Selected Topics in Integral Geometry

Selected Topics in Integral Geometry

Translations of

MATHEMATICAL MONOGRAPHS Volume 220

Selected Topics in Integral Geometry I. M. Gelfand s. G. Gindikin M. I. Graev

EDITORlAL COMMITTEE AMS Subcommittee Robert D. MacPherson Grigorii A. Margulis James D. Stasheff (Chair) ASL Subcommittee Steft"en Lempp (Chair) IMS Subcommittee Mark I. Freidlin (Chair) H. M. reJILcfl8Jlll, C. r. rHllJlHKHB, M. H. rpaeB H3GP AHHbIE 3A.nA '4H HHTErPAJIbHoiit rEOMETPHH 1I0BPOCBET. MOCKBA, 2000

Translated from the Russian by A. Sbtern 2000 Mathematics Subject Classification. Primary 53C65j Secondary 42A38. 42BI0, 43A32, 44A12, 46F12, 60005, 6OEOS, 6OElO, 65RlO, 92C55.

For additional information and updates on this book, visit www.ams.org/bookpages/mmono-220

Library of Congress Cataloging-in-Publlcatlon Data Gel'fand, I. M. (Izrail' Moiseevich) [Izbrannye zadachi integral'noi geometrii. English) Selected topics in integral geometry / I.M. Gelfand, S.G. Gindikin. M.I. Graev ; [translated from the Russian by A. Shtem). p. cm. - (Translations of mathematical monographs, ISSN 0065-9282 ; v. 220) Includes bibliographical references and index. ISBN ~21S.2932-7 (acid-free paper) 1. Integral geometry. I. Gindikin, S. G. (Semen Grigor'evich) 11. Graev, M. I. (Mark losifovich) Ill. Title. IV. Series. QA672.G4513 2003 516.3'62-dc21

2003052222

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08 07 06 05 04 03

Contents Preface to the English Edition

xi

Preface

xiii

Chapter 1. Radon Transform 1. Radon transform on the plane 1.1. Radon transfoml on the Euclidean plane 1.2. Inyr.xsion formula 1.3. Remarks 1.4. Radon transform on the affine plane 1.5. Relation to the Fourier transfoml and another proof of the imersion formula 2. Radon transform in three-dimensional space 2.1. Radon transform in Euclidean space 2.2. Radon transform in the affine space 2.3. Radon transform for a space of arbitrary dimension 3. Wave equation and the Huygens principle 3.1. 1'wo-dimeruiional case 3.2. ThJWKlimetwjonal CJI!!C 4. Cavalieri's conditions and Palcv-Wiener theorems for the Radon tra.ru;form 4.1. Cavalieri's conditions for rapidly decreasing functions 4.2. Pn.ley-Wiener theorem for the space S(R2) 4.3. PaJey-Wicner theorem for the space 1>(R2) of compactly supported infinitely differentiable functions 4.4. Inversion of the Radon transform of a function I e 7>(R2) using the mO~nL~

5.

6. 7.

8.

1 1 1 2 4 '" 5 7 7 9 10

12 12 13 14

14 IS 16 17

4.5. Reconstruction of unknown directions from known values of 'RI Poi'fflD fonnula for the Radon transform, and the discrete Radon transform 5.1. PoillHOn formula for the Radon transform on a plane 5.2. Dj.'!Crt1e Radon t.ranat"orm: relation to the FOllrjer series 5.3. Problem of integral geometry on the torus Minkowski Funk transform Radon transform of differential forms 7.1. Radon transform of I-forms on the plane 7.2. Radon transfoml of 2-forms on the plane 7.3. Radon transform of 2-forms in three-dimensional space 7.4. Radon transfoml of 3-forms in three-dimensional space Radon transform for the projective plane and projective space v

17

19 19 21

21 22 25 25 26 28

29 30

vi

CONTENTS

8.1. Spaces pa and (pa)' 8.2. Radon transform for p3 8.3 R.f>lsrtion t,o t,hp Rffinp Radnn trRnRfnnn fnr R3 Rnd tn thp Minkowski-F\mk transform for the three-dimensional sphert' 8.4. Inversion formula for the Radon transform on p3 8.5. On thp inversion formulas for t.hp Rffinp Radon transfnrm nn R3 and the Minlrowski-F'unk transform for S3 8.6. Description of the image of the Radon transform for p3 8.7. Radon transform for the projective plane p2 8.8. Radon transform for the projective space of an arbitrary dimension 9. Radon transform on the complex affine space 9.1. Definition of the Radon transform 9.2. Relation to the FouDer transfonn 9.3. Inversion formula for the Radon transform 9.4. Case n - 2 9.5. Relation to Paley-Wiener theorems for the affine Radon transform in 22 and R3 Chapter 2. John Transform 1. John transform in the real affine space 1.1 John transform in R3 1.2. John transform and the Gauss hyPergeomt'tric function 1.3. Theorem on the image of the operator .1 lA. Space S(H') 1.5. Description of the image of S(R3 ) in the space S(H') 1.6. Proof of Theorem 1.1 on the image of the John transform 1.7. Analogs of the operator K 2 John transform nf differential fnrms no R3 2.1. Definjtjon of the John tran'ifonn of differentia] forms 2.2. John transform of 3-fortJl5 on R3 2.3. John transform of 2-forms on R3 24 John transform of 1-forms on R3 3. John transform in the thretHlimensional real projective space 3.1. Manifold of lines in pa 3 2 Jnhn transform in p3 3.3. Relation to the John transform in the affine space 304. Description of the image of the John transform 3.5. Another way to define the John transform 3.6. Proof of the theorem on the image of the John transform 3.7. John transform 88 an intertwining operator 4. John transform in the complex affine space 4 1 Jnhn transform in C3 4.2. Differential form KI{) and the theorem on the image of the John trMsform 4.3. Inyemion formula 404. Analogs of the operator K 5. Problems of integral geometry for line complexes in C3 5.1. Problem of integral geometry for a complex of lines in C 3 intenleCting R cnryp

30 31 32 32

34 35 35 38 38 39

39 40

40 40

43 44 44

45 46 48 50

50 51

53 53

54 56 58

59 59 60

61 62 63 64 65 67 67

68

68 69 71 71

CONTENTS

5.2. 5.3. 5.4. 5.5.

2.

3.

4. 5.

6.

7.

8. 9.

ca

Definition of admissible line complexes in Necessary and sufficient conditions for a complex K to be admissible Geometric structure of admissible complexes Description of admissible complexes

72 73 75 76

Integral Geometry and Harmonic Analvsis on the Hyperbolic Plane and in the Hyperbolic Space 79 Elements of hyperbolic planimetry 79 1.1. Models of the hyperbolic plane 79 1.2. Horocycles 81 1.3. Geodesics 82 Horocycle transform 83 2.1. Definition of the operator 'Rh 83 2.2. Inversion formula 8.3 2.3. Asgeirsson relations 85 2.4. Symmetry relation 86 2.5. Inversion formula for the borocycle transform in another model of the hyperbolic plane 86 Analog of the Fourier transform on the hyPerbolic plane and the relation between this analog and the horocycle transform 86 3.1. Founer tran.'ifoOD OD R2 86 3.2. Fourier transform on the hyperbolic plane 88 3.3. Relation to the horocycle transform and the invemion formula 88 3.4. Symmetry relation 90 3.5. Plancherel formula 91 Relation to the representation theory of the group 8L(2, R) 91 Integral transform related to lines (geodesics) on the hyperbolic plane £2 93 5.1. Definition and the inversion formula in the Poincare model 93 5.2. Relation to the Radon tr-dJlSform on the projective plane 95 Horospherical transform in the three-dimensional hyPerbolic space £3 96 6.1. Models of the hyperbolic space 96 6.2. Horospheres 97 6.3. Horospherical transform 98 6.4. Inversion formula 99 6.5. Symmetry relation 101 6.6. Inversion formula for the horospherical transform in another model of the hYperbolic space 101 6.7. Integral transform related to completely geode8ic surfaces iD /:.3 102 Analog of the Fourier transform in the hyperbolic space, and its relation to the horospherical transform lOO 7.1. Definition ofthe Fourier transform 103 7.2. Inversion formula 104 7.3. Symmetry relation and the Plancherel formula lOO Relation to the representation theory (01" the group 8L(2. Cl 105 Wave equation for the hyperbolic plane and hyperbolic space, and the Huygens principle 106 9.1. Two-dimensional case 106

Chapter 3. 1.

vU

CONTENTS

viii

9.2. Thre&dimensional case

108

Chapter 4.

