie-mannian symmetric spaces of the noncompact type, we obtain a new proof of Helgasons inversion formulas (Theorem 13 and Corollary 20). In Theorem 14 the same method.is applied to the totally geodesic transform, thus extending to all classical hyperbolic spaces known results for the real ones. c.It is now an intriguing question to compare the results (*) and (**) of methods a and b. For the2-dimensional totally geodesic transform onX=H3(R), Helgason ([10]) obtained a curious amusing formula by equating the right-hand sides of (*) and (**). In sections 6.4 and 6.5 we give direct proofs of such formulas, for the Laplace operator rst (Proposition 16), then for general invariant di¤erential operators (Theorem 17). The content of these results is easily understood on the example of the Radon transform on all hyperplanes ofX=R2k+1 the inversion Here(see section 6.4 for more details). formulas (*), resp. (**), are Cu(x) =LkxRRu(x), resp.Cu(x) =@t2kRtRu(x)t=0, whereCis a constant factor andLis the Euclidean Laplacian. Passing from one to the other is thus an immediate consequence of thewave equation LxRtv(x) =@t2Rtv(x) forv=Ruand allxandt. In Proposition 16 and Theorem 17 we construct solutions of some generalized wave equations, some of them only valid whentis the identity (but this su¢ ces for our purpose). Such results may have independent interest, providing explicit solutions of certain multitemporal wave equations by means of shifted dual Radon transforms, which appear as integrals of elementary plane waves (Proposition 19, for horocycles). One last remark : explicit inversion formulas for the totally geodesic Radon transform seem rather di¢ cult to obtain, and most of them in the literature are only given for spaces of constant curvature. We obtain here some results forX=Hn(F), withF=R,Cor H, provided that the tangent spaces to the geodesic submanifolds under consideration are
INVERTING RADON TRANSFORMS : THE GROUP-THEORETIC APPROACH
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F This-vector spaces (section 4.3.c, Theorem 14, Proposition 16). seems to be the simplest case afterRnandHn(R).
Acknowledgements.work is clearly and strongly inuenced by Sigurur Helga-This sons books and papers, and I take this opportunity of expressing all my debt to him. I am also deeply grateful to Mogens Flensted-Jensen, Henrik Schlichtkrull and Fulvio Ricci for allowing me to present part of these results in Copenhagen and Torino respectively. I am indebted to the referee for remarks leading to several improvements in section 6.3.
Notations. a. General notations.As usualR;C;Hrespectively denote the elds of real numbers, complex numbers and quaternions. When considering vector spaces onH, the scalars will act on the right. IfXis a (realC1) manifold,C(X)is the space of complex-valued continuous functions onX,Cc(X)the subspace of compactly supported functions andD(X)the subspace of compactly supportedC1functions;D0(X)is the space of distributions andE0(X)the subspace of compactly supported distributions. IfT on a space of functions on di¤erential)is an operator (e.g.X, a notation like T(x)f(x; y)means thatTacts on the variablex, noty. IfGis a (real) Lie group, lete,g,exp,Ad,adrespectively denote its origin, Lie algebra, exponential mapping, adjoint representations ofGandg. WhenGacts onX, we shall writegx, or sometimes(g)xor evenX(g)x, for the point obtained wheng2G acts on the pointx2X. In particular, forV2g, it is convenient to writegVfor Ad(g)V this context,. InD(X)is the algebra of linear di¤erential operators onXwhich commute to the action ofG, andD(G)refers to the special case whenGacts onto itself by left translations. IfXis a Riemannian manifold,(x; r)will denote the sphere with centerx2Xand radiusr0. Also!n= 2n=2=(n=2)is the area of the unit sphere in the Euclidean spaceRn. b. Riemannian homogeneous spaces.LetGbe a Lie group,Kacompactsubgroup andg,k homogeneous manifold Thetheir Lie algebras.X=G=Kcan be provided with a G Indeed a scalar product can be taken on-invariant Riemannian structure.g, invariant under the compact groupAdG(K); theng=kpwherep, the orthogonal complement ofking, is aK stable under-invariant (i.e.AdG(K)) vector subspace which can be identied with the tangent space toXat the originxo=K by the action of. Carrying GtheK-invariant scalar product onpwe thus obtain a Riemannian structure onX, and elements ofGare isometries. We shall also considerY=G=H, whereHis another Lie subgroup ofG. c. Riemannian symmetric spaces XI for their basic(see [8] chap.IV or [15] chap. properties). A special case of the previous one, they are the homogeneous spacesX= G=K, whereGis a connected Lie group provided with an involutive automorphismand Kis a compact subgroup which lies between the group of all xed points ofinGand its identity component. The di¤erential ofateinduces a Lie algebra automorphism of gand the eigenspace decompositiong=kp(same notations as before). The exponential mapping of the symmetric space isExp :p!X, related toexp : g!GbyExpV= (expV)KforV2p curve. TheExpRVis the geodesic ofXwhich is tangent to the vectorVat the originxo=K. d. Riemannian symmetric spaces of the noncompact type.Assuming further thatGis a connected non compact real semisimple Lie group with nite center andK a maximal compact subgroup, we obtain the subclass of Riemannian symmetric spaces of the noncompact type, particularly interesting because of their rich (and well-known)
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