Integral Geometry and Harmonic Analysis on the Group G = SL(2, Cl 111 1. Geometry on the group G 111 1.1. Group G as a homogeneous space 111 1.2. Plane sections of the hyperboloid G 112 113 1.3. Manifold of horospheres 1.4. Embedding the manifold of horospheres H in the projective space 116 117 1.5. Line complex in C 3 associated with the manifold of horospheres 1.6. Manifold of paraboloids 117 2. Integral geometry on the grOUp G = SL(2, Cl 119 2.1. Integral transforms related to the Space H of horospheres and the complex of lines K 119 2.2. Symmetry relations for the borospherical transform 121 2.3. Inversion formula for the integral transform 14> related to the line 121 complex K in C3 2.4. Inversion formula for the horospherical transform 123 2.5. Inversion formula for the horospherical transform on the hyperbolic space £,3 124 2.6. Integral transform related to paraboloids on G 125 3. Harmonic analysis on the group G = SL(2, C) 128 3.1. Laplace-Beltrami operator on the group G 128 129 3.2. Horospherical functions on G 3.3. Fourier transform on G 131 3.4. Relation between the Fourier transform on G and the horospherical transform

132

3.5. Symmetry relation for the Fourier transform 133 3.6. Inversion formula for the Frulrjer transform 13.1 3.7. Analog of the Plancbere1 formula 135 3.8. Relation between the Fourier transform on G and the representations of the grOUP G x G 136 3.9. Relation to the representations of the group G 137 4. Another version of the Fourier transform on G = SL(2, Cl 139 4.1. FUnctions \IIy(g;{,() 140 4.2. Fourier transform on G 141 4.3. Relation between the above two versions of the FOllrier transform

141

4.4. Symmetry relation 142 45 Inversion formllla and plancherel formula for the FOllrier transform 1"142 4.6. Relations with representation theory 143 Chapter 5. Integral Geometry on Quadrics 1. Integral transform related to the hyperplane sections of a hyperboloid of two sheets in Rn+! 1 1 Definition

145 145 145

1.2. Admissible submanifolds in the manifold of hyperplane sections of Iq

~

1.3. Operator ICE 1.4. Local and nonloca1 operators

IC

148 151

CONTENTS

2.

~

1.5. Invmion formula 1.6. Examples

J 52 153

Integral transform related to spheres in Euclidean space E"

158

2.1. Definition 2.2. Operator It;r 2.3. Inversion formula

2.4. Examples

157 157

159 159

Bibliography

165

Index

167

Preface to the English Edition This is the English translation of a book published in Russia in 2000. The hook i.~ " realization of our old plan to write a small book explaining the main ideas of integral geometry in the context of several simple examples, and it follows our point of view that even now certain specific examples play a bigger role in integral geometry than general results. For these simple examples we selected the classical Radon tJ"8llSform, itB generalization suggested by F. John, hyperbolic ,,-ersions of the Radon tr8llSt'orm, and tilt> horospberical transform for the group SL(2, C). In discussing the Radon transfono, whkh, of course, is treated in other books. we emphasize sewral circumstances, which are usually not considered. Ooe example is t,be projective Invariance of t.he Radon transfono. This allowed us to regard the afIine Radon transform, the Minkowski-Funk transforDl, and the geodesic hyperbolic Radon t.rans&mn asdUl'ercnt realizations of the projective Radon transfOrm. We also considered it important t.o illustrat(l. by simple examples, the central role played in brtegral geometry by the operator K., which is responsible for the universality of explicit inversion formulas. The English edition of the book contains some modificatioos and corrections. In particular. \'Ire added Chapter 5 devoted to integral geometry 011 quadrlcs, or, in other words, to a conformally invariant version of the Radon transform. This approach allows us to combine the previously mentioned version of the Radon transform with the hyperbolic horospberical transform. We are grateful to AIik Shtem. for his excellent translation and to Sergei Gelfand for a number of useInl COJDlDPRts.

April2003

The authors

xl

Preface Integral geometry studies mainly integral transforms assigning to a. fWlCtion on a manifold X the integrals of this function over submanifolds that form a. family M. It is BBSumed that the £amily M itself is endowed with the structure of a manifold. This establishes a correspondence between functions on the manifold X and functions on some manifold M of submaoifolds of X. For instance, to functions on EuclideaD space En ODe can assign the integrals of these fWlctions over all possible lines: this rule defines all integral transform sending functions on E" to functions on the manifold of lines. Along with the integration of functions on X over 8ubmanifolds. integral geometry considt>fS similar integral transforms of other analytic objects on X (densities, differential forms, sections of bundles, etc.). The main problems are in the description of tlx> images and kernels of these transforms and in the construction of explicit inversion formulas recm-ering the original objects from their images. The first book devoted to t.his area of mathematics was the monograph by I. M. Gelfand, M.I. Graev, and N. Va. Vilenkin (8). Integral geometry interacts with the classical direction in geometry originating from PlUcker. Klein, and Lie; the cornerstone of this direction is dualities between pairs of manifolds s1lch that points of one manifold are realized as submanifolds of the other. Manifolds whose points have such geometrical nature, carry specific structures used. in integral geometry. Transforms in integral geometry are studied in the language of these geometric structures. The typical examples of such manifolds are the four-dimensional variety of all lines in the three-dimensional projective space (the Pliicker-Klein quadric) and, more generally. the Grassmann manifold of k-dimcnsional planes in the n-dimensional projective space, the Lie manifold of spheres, and 80 on. All these manifolds play a significant role in integral geometry. The best known example of an integral geometry transform is the Radon. transform which appeared in 1917 in Radon's paper (311 staying aloof &om his other mathematical heritage. Radon considered the operator of integration over hyper. planes in Euclidean space and gave the exposition a surprisingly perfect form, which, uncommon at that timl.", rombined analytic and geometric considerations aud anticipated possible analogs of this transform in other homogeneous spaces. The attention was focused on the inversion formula reconstructing a function from its integrals over the hyperplanes. It is remarkable that the resulting formula is absolutely explicit and has principally different form depending on whether the dimension of the space is even or odd. In the odd-dimensional case the formula is local; namely, some differential operator is averaged along a family of parallel hyperplanes (and to reconstruct the function at a point, it suffices to know the integrals of this function only along the byperplanes close to this point). In the e\-en-dimensional case the inVl'rsion formula is nonloca1; namely, some integral operator is averaged xiii

xlv

PREFACE

along a family of parallel hyperplanes (in contrast to the differential operator arising in the odd-dimensional case), and we need integrals over distant planes. This corresponds to the fact that the Huygens principle holds for wave propagation in odd-dimensional spaces and fails in even-dimensiooal ones. Already before Radon, Minkowski and FUnk considered an analog of this transform for the spheres in which an even function on the sphere is reconstructed from the integrals of this function over the great circles. This reconstruction can be carried out by using spherical polynomials. Radon knew about these results, but he seemingly did not know that these transforms are projectively equivalent (see Chapter 1). In 1938, F. John [28] considered a natural generalization of the Radon transform in which a function on three-dimensional space is integrated over all possible lines rather than over planes. A new exceptionally important feature of this construction is that the family of lines depends on four parameters. Thus, the John transform acts from functions of three variables to functions of four variables, and hence it is natural to expect that functions in the image satisfy an additional condition. The main observation of John was that these functions satisfy the ultrahyperbolic differential equation which completely describes the image. This observation, which was of great importance for the future of integral geometry. not only put overdetermined problems into circulation but also determined the relations between integral geometry and differential equations. Starting from the 19405, one of the central problems in mathematics was to develop an analog of the Fourier integral for noncommutative Lie groups. Among the first groups under investigation (mainly because of physical applications) was the Lorentz group 5L(2, C), i.e., the group of second-order unimodular complex matrices. For this group, I. M. Gelfand and M. A. Naimark succeeded in constructing a theory in which the role of exponential functions was played by irreducible infinite-dimensiooal unitary representations of the Lorentz group. and, at the same time, the main features of the classical Fourier integral were preserved. Obtaining analogs of the inversion formula and the Plancherel formula for the Fourier integral was the culmination of this theory. Later on it became clear that the main point in the proof of these formulas was in the reconstruction of a function in C3 from the integrals of this function over the lines intersecting a hyperbola. Thus, a complex version of the John transform was considered; however, to recover the function, the integrals over the lines of the three-parameter family of lines intersecting the hyperbola were used rather than the integrals over all complex lines (forming a four-parameter family). This setting of the problem is natural because the reconstructed function depends on three variables. The analog of the Fourier transform on the Lorentz group is related to the above transform of integral geometry by an ordinary one-dimensional Fourier transform. This observation opened new possibilities for integral geometry. First, the following problem arose: To what extent the phenomenon noticed in the case of Lorentz group is general? It turned out that, for a broad class of homogeneous spaces including complex semisimple Lie groups and Riemannian symmetric spaces, there is a transform of integral geometry, namely, the horospherical transform, which is related to an analog of the Fourier transform for these homogeneous spaces by means of the ordinary Fourier transform. This connection is similar to

PREFACE

ICV

the relation between the Radon transform and the classical n-dimensional Fourier tmnsform carried out by the o~mensional Fourier tnwsform. On t.he other hand, the context of integral geometry io; much broadt'.r than that or representation theory. For instance, instead of the family or liDf.'S in CS intersecting a hyperbola one may collbider the family or liJ1eJ intersecting any other algebraic curve and 0btain a similar explicit inversion formula, although here there is no relation to groups or homogeneous spaces. This suggests the idea that perhaps the natural setting for multidimensional harmonic analysis contaius not only groups or homogeneous Space8 but also some more general geometric structures, in particular, "good" manifolds of submanifolds for wbidl there are explicit inversion formulas. It then seems natural to develop methods or geometric analysis meant for integral geometry and enabling one, U1 particular, to c:oostruct bannonie analysis on a broader class of homogeneous spaces induding symmetric spaces. This became the mainstream for the further development of integral geomet.ry. Much has already been done in this direction. but many things remain unclear. Tbere is a rather complete picture for the problem of "good" (admissible) families or complex conl'cs for which there is an explicit local im-ersion formula (see [4, 1, 22)). However, the results on Camilie8 or complex sub manifolds of dimension greater than one are far less complete (although the known result.s are sufficient to obtain the Plancberel formula for complex semisimple Lie groups by using the technique or integral geometry: see 114)). There are many problems that remain unsolved in the real casei this concerns especially the construction or nonlocal inversion formulas and the clarification of relation between the discrete series representations and integral geometry. Some results in this direction were obtained recently [17). On the other hand. interesting and deep connections or int~ geometry with multiduneJ1Sional complex analysis, symplectic geometry, and noulinear differential equations explicitly integrable by methods or the imwse problem were disco\wedj see [19, 16, 12, 13J. W(> also note that there are applied aspects of integral geometry, see [2, 18, SOJ. The in..-ersion of the Radon transform is the background of computer tomography. There are other physical problems (in astropbysics, geophysics, electron microscopy, etc.) in which the data can be interpreted as t-he Radon transform. The aim of this book is quite modest. We intend to show some ideas and 000structions of integral geometry by the example of the most elementary problems. We hope that the main part of the material is accessible to university students, who can see by these e.ulDpies how the interaction or elementary analysis and geometry leads to beautiful and import.ant result-s. The Jiw chapters of this book are devoted to five particular transforms. We begin ",.ith the Radon transform (Chapter 1). An important point here (which is often neglected) is that the Radon transform is of projective nature, and in the projecth,'e version, it includes not only the affine Radon transform but also the Minkowski ·Funk transform on the sphere. In Chapter 2 we study t.he John transform (the X-ray transform). It has already been noted that the inversion problem for this transform is overdetermined, and therefore it is of importance to lmd(,J'b"tlmd how to pose this problem and describe a class of ill'o'ersioll fonnulas for appropriate three-parametric families of lines. This problem is solved here in the complex case. III Chapter 3 wc present integral ~ ometry on t\\>'O-dimensional and three-dimensional hyperbolic spaces. The nonzero curvature or these spaces leads to a greater diversity or problems. In particular,

xvi

PREFACE

there are two versions of the Radon transform, namely, geodesic and horospherical. However, the inversion formulas are similar to those in the Euclidean case, and their proof is only slightly more complicated. In general, it is typical for integral geometry that explicit inversion formulas have some standard structure and only slightly change if the geometry becomes more complicated (contrary to the formulas of group representation theory). This phenomenon has grave reasoDS. which have mainly been understood, but their discussion is beyond the framework of our book. We also consider the analog of the Fourier transform in the hyperbolic space; the results for this transform are obtained by using the horospherical transform. In Chapter 4 we develop integral geometry on the group 5L(2, C); as we have already mentioned, this example stimulated the modem development of integral geometry. Finally, in Chapter 5 we present a conformally invariant version of the Radon transform (the projective Radon transform). We give three variants of this transform, which are convenient in different applications; they are associated with the family of hyperplane sections of the sphere in an+l, the family of hyperplane sectioDS of the n-dimensional hyperboloid of two sheets, and the family of all spheres in an, respectively. As in Chapter 2, we have an overdetermined problem of integral geometry and we give a universal inversion formula (the operator It). Among interesting consequences is the conformal equivalence of the Radon transform and the horospherical transform in the hyperbolic space. In view of the elementary nature of the book, we do not present a detailed bibliography on integral geometry and restrict ourselves to references to some publications mentioned in our exposition. The authors are deeply indebted to M. M. Graev (Jr.) for his help in preparing the manuscript and useful comments.

CHAPTER 1

Radon Transform 1. Radon transform on the plane

1.1. Radon transform on the Euclidean plane. By the Radon tTrm$/orm 011 the EllClideau plane we mean the integral transform sending a function I on the plane to the integraL~ of thi.<J function over all possiblp lines (with respect to the Euclideau length on these lines). Let us repm;ent. tbe Radon trantd"onn by an explicit formula. We defuae the Iines on the plane by equations of the form Xl

c:DSr.p + X2 sin:p - p

=0

or by the parametric equations (in which p is fixed and t is a parameter)

= -t sin If + P cos:p, X2 = t cos I{) + P sin :p;

Xl

Euclidean measure on such a line is equal to dt. We thus introduced the parameters (:p, p) on the manifold of lines, and it is obvious that a line corresponds to two pairs (:p,p) nnd (.p/.y) of parameters if and only if 'P' I{) + 1fk and p' (_l)lrp for some k E Z. This l:NU'smetrization shows that the manifold of lines on the plane is obtained from the infinite circular cylinder with cylindrical coorc:UnateJ (IP,p) by identifying the points (:p,p) and (if' + IT, -plo The Radon transform of a function I(Xl. X2) is given by the following formula: +X> (Ll) 'R/(if'.I') = -00 I(-t sinif'+P COt; If. t cuscp + IJ8in ip) dt.

=

=

f

The integral dt'pCndli on the parameters if' and p; however. it follows &om the fonnula 'R/(I{) + 11', -p) = 'R/(rp,p) that the function 'R.I can be pushed down to the manirold of lines. 'fht> Radon transfonn can be considered OD various function spaces. For sim· plicity. \\'0 first treat the t.ransfoml on the space D(JR2) of all compactly supported infinitely differentiable functions. [n this case, the Radon transfonn 'R I is a com· pactly ~upported infinitely differentiable ftmction on the manifold of lines, i.e., an infinitely differentiable function uf rp and p compactly supported with respect to p. In what follows we also consider this transform on the Schwartz space S(1R2) of all infinitely differentiable functiolL'" on JR2 decrt'8lling as r ...... 00, together with all their derl\'''3tives, more rapidly than an arbitrary negative power of r. REMARK.

Equation (1.1) is frequently represeDtvd in the Corn.

'R/(r.;,p)

=f Ja

!(xt.x2)6(xt cosr.p + X2 sin", - p)dz1 ti:e2 l

= (6(.rl

cos:p +:£2 sin

1. RADON TRANSFORM

2

where 6(·) is the Dirac delta function of one variable. Here one must view the expression 6(Xl CQ8cp + X2 sincp - p) as a distribution (a "generalized function") of two variables Xl and X2. To this end, we pass to the new coordinates u. v on the plane, where u = Xl CQ8cp + X2 sincp - p, and apply the distribution 6(u) for any fixed v. One can immediately see that the result does not depend 011 the choice of the coordinate v. 1.2. Inversion formula. Consider the following problem: Reconstruct the function I E V(R2) from its Radon transform 'RI. The inversion formula was first obtained by Radon (see the Preface), and in fact we reproduce his argument. It uses the observation that the Radon transform commutes with the motions of the Euclidean plane. This implies, first, that it suffices to find a way to reconstruct the function I at a single point, say, 0 = (0,0), and second, we can restrict ourselves to radially symmetric functions. We proceed with the calculations. Let the function I(xl. X2) = F(r) depend only on the distance r = (X1 2 + X22)1/2 from the point 0 = (0.0). One must reconstruct the value 1(0,0) = F(O) from the function 'R 1("" p). Since the Radon transform commutes with rotations, it follows that 'R I depends only on p. 'RI(cp, p) = F(P). The problem finally reduces to the inversion problem for some integral transform F - F for functions of one variable. It immediately follows from (1.1) that F(P)

=

1

+00

F( ../t2 + p'l) dt,

-00

i.e.•

(1.2)

F(P)

= 21."" Ipl

F(r)

../r2 -

rdr

p'l

=

1"" ../t p2

F(v'i) dt. p2

Thus, the passage (transformation) from F to F reduces to the Abel transform (integration of order 1/2). It is known (and can readily be verified directly) that formula (1.2) implies (1.3)

F(r)

11

= --11"

00

r

F'(P) ~ dp. vp'l - r2

In particular, (1.4)

F(O)

= --110&: -F'(P) dp. 11" 0

P

Note that F(P) is an even function. Therefore, since F(P) is smooth, it follows that the integrand is a regular function, and hence the integral exists. We pass from (1.4) to the inversion formula for an arbitrary function f. Averaging the function lover a circle centered at the point 0 = (0,0) we obtain a function depending only on the distance r from the point 0,

F(r)

1

f'h I(r coscp, r sin cp) tLp.

= 211" 10

1. RADON TRANSFORM ON THE PLANE

3

Note that F(O) = f(o.O). Let F be the Radon transform of the function F. Since the Radon transform commutes with rotations, it follows that 1

F(P)

{21r

= 21r 10

R./(..", p)d..",

i.e., F is the mean value of the function R. / from the point O. From (1.4) we obtain

Ovel'

the set of all lines equidistant

= _! [

F'(p) tip. P Since the Radon transform commutes with the translations, we thus obtain the following assertion.

f(O,O)

rr 0

THF-OREM Ll. If 'R f is the Radon transform the following inversion formula 1w1d8:

0/ a function

/ E 1>(R2), then

(1.5) where

-

(1.6)

F(Xl' X2,

p)

1 {2ft

= 2;r Jo

Rf(..", P+Xl

COSCP+X2

sin..,,)dlp,

i.e., F is the mf'.an value of the /unction 'RI over the lines equidistant point x = (Xl. X2).

from

the

Using integration by parts. we can represent (1.5) as follows: (1.7)

F is an even infinitely differentiable function of p. To in\'Crt the Radon transform, we have used (1.3) for r = 0 only. For r > 0, this formula contains an additional information concerning the relation between the averages of the function I and the Radon transform 'Rf of f. Namely, Recall that

THEOREM

1.2. The follmumg rel4tion hold8 (the Asgeirsson relation):

(1.8) where F (Xl.

'IS

tile m.ean value of the function

J over the circles centemd at the

point

X2), i.e.,

(1.9)

1 {2w

F(xJ,

X2;

r)

= 2,.. 10

I(x)

+r

cos..",

X2

+r

sin..,,) dlp,

and F IS the mean value of the junction 'R / over the lines equidistant from the point (XloX2) (see (1.6». We also have

(LlO)

1. RADON TRANSFORM

Formulas (1.10) and (1.8) can be obtained from (1.2) and (1.3), respectively, by using the corresponding translation.

For any x = (XIt X2) and a > O. the condition that F(x. r) lor r > a is equivalent to the condition that F(x,p) = 0 lor Ipl > a. COROLLARY.

=0

1.3. Remarks. 10. In the inversion formula one can substantially weaken the assumption that the function I is infinitely differentiable. It is clearly sufficient that the function 'R. I be twice differentiable with respect to p and, to this end. it is sufficient that the function I be twice differentiable. Moreover. it suffices to assume that the function I belongs to the space Cl of differentiable functions with continuous first partial derivatives because the Radon transform of radially symmetric functions is smoothing at the origin, namely, if FECI, then F'(P) = O(P) as p -+ O. This follows from the relation F'(P) =

2Pl:X) p

F'(r)

Jr2 -p2

dr =

pl:X) F'( ~) ds. 0

.;sJs+p2

The interest to the Radon transform has grown in connection with tomography problems; see [30,2, 18]). To study these problems, one must consider the Radon transform not only for smooth functions but also for piecewi.'!(> smooth functions whose discontinuity lines are piecewise smooth curves (for instance, the Radon transform of the characteristic function of an open set). The inversion formula works well at the points of smoothness of such functions because the Radon tran.... form is a smooth function in a neighborhood of the lines passing through the points of smoothness. However, the inversion formula is poorly adjusted for recovering the function in a neighborhood of a point of nonsmoothness. Special regularizing procedures are used to overcome these difficultit>S. The development of computational methods based on the inversion fommla requires getting over some additional difficulties. 2°. For the validity of the inversion formula, we do not need to assume that the function I is compactly supported. It is enough to assume that I is rapidly decreasing, and even the weaker condition I = 0«1 + x¥ + x~)-l) i.o; sufficient. In particular. the inversion formula holds for the functions I in the Schwartz space S(R2). Recall that S(R2) is the space of infinitely differentiable functions on R2 decreasing as r -+ 00, together with all derivatives, more rapidly than any power ofr. 3°. For any function in Cl, one can change the order of averaging 'R. I = F(XI.X2:p) and the integration over P in the inversion formula and represent the inversion formula in the following form:

HerE.' the integral with respect to p must be taken in the principal value sense.

1.4. Radon transform OD the aftlne plane. In fact, the Euclidean structure in not needed to define the Radon transform on a plane, and the affine structure is quite sufficient. Let us define the lines by the equations ~l Xl

+ ~2 x2 - P = 0

1 RADON TRANSFORM ON THE PLANE

=

and endow every line ~l Xl +{2 X2 - P = 0 wit.b the measure dpt sucb that. hI dX2 d(~l XI + ~2.l'2 - p) dp(. In the coordinates Xl. X2 this IDe08ure can be written as follows:

dx]

U2

dlt( 1{21 I~d' Wc dt!fine the Rndon translonu of a (unction I on the affine plane by the relation

=

(1.12)

"R/(~1.{2;P)= f J(l

=

I(X lt X 2)dl'(=j+oo 1(.l'l,P-~IX])dl:ll' 2'1 +(2 2'2="

2

-0(;

..

The function R. I tbU8 obtained satisfies the condition

"R/(>'(.]. >'(.2: >.p)

(1.13)

= 1>'1- 1 "RI(elo~;p).

=

Thus, this (wlction d(!pclld'4 not only on the line {I Xl + ~ %2 - P 0 but also on the choice of the 1>lIl'8mcters ~l. ~2. p. III t.be case of (~I.e2) = (cosY'. sinY'). the fWlction 'RI coincides with the ftmetion 'R./(t;.p) introduced in the EucUdean case. Oue frequently writES the expression for 'R. I in the fonn "R I({I.

(.2;

p)

= Jfa

where the definition of 6({. Xl

+ 6 X2 - I»

+ ~ X2 - p) dx l dx2•

is similar to that given in 1.1.

Tile inversion fomnda Jor the Radon transJo"fn (1.12) ;s

TUEOREM 1.3.

(1.14)

l(xI. X2) 6(el Xl 2

I(Xl •.r2) 12 = --4 7r

111-""('R.f)~(el'{2.p+el.l'.+{2X2)P-ldl})({1t1!.2-{2d(d. r

(

_<Xl

wllere r is Ofl arintmry contour in JR2 \ 0 intersecting once almost every my iSS'Uing from the point O. The validity of this formula follows from t\\'U simple facts. 1) The di.fft"rcntial fono in the integrand can be pushed down to the manifold S of rays issuing from tht' point O. i.e.. to the base space of the bundle R2 \ 0 - S. This follow~ from the fact that this form is homogeneous with mJpect to { of degree zero. and the form wnishcs on the vectors tangent. to thfo libels of tht' bundle ((~l' ~2) = ({t· ~ll) d>.). 2) If for r we take the twit. (ircle oontered at the point 0, then (1.14) coincides with fonnula (1.11) proven above.

1.S. Relation to the Fourier transform and another proof of the inversion formula. Let F(el.{2) be the Fonrier transfoml oftbe fuoction I(Xl,X2), Lt'.. (1.15)

F({I.{2)

= (2",)-1 f

JR3

I(ZI,Z2)C·«(lzl+E;a"'2)dxldx2.

To establish the relation betwet.'Jl the Fonrit'I' trnnsform F(el. {2) and the naclon trall.'4fonn R./({1.{2;P) of the fUllction/. wc reprcm.>nt till! intt-gral (1.15) for ({1.{2) :f:. 0 08 an iterated integral. where the integration it; t.aken lUst over the lines {I Xl + {2 Z2 = P and then over p.

F({I'{2)=(27r)-·j+~(l. -OQ

tl

%1

-rEl Z2=P

J(Zloz2)d~)eiJldp.

1. RADON TRANSFORM

6

Thus. we have THEOREM 1.4. The Radon transform 'R. [ and the Fourier transform F o[ a function [ are related as follows:

F(~I'~2) =

(1.16)

00

(271")-11: 'R.[({I.{2;P) e ip dp

«{1'{2)::/: 0).

Since tbe function 'R. [ is homogeneous. one can represent (1.16) in the fonu +00 (1.17) F(A{I. A~2) = (271")-1 -00 'R.[({I'{2;p)ei~Pdp.

1

Thus, the two-dimensional Fourier transform is the composition of the Radon tr~ form on th(' plane and the one-dimensional Fourier transfonn. Let us now present another proof of the inversion formula for the Radon tran.'iform. This approach is based on the inversion formula

= (271")-1 f F f(~.. ~2)e-i(~IZI+~2Z2) ~I ~2 JR = (471")-11+ 2 F [(A COS". A sin")e-'~(zlCU9,,,+z2sinop!) IAI d", dA

[(x .. X2)

2

00

1

11'

-00

0

for the Fourier transform. Substitute the expression in (1.17) (in which th(' function = 'R.[(",:p» into this inversion formula. Wc obtain

F [(A cos",. A sin",) is represented in terms of 'R.[(cos..p. sin,,: p)

+001+01) -, 1 .~ =--4' 1 Fp (Xl,x2,p)e' PsgnAdpdA 71" I -00 -00 1 +00 = --2 ~(XI. X2. p)p -I dp. 71" -oc

1

Here the function F is given by (1.6). At the last step we used the following fact: the Fourier transform of the distribution sgnA e S'(R) is the distribution 2ip-1 (see 110», i.e., if'" e S(R) and .p(A) = r~: "'(p)ei~p dp. then

1::x. ~(A)

sgnAdA = 2i

1:

00

tJ'(p) p-I dp.

In general. the integral on the right-hand side is understood in th(' principal value sense. However. if '" is a smooth odd function, then this integral converges in the ordinary sense. We finally obtain the inversion formula (1.5). REMARK. Note that the Radon transform sends the Laplace operator

~

=

-£ + fir on the functions [(XI. X2) to the operator /i;s on the functions 'R.[(",:p).

More generally. ifthere is a distribution K (r 2 ). where r2 = xi +x~. then the function K(~) K( -r2) * [ becomes the convolution K( -PZ) * 'R. [ (with respect to the

t:l

2. RADON TRANSFORM IS THREE-DIMENSIONAL SPACE

7

variable p). These reasonings can Ix> used to invert the Radon transform of a nonsmooth function. If I is a compactly supported distribution (in particular, a nonsmooth function). then we can represent I in thf:' form I = {I - A)k 'P, where 'P is a smooth function and k is a positive integer. The Radon t.ransfonn of the function I can be defined 88 the convolution K (1 + y)" .. 'R..p. This definition agrees with that given aboVE' for compactly supported continuous functions. If I is a nonsmooth function, then \W take the convolution of 'R I with (I + y) -k and obtain a smooth function 'R rp. Applying the inversion formula. for the Radon transform to the function 'R 'P, we find the function 'P. Finally, applying the differential operator (1- A)1r to 'P, we obtain the desired function I.

2. Radon transform in three-dimensional space 2.1. Radon transform in Euclldean space. By the Radon transform in three-dimensional Euclidean space W(> mean the integral transform sending a function on the space to the integrals of this ftmction over all possible planes (with respect to the Euclidean area E'lement on each of these planes). All considerations bere are carried out along the same schCDle as for the Radon transform on the plane. Let us deBnt> the Radon transform by Bn explicit formula. We present every plane by an equation (w, x) - p = O. «w, x) =

WI Xl

where

Iwl = 1

+ W:2 X2 + ""'3 X3), or by parametric equations r = tl 0 1 + t20:2 + pw,

where 0: 1 , 0/2 is a pair of mutuaUy ortbogonal unit vectors, i.e., (0: 1 , 02) = (w, Q I ) = (w, Q2) = O. Euclidean measure on such a plane is equal to dtl dt2. Thus, every plane is defined by a pair (w,p), where vJ ill a point of the unit sphere and p is a number, and it is clear that t.he planes r.orresponding to different pairs (w,p) and (w'.P') coincide if and only if (w',P') = -(..."p). The Radon transform of a function lex) i~ giw.n by the relation (2.1)

'R. I(w,p) =

I I(tl a 1 + t2 0:2 + pw) tit1 dt2, 1",2

where 0 1 • 0 2 are arbitrary unit vectors such that (0/ 1 , 02 ) = (~', ( 1 ) = (101, a2) = 0 (and the integral does not depend on the choice of these vectors). Since 'R. I( -101, -p) ='R/(..."p). it foUoM! that the function 'RI can be pushed down to the manifold

or planes.

As in the two-dimensional case, one frequently rt"presents (2.1) in the form 'R/(w,p)

=

s la11(:z:)6«w,x)-p)dx

where c5(.) is a function on R. For simplicity, we ronsider the Radon transform on the functions I E'D(R3). In this case, the functions 'R I are rompactly supported and infinitely diflere.ntiable on the manifold of planes, i.e., these functions are infinitely differentiable when regarded 88 function.'i of w and p and are compactly supported with respect to p.

l. RADON TRANSFORM

8

Let US find the inversion formula for the Radon transform on three-dimensional Euclidean space. We first consider a function I(x) = F(r) depending on r = (X1 2 + X2 2 + X3 2)1/2 only. The Radon transform of I does not depend on w. i.e., 'R/(w,p) = F(P). According to (2.1) we have (2.2)

F(P)

=f

1R2

F(JtI2

+ t22 + p2)dtl dt2 = 271'1.x F(r) rdr. 1,,1

Thus, in contrast to the two-dimensional case where we obtained the integral of order 1/2, here we simply obtain the integration operator. Therefore. the inverse is the differentiation operator, and hence F(r)

(2.3)

= _..!.. F'(r) . 271'

r

In particular,

(2.4)

1(0)

= F(O) = - 2~ F"(O).

To obtain the inversion formula for an arbitrary function I. we use the fact that. as in the two-dimensional case, the Radon transform commutes with the motions of Euclidean space. Introduce the mean value of the function lover the spheres centered at a fixed point x, (2.5)

F(x; r)

=..!.. f I(x + rw) dw. 471' 11",1=1

where dw is the surface area element of the unit sphere. Further. let F(.r; p) be the mean value of the function 'RI over the planes equidistant from the point x, i.e., (2.6)

F(x;p)

= 41 f 'RI(w. p+ (w. r.»dw. 71' 11",1=1

Then it fonows from (2.4) and from the fact that the Radon transform commutes with the motions of Euclidean space that

1 :;" I(x) = F(x.O) = - 271' r" (w:O). Thus, we have proved the following assertion. THEOREM 2.1. The inversion lormula lor the Radon tronslonn on three-dimensional Euclidean space has the lollowing lorm:

(2.7)

I(X)=-SI 2 f ('Rf);(w;(w,r.»dw. 71' 11",1=1

We see that the three-dimensional case substantially differs from the twodimensional one. In contrast to the two-dimensional case, here the inversion formula is local. In other words, to reconstruct a function I at a point x. one must know only the integrals of I over the planes close to x (in fact, over the planes in an infinitesimal neighborhood of the set of all planes passing through x). In passing, we have obtained the following relations between the averages F

2. RADON TRANSFORM IN THREE-DlMKNSIONAL SPACE

P of the fuu('tions f

and

aud

'RI.

respectively «2.8) is the S()-called

9

Asgeirsson

relation for three-dimensional spare): F(x;p)

(2.8)

= 2r. I~Q F(x, r) rdr, ;pl

F(r) ;::

(2.9)

_..!.. F;(z: r) . 2r.

r

(Tbese equations are obtaiood frolD (2.2) and (2.3) by traMlation.~) The Radon transfonn of a function I is related to its f'(mricr trallsform:F f by the following formula:

:F I(Aw)

(2.10)

= (211')-3/2 /

+00 -<>C

(lwl = 1).

'R/(w, p)ei).Pdp

Starting from this equation and using the inversion formula for the Fourier trailSform. one cau obtain another proof of the inversion formula for the Radon transtonn, namely, I(x);:: (2r.)-3/2

= ~ (21r)-3 / 2

f :F/(~)e-i{(.~} d{

Ja

3

21.1... /+CIO :F I(Aw)e-'>'(""~) A2 dw dA /+00 f /+00 'R I(w. p) c..\ z» A2 tip dw dA 1=1

-oc;

;:: ! (211') -3/2 2

= 4\ 1r

(p- (....

-oc;

Jf..JI=1

-<>0:

/1'00 j+OCl F(x, p) et>.p A2 dp dA = -:!. F;'(x, 0). -0<)

-O
-

Comparing this proof with similar arguments for the plane (ill 1.5), we note that the difference bet\\'Een the inversion formul8ti for the plane mad for the spore is caused by the distinction bctwoon the expressions for volume elements in the polar coordinat.es, namely, IAI d>' dw for the plane and A2 d>' dw for the space. REMARK. In the l.wo-dimerunOlUll case, the original functioD I is recoust.ructed from 'RI by a uonlocal formula. H~-er, the function ..rA/. where ~ stands for the Laplace operator, is reconstructed from 'R I by a local inversion formula:

../is. is an t>lliptic pseudodifl'erential operator, the singularities of the fmIC../is. I and I coincide. This fact is sometinle9 tlsed in tomography to n."CODstruct the singu1arities of the fWlction I from the kJ1oy.·n Radon transform 11. f. SiJl(~

tioll8

2.2. Radon transform iD the affine space. To define the Radon transform in three-dimensional 8p8(.'C, it suffiCCtl, as iD the two-dimen.'lional C88t", to have an affine rather than Euclidtlall structure. Let us define planes by equations of the form {IXI +~2X2 +~3Z3 -

and equip every plane ~1 Zl dXI d.r2

- P = 0 by a measure d#.~ sudl tbat p) dp~. For the coordinates on t.he plane one can

+ ~ X2 + e3 Z3

= d({l Xl + {2 X2 + {J Xa -

P= 0

I. RADON TRANSFORM

10

take any pair written as

Xi, X j,

i, j

= 1. 2, 3. and the measure dp.( in these coordinates can be (k"" i.j).

i.e., it depends on the equation of the plane. We define the Radon transform of a function relation

I

OD

the affine space by the

(2.11)

'RI(>'{; >.p)

= 1>'1- 1 'RI(e:p)·

e,2',

Thus. the function 'R I depends not only on the plane el XI + + e3 X3 - P = 0 but also on the choice of the parameters {I. e2, {3. p. For lel = 1, the function 'RI coincides with the function 'RI(w.p) introduced earlier in the Euclidean case. The expression for 'R I is frequently represented in the form

= JR3 ( l(xI. x"X3) 6(el.EI + e,x, + {3 X3 -

'RI({I, e"e3; p)

where the definition of 6({1 XI dimensional case. THEOREM 2.2.

+ {, X2 + e3 X3

-

p) dx l dx, dx3,

p) is similar to that in the

The inversion formula for the affine Radon transform

t~

(2.11) is

(2.12)

where

wee) = el d{, " ~ + e, d{3 "d{1 + e3 d{1 d{,. and r is an arbitmry surface in R3 \ 0 whose intersection with almost every my starting from the origin 0 consists 01 one point. As in the t'flo'lHlimensional case, the inversion formula (2.12) follows from two simple facts: first. the integral (2.12) does not depend on the choice of the surface r, and second. if r is the unit sphere, then (2.12) coincides with the inversion formula (2.7) for the Euclidean space. 2.3. Radon transform for a space of arbitrary dimension. The definition of Radon transform for R' and R3 can be extended to spaces of arbitrary dimension. Namely. the Radon transform of a function / e S(Rn) is defined by the formula (2.13)

'R I(e. p)

=

1 R"

I(x) 6( (e, x) - p) dx,

where (e. x) = EeiXi. dx = dx l one variable. The definition of 6( and n = 3.

··

ee(Rn),\O.

peR.

·dxn , and 6(P) stands for the delta function of p) is similar to that given above for Tt = 2

(e. x) -

2. RADON TRANSFORM IN THREE-DIMENSIONAL SPACE

11

We present without proof the inversion formulas for the Radon transform. H

!.p

= 'R. I, then

(2.14)

=

for odd ri, where (7«() ~""l(-I);-l{.Aj;ei~j and"Y C R" is an arbitrary surface intersecting once every ray in Rn issuing from the origin 0;

(2.15)

f(x)

=

(-1)" /2 1!~ a,,-l
(2r.) 11

,

88"-1

-00

L.=(~.Z)+p p

_ 1 dpAO"(e)

for even n, where the integral over p is understood in thl" principal value sense. Thus. similarly to the CaseH n 2 and n 3. the iUVlmIion formula is local for odd nand nonlO<'.a1 for even n. The inversion fonnulas (2.14) and (2.15) can be combined by using the distribution (p - iO)-I. By definition, this distribution L'i the limit as IJ -+ +0 of the distribution u .. (P) (p - is)-l. It is known [10] that the function (p - iO)-1 ('an be represented as a linear combination of the distributions p-l and 6(P),

=

=

=

(p - ;0)-1 = p-I

(2.16)

+ i1i6(p).

For any n we haw the following inversion formula: (2.17)

I(x)

= (2~r i1:ooan-;;'~~,p)(P-({'X)-iO)-ldPA(7({).

Indeed. it follow8 from (2.16) that relation (2.17) coinddes with (2.14) for odd n and with (2.15) for even n. Using the integration by part.s (n - 1) times \'Iritb respect to p, we can rewrite this fomlula as follows: (2.18)

I(x)

= (n -

I)!

. "11+"" .,

(2~)

-00

ip({.p}(p -

(~.x) -

iO)-n dp A (7(0·

We can represent the im-ersion formula differently by passing from the Radon transfonn 'R. to the "complex" Radon transfonn -R.. Namely. dcflne the operator -R. by the foUowing fonnula: (2.19)

R/({. p)

=(

la. I(x)( ({. x) -

One can readily see that the fllnction.~ Rf({,p)

=

1:

00

p - iO)-1 dx.

RI aDd 'R. I are related as follows:

'R/«(. 8) (8 -

P - iO)-l d8.

This, together with (2.17). implies the inversion formula for the "complex" Radon transfonn -R. : if tP RI. then

=

(2.20)

I(z)

= (.!...)" 21i

(

an-I .p(W.8)

88n -

1,,,,1,,,,1

Note that this formula is local for any

fl.

1

I

,,={.... .z:)

dt1.

12

I. RADON TRANSFORM

3. Wave equation and the Huygens principle Here we shall show how one can obtain the classical formulas for the solutions of the wave equation by using the Radon transform. The proof is based on the method of plane waves.

3.1. Two-dimensional case. For the two-dimensional wave l'quation (flu

(flu

(flu

= -OXl -2+ OX2 - -2 ' (Jt2 consider the following Cauchy problem: (3.1)

u(O. x)

= O.

u~(O. x)

= I(x).

and assume for simplicity that I E 'D(1ll2 ). The method of plane waves is based on the following fact: if rp(s) is an arbitrary smooth function of one variable. then the function ..... (t. x) = ;pet + (w. x). where Iwl = 1. is a solution of the wave equation for any w (this fact can be verified directly). The solutions of this kind arc called the plane waves. The method of plane waves consists in representing an arbitrary solution of the wave equation as a superposition of plane waves. We proceed with constructing a solution ofthe Cauchy problem (3.1). Consider the following function: u(t. x)

(3.2)

= 4\ 1r

1+

00

-00

2 101 .. 'R/(;p. p - t + XI COS"" + X2 sin;p)p-I d."dp.

where 'R/('P, p) is the Radon transform of the function f. Since the function u is obtained as a superposition of plane waves. it follows that this fimction is a solution of the wave equation. Let us show that u is the solution of the Caucby problem (3.1). Indeed. for t = O. the integrand in (3.2) changes sign tmder tht.' change of variables ('P. p) ...... ('P + 1r, -p), and hence u(O.x) = O. Furthermore. we have u:(O. x)

= -4\ 1r

1+

00

-00

12"('Rf)~('P. P+Xl COSCP+X2 sinop)p-Id'Pdp 10

= I(x),

where the last equality follows from formula (1.11) for the Radon transform on the plane. We intend to return in (3.2) from the function 'R I to the original function I. To this end. we use relation (1.10) for the mean values of the functions I and 'R I.

102.. 'RI('P, P + XI cosry + X2 sin.,,} dry = 2/ 12.. l(xI + r cos"', X2 + r sin",) rd;pdr 1,,1 10 Jr 2 - p2 = I(x + y) dy (dy = dYI dY2). h112>r JlyI 2 -Tfl 00

21,

where

lyI 2 = Y1 2 + Y22. We obtain u(t. x) = 212 1r

1 K(t. y) I(x + y) dy. 1R2

;J WAVE EQt:ATION AND THE HUYGENS PRINCIPLE

13

where K (t. 11)

HIIII

=[

(lp

_r.=::;:;:='=;==~

Hrl P Vlyl2 - (p - t)2 (for t < 1111. the last integral must be tmderstood in the principal value sense). Elcmcutary manipulations show that K(t, Y) = 0 for t < 1711 and K(t, y) = . / 111 :I v t -111'1 for t > 1,,1. Finally. we see that

u(t, ,r) ;; ~

(3.3)

21f

1<' Jt

f(x

111 1

+ 1I) dy lyl2

2 _

(the Poisson fomltlla). Thus, the value of a solution of the "'8ve equat.ion at a point x and at a moment t depends only on the initial data in the disk of radios t c:eoten!d at x. This is the so-called effect of "finite domain of dependence" for the existence of the forward front of the wave; namely, the signal moves with uuit speed, and at time t it can come to x only from the points whose distance from ;r. is not grcatel'than t.

3.2. Three-dimensional case. Now we find the solution of the Caucby problem (cf. (3.1» for the three-dimensional wave equation. (3.4)

lJ2u n.2

UI.

lJ'lu

{flu

{jl u

= IT.''! ~ 2 + li"2" + 2' V.T2 IT'''3

u(O, x)

0-

= 0,

u~(O, ;r.)

= I(x).

(',onsid('r the following (miction:

(3.5)

ll(t.X)=-8: 2 11

f

)10..:1 .. 1

(Rf)~(w,t+{w.x)dw.

where! R/(w. p) ill the Radon tral1sfonn of the fuJl(,'tiou/(x) = I(x., X2, X3)' This function is a 8uperposition of plane waves, and hence it satisfies the wave equation. We show that it hi the solution of the Cauelly problem (3.4). hldeed, £Or t = 0, the integrand in (3.5) u. an odd function on the sphere ~'I = 1. Hence. u(O, x) = O. Furthermore,

u~(O, x) = -8\ f 11'

)I,wl>=
(RJ)~(,",,,

(w, x»dw

= I(x),

where the last equality follows from the inw.rsion fonnula for t.he Radon transform on three-dimensional 8p8(',c. We return in (3.S) from the function RI to the original function I. To this cnd. we use relation (2.9) bet\\'t!Cll tbe mC8l1 values of the functions I and RI:

1.

f(x+tw)dW=--21

WI~

f

1ft )'",1=1

1.... /=1

(Rf)~(w,t+(w,x)dw.

obtain

(3.6)

u(t, x) =

4~ f

" )1",,1=1

f(x

+ tw)dw

(the Kirchboff Connula). Wc HOO that thl! value of a solution uf the tbree-dimensional wave equation at a point :r and at a moment t depends only on the illitial data at the points whose distance from x is equal to t (the Huygens principle). Thus, here wc face not only the effect of finitt' domain of dependence but also a stronger effect, namely, the wave has a sharp hack front: at, the moment t the signal issued by any point placed

1. RADON TRANSFORM

14

at a distance less than t from the point x bad already passed through the point x at some previous moment and cannot be observed at this point. This distinction between the two-dimensional and the three-dimensional case corresponds to the fact that the inversion formula for the Radon transform is local in the three-dimensional case and nonlocal in the two-dimensional case.

4. Cavalieri's conditions and Paley-Wiener theorems for the Radon transform 4.1. Cavalleri's conditions for rapidly decreasing functions. Consider the Radon transform 'R/(~I' ~2; p) (in the affine version) on the functions I E S(R2); see (1.12). We are interested in the family of functions 'R I obtained in this way. Recall that the functions 'R I satisfy the homogeneity condition (of degree -1) 'RI(>'~, >.p) =

(4.1)

1>'1- 1 'R/(F..

p).

f\u1;her, these functions are infinitely differentiable on (R2 \ 0) x R and rapidly decreasing with respect to p 88 P - 00 together with all derivatives. and this decrease is uniform with respect to F.. IF.I = 1. The following question arises: Are there other conditions on the functions 'RI in the image of S(R2)'! First we note that

I

(4.2)

+

CXl

'R. I(F., p) dp =

-00

( JR

I(x) dx.

2

i.e.• the left-hand side does not depend on F.. This is closely related to the classical Cavalieri principle. which enables one to calculate the area of a plane figure in terms of the lengths of the intersections of this figure with a pencil of parallel lines of any given direction. Namely, if / is a characteristic function of a plane figure. then 'RI(~. p) is equal to the length of the intersection of this figure with the line ~IXl + ~2X2 - P = 0 for any I~I = 1. and thus relation (4.2) expresses the area of a plane figure in terms of the lengths of the intersections of this figure with a pencil of parallel lines. In particular, the left-hand side of this relation does not depend on the direction of the pencil. For this reason, we refer to both the condition that the integral r~= 'R I(F.. p) dp does not depend on F. and some generalizations of this condition as Cavalieri's conditions. Consider the moments of the function 'R/(F.. p) with respect to p. (4.3)

l,,(F.) =

i:oc, 'R/(~.

p) plc dp.

k

= 0.1.2, ....

Substituting the expression for 'RI in terms of the original function I(x) into (4.3). we obtain (4.4)

1

+00 'R I(F.. p) pt dp =

-00

(

~2

l(xHF.. x)1c dx.

Relation (4.4) implies the following property.

Cavalierl's conditions. The integralllc(F.) is a homogeneous polynomial 01 degree k in F.1o F.2 (k = 0.1.2 •... ).1 lTbeIIe CODditions (for affiDe apaces or arbitrary dlmensioD) were first formulated iD [8).

4. CAVALlER1'S CONDITIONS AND PALEY. WIENER THEOREMS

IS

We gi\'e an interpretation of the integrals J,.({) in the language of the Fburier tran.'lfonn F I of thc ftmction I. It was shown in § 1 that the ftmctions :FI(~) and 'R. I(f.. p) are related as £0110\\'8:

F I().f.)

= (2r.)-1

f ..

oo

_'.Xl

'RI(E.. p)ei~p dp

(f. :/: 0).

This implies tbe fonnula

1,,(0 = 2r.i-"

(4.5)

8*~~~).f.) I~=o'

Assume now that the function F I is rapidly decreasing and infinitely diJl'ercntiable at the point 0 Dl any direction, i.c.. the ftmction F I().£.) is in6nitely cWW.rentiablc with respect to ). at the point ). = 0 for any ~ :f. O. We show that Cavalierfs conditions are equivalent to the condition that the function :F I(() Is infinitely differentiable at the point f. = O. Indeed, let :F f(f.) be infinitely differentiable at the point f. = O. Then the right-hand sidt> of (4.5) is equal to the homogeneous component of degree k in the ThyJor expansioll of F f in a neighborhood of the point f. = 0; thus. relations (4.5) imply Cavalieri's conditions. Conversely, assume that the function F I is infinitely diff(!ccntiablc at tbe point 0 in every direction and that Cavalieri's conditions are satisfied. Then it follows from the TayJor forJbllla for F I(). f.) regarded as a function of'\ that

F I().f.)

=

t

l)k

~~~).f.) I _+ 0(1).1") = (2r.)-1 ~=O

"'=0

t i:~k lk(~) k=D

+ 0(1).1")

•

for any positilo"C integer n and Bny { with 1(1 = 1. Since Ik<el is a homogeneous polynomial of degree k in ~I. ~2' tlu.'l equation can be repn>seoted in the fonn n.

F f«()

(4.6)

= (2r.}-1 L 1t=O

.1:

k' ,l,,«() + oOel"), •

We finally obtain a Thylor formula of order k for the function F ICe) at tbe origin O. Hence, F f(f.) is infinitely differentiable with respect t~ ~lo Q at the point O.

4.2. Paley-Wiener theorem for the space S(1R2 ). The assertions desai~ ing th£. images of various runction spaces under the Radon transfonn are often called Paley- Wiener t/u>orems for this trnnsfonn. TUEOREM 4.1. A functum CP({1o (2; p) on (R2 \ 0) x R is the &don tramlonn of some functiOlI I e S(lR2) if and only if

1)

ip sati~fies

the homogenetl.y condition (4.1);

2) '" is infinitely differentaGble;

:l) every denlJatire of the function t.p decrwucs for raptdly than any negative power of Ipl; 4) t.p satisfies Cavalieri's con4itions. PROOF.

I{I

= 1 as p -

00

more

Wc must prm-c the sufficiency of these conditions only. Let a function

Ip?({, p) satisfy the conditions of the theorem. Define the function F(~) on R2 \ 0 by t.he foml1lla

1 F(f.) = -2 11"

f+oc "'({, p)e -00

ip

dp.

I. RADON TRANSFORM

16

It follows from the homogeneity condition 1) that this relation is equivalent to the formula F(~~)

1 = -2

j+x

'P(~. p)ei>'p dp.

-oc

1r

Conditions 2) and 3) imply that the function F is infinitely differentiable at any point ~ ::F 0 and infinitely differentiable in any direction at ~ = 0: hence. it is infinitely differentiable at the point ~ = 0 by Cavalieri's conditions. Further, by conditions 2) and 3). the function F rapidly decreases together with all its derivatives. Thus. F E S(R2). and hence there is a function I E S(R2) such that F = F I. Then "'(~. p) coincides with the Radon transform 'R/(~. p) of the function / because the Fourier transforms of these functions with respect to p are equal to the same function F /(~ e).

4.3. Paley-Wiener theorem for the space "D(R2) of compactly supported in8nitely differentiable functions. THEOREM 4.2. A function 'P(~I. {2: p) on (R2 \ 0) x R is the Radon trons/onn 0/4 function / E "D(R2) i/4nd only i/ 1) 'P satisfies the homogeneity condition (4.1); 2) 'P is infinitely differentiable on (R2 \ 0) x R and compactly supported with respect to p: 3) ." satisfies Cavalieri's conditions. PROOF. If / E "D(R2), then the Radon transform 'RI of I clearly satisfies conditions 1) and 2). Cavalieri's conditions follow from the relation 1)(R2) C S(R2). Colwersely. let a function '" satisfy the conditions of the theorem. Then." satisfies the conditions of Theorem 4.1 as well. and hence 'P is the Radon transform of some function I E S(R2). It remains to prove that the function I is compactly supported. Let us present the proof which is due to Helgason 126). By assumption . .p is a compactly supported function. i.e ..

(4.7)

'R/(9. p) = R,/(cos9. sin 9;

p) = 0

if

Ipl > a

for some a > O. We prove that (4.8)

F(x. r) ==

L

2W

'R/(XI

+r

0088,

X2

+r

sin8: p)d9

=0

if r>

Ixl + 4.

Indeed. let F(x, p) be the mean value of the function." on the set of lines placed at the distance Ipl from the point x. It follows from (4.7) that F(x, p) = 0 for Ipl > Ixl + 4. Then F(x. r) = 0 for r > Ixl + 4 by the Asgeirsson relation (§ 1.2). Next. let us prove that if the function I satisfies condition (4.8). then the functions x, I. i = 1.2. also sati.olfy this condition. Indeed. it follows from (4.8) that

1.

I(x + y) dy

11I1>r

and hence

1.

11I1>r

/~. (x + 7/) dy =

=0

for r>

0 for r>

Ixl + a.

Ixl + a.

i

= 1.2.

.&. CAVALIERrs CONDITIONS AND PALEV-WlENER THEOREMS

17

By the Stokes theorem. this implies that

1,

fz.(x+y)dy=r 1

1111>"

2~

l(x+rw)",·;d8.

0

where w = (cos9, sinS). Thus,

12ft I(x + rw)(.l + TW), dB == Xj 12"ff I(x + rw) dJJ + r 88

12ft I(x + rw)

W;

dfJ = 0 for r >

Ixl + a,

required.

It follows from the abc»'E' that the nmction P(x) I(x) has zero mean value 0\'Cl'

the- circle (ccntered at the point 0) of every radius r > a for any polynoolial P(x). Hence, I = 0 011 this circle, and thus I vanishes in the entire domain 1.1:1 > a. 4.4. Inversion of the Radon transform of a function I e V{R2) using the moments. Let I be a compactly supported function. Then the Fourier tra.n. form :F I of I is all entire nmctiou and, by (4.6), the Ta-ylor series of the function :F f is given by the fonnula (4.9)

F

I(~) =

<Xl

(21i) -

I

L

.k

~! ft..«().

10=0

where I,,«() is the kth moment of the fulK'tion 'R. /(E., p) "Titb respect to P. which is a poll''Ilomial of degree k. TakiIJg the inversE> Fourier transform of F /, we obtain the way t.o reconstruct t.he fullction I from the moments 1". Thus, for a function I E V(JR2), we obtain an invennon formula usilJg the luomcnts I" only. TIJ.is formula substantially differs from that in § 1. Sinet' the moments I" are bU1nogencous polynomials, in order to rec:onst.ruct t.he function f, it suffices t.o know the values of I,. on an arbitrarily small arc of the circle 1f.1 = 1 or even on an infinite sequence of points (in general position). REMARK. It is worth mentioning the difI'erence bet\\'een various iDversion fo.... mulas. In an inversion fonnula v.-e have an operator defined on some space (which is generally larger than the image of the Radon transform) and determined by some kernel (generally, a distribution kernt>l). The operators in different iuversion formulas coincide on the image of too Radon troDSfonn. It follows from the Paley-Wiencr theorem for tht> space S(]R2) that every functional vanishing on the image of S(R2) is obtained from Cavalieri's CU11ditioDS, and the inversion formula for S(R2) is essentially wtiquc. Howe\'CI', the image of 1)(22) consists of compactly supported functions. and the family of annihilating functionals is richer in this case. Therefore, for V(1R2 ), it is possible to constrt1('t inversion formulas different from that in § 1. A version of such a formula, which uses the moments, was obtained in this subsc.>ction; for anotru-.r version, see 5.1.

4.5. Reconstruction of unknown directions from known 'YBlues of 'R. I. Here we present. another example in which Cavalieri's conditions arc used \"Tben inverting the Radon transform. IT the Radon transform 'R./(;p, p) of a compactly supported function f is known for some set of angles 'Pt. i = 1,2, ..• , n, then tIle function I ('an t~ reconstructed approximately.

1. RADON TRANSFORM

18

The following situation arises in electron microscopy of ribosomes: the values of 'R J( I{) , p) are known for some angles I{) 1 , 1P2, ••. ,I{)n and for all p, i.e., n functions of p are given, but the values of the angles I{)i themselves are unknown. Therefore, to reconstruct J, one must first reconstruct the unknown angles. or. to be more precise, to reconstruct these angles up to a common rotation. Goncharov (21) showed that using the fact that J is compactly supported and applying Cavalieri's conditions, one can solve this problem in the generic case provided that n is sufticiently large. Let J be a compactly supported function, let 'R J be the Radon transform of J, and let 1,,(1{) be the moments of the function 'RJ (k = 1,2, ... ). By Cavalieri's conditions, the moment I" is a homogeneous polynomial of degree k io COSI{). siol{); in particular, 11(1{) = ~1 COS I{) + ~2 sinl{),

(4.10)

12(1{)

= JJl cos2 I{) + 1'2 cos I{) sin I{) + 1'3 sio2 1{).

Note that the expressions for 11 and 12 can be represented in the fonn 11(1{) = a cos(1{) - a), (4.11) 12(1{) = b cos 2(1{) - /3) + c. All parameters in (4.10) and (4.11) depend on J. Suppose that 11(1{) ~ 0 and h(l{) ~ 0 and consider the following parametric curve r / in the plane (YIt 112):

Yl = /1 (I{) , LEMMA.

r/

1/2 = 12(1{).

is an algebraic cuMle oJ degree Jour.

Indeed. the degree of the curve r J is determined by the number of points in the intersection of this curve with a line in general position A Yl + B 1/2 + C = 0, i.e., the number of solutions of the system of equations

{

A (~1 tl + ~2 t2) + B (1'1 t~ + 1'2tlt2 + 1'3t~) t¥+t~=1.

+ C = 0,

We emphasize that different curves correspond to different functions J. A curve of degree four is determined by 15 points in general position on this curve (the number of coefficients of a polynomial P(Yl. 112) of degree four is 15). Therefore, if the functions 'RJ(I{), p) are known for 15 values of I{). then 15 points on the curve r / are known, and hence the curve is known. For a known curve r /' the coefficients a, b, c in (4.11) are defined by the formulas 1 1 a (Yl)max, b 2«Y2)max - (Y2)mln). c = 2«1I2)max + (lI2)mln).

=

=

Since a shift of the parameter I{) does not change the curve r /. we can assume that I{)I = O. Thus, we know cosa, cos2/3 and COS(l{)i - a), COS2(l{)i - (J) for i = 2,3, ... , 15. Hence, we can readily see that iJ a, /3, and I{)i are in general position, then the angles f/J2. f/J3 ••••• 1{)1Ii are detem&ined uniquely. REMARK. The number of angles n = 15 is excessive because the above construction can give only curves r / of special form rather than all curves r of degree four. In fact, it is enough to know only five angles I{)It 1{)2, .•. , 1{)5 (1{)1 = 0) for the

5. POISSON FORMULA AND THE DISCRETE RADON TRANSFORM

19

following reason. Since the angle I{)l is equal to O. it follows that the coefficients ~l and III in (4.11) are known. As a result. we obtain an overdetermined system of 2·4 = 8 equations in seven unknowns ~2. 112. 113 and 1{)2, 1{)3. 1{)4. 1{)5. One can show that this system has a unique solution in the generic case. REMARK. The results of this section can be immediately extended to the multidimensional Radon transform.

5. Poisson formula for the Radon transform, and the discrete Radon transform2 5.1. Poisson formula for the Radon transform on a plane. Along with Fourier integral, we have its discrete analog, namely, Fourier series. We shall see in this section that there is a discrete analog of the Radon transform, which is related to Fourier series in just the same way as the Radon transform is related to the Fourier transform. Recall that the Fourier series can be obtained from the Poisson formula for the Fourier integral. Namely, let I be a smooth compactly supported function on R2 and let F be the Fourier transform of I. Then, by the Poisson formula, +00

(5.1)

+00

L

L

I(Xl+k l .X2+ k2)=211' FI(211'k},211'k2)e-21ri(kl~I+k2%2) k l .k2 =-oc "1.k2=-OC

(both series are absolutely convergent). If a function I is supported on the square 0< X}, X2 < 1, then only the term with (klo k2) = (0, 0) on the left-hand side can be nonzero for points (Xl. X2) in this square, and we obtain a representation ofthe function I by a Fourier series. +00

(5.2)

I(x}, X2) = 211'

L

F/(211'k lo 211'k2) e- 21r i("1 %1+"2%2).

Thus, a function supported on the above square can be represented by both the Fourier integral and the Fourier series. Let us present an analog of the Poisson formula for the Radon transform. Introduce the subset A of points (k l • k2 ) E Z2 such that either kl > 0 and the numbers kl' k2 are coprime or (klo k2) = (0, 1). THEOREM 5.3. Let I be a smooth compactly supported function on the plane R2 and let 'R./(~I'~2;p) be the Radon translorm 011. Then

+00

(5.3)

L

I(XI

+ kl' X2 + k2) +00

=I('R.f)+

L (L

'R./(klok2;kIXI+k2x2+m)-I('R.f)),

("I."2)EA m=-oo

where (5.4) 2 See (11) and the computational applications in (20).

I. RADON TRANSFORM

20

PROOF. Recall that the Radon transform is obtained as a composition ofthe two-dimensional and onMimensional Fourier transfonns, namely.

'R1(~1'~2;P) = i~ FI().~)e-i>'P d)'.

(5.5)

where F 1 is the Fourier transform of I. Therefore. formula (5.3) can be obtained by combining the Poisson formulas for the two-dimensional and one-dimensional Fourier transfonns. Note first that each nonzero point of Z2 can be represented in the form m(klo k 2). where m E Z \ 0 and (klo k 2) E A. Thus. the Poisson formula (5.1) can be rewritten as +00

E

(5.6)

1(%1 +klo %2+ k2)

= 211'{FI(0)

kl.k2=-00 +00

+

E (E

FI(211'mk lo 211'mk2)e-2.. im(kl ZI+k2 Z 2 )

-

FI(O»)}.

(kl.k2)EA m=-OCl

It follows from (5.5) and the Poisson formula for the one-dimensional Fourier tra.nsform that +~

L

'RI(~; p+ m)

=

+~

L

FI(211'm~)e-2#i"'P.

m=-oc

Hence, +00

211' (5.7)

L

FI(211'mk lo 211'mk2)e-2 .. im(klzl+k2"2)

m=-oc

+00

L

=

'R/(k lo k2;kl Xl

+ k2%2 + m).

m=-oo

Since

FI(O) = (211')-1

i::x. 'R1(~.p)dp

for any {. relation (5.3) follows from (5.6) and (5.7). COROLLARY. 11 a function 1 then

u supported on the unit square 0 < XI.

%2

< I,

+00

(5.8) I(XI.%2) = I('R/)

+

L (L 'R/(klok2;kl%l+k2x2+m)-I('Rf))

«klo k2)EA m=-oo

lor any point (XIo X2) 0/ thu square, and the series on the right-hand side is absolutely convergent. Thus. we have obtained another inversion formula for a smooth function 1 supported on the unit square. Moreover. if I('RI) = 0, then the function 1 can be reconstmcted at a point % provided that only the integrals of lover the lines with rational angular coefficients passing through X and the parallel translations of these lines corresponding to the integral values of Il.p are known. Note that the distance between the parallel lines kl XI + k2 X2 = P and kl XI + k2 X2 = P + 1 is equal to (k12 + k22)-1/2 (in the ordinary Euclidean metric). The presence of integrals over shifted lines re8ects the fact that the inversion formula is nonlocal.

5. POISSON FORMULA AND THE

DL~

RADON TRANSFOR1..t

21

REMARK. Formula (5.8) can be used to munerically invert the Radon transform for not DeCeSIlMily smooth functions. &.2. Discrete Radon transform; relation to the Fowler series. Let I be a function supported OD tbe unit square 0 < Xl. X2 < 1 and let 'RI be the Radon transform of /. Starting from (5.8), it is natural (by analogy with the Fburier series) to refer to the nmctioD F on A x R given by the formula

F(klo ~; p) =

(5.9)

+oc

L

'R/(k 1• k,;p + m) -l('R1)

m=-oc

as the discrete Rtulon tnms/oma of f. This function is periodic in p with period 1. By (5.8), the function f can be expressed in terms of its discrete Radon tramrform as fOllows:

(5.10)

f(Xl, X2)

=[('RI) +

L

F(k" k,;kl Xl

+ 'v.lX2),

0 < Xl, X2 < 1.

(k:."z)EA

The ftmction F is related to tbt> FOurier st'Jies of the function f iD exactly the S8Jlltl way in which the Radon transform 'R! is related to the Fourier tnmsfonn :F!. Namely, let f(Xl. X2)

=

+00

L

a(ult n,)e2I1'j(ftl:r:I+"2~)

be tbe expansion of / in the Fourier series on 0 < Xl. F(kl, k,i p)

X2

< 1 and let

+00

L

=

b",(kt. k:z) e2ll'i mp

'n=-OQ

be the expansion of F in the Fourler series with respect to P OIl the iuterval 0 p < 1. Note that bo(ka. k2) O. Substituting these series into (5.10). we obtain

=

+00:: E

<

a(na, '12) e21r i (n. %1 +n2 %,)

E L b,.,,(klt k2)e2II'im(.IIZI+~Z2).

=1('R1) +

("" '-2 )EA ~

Hence, (5.11)

(5.12)

a(mk •• m~) = b",(klt k,) 0(0,0) = [('Rf).

for

m;' 0,

(ki. k2) E A,

Relation (5.11) is a discrete analog of {5.5}.

5.3. Problem of Integral geometry OD the torus. One can interpret ~ dimeusionaJ Fourier series in two ways, namely, as a representation on the square 0< Xl, X2 < 1 of a function supported on this square and as a representation of a periodic fullctioll f all the mwre R2, i.e., 8 function I such that !(x+n) = !(x) for any n e Z2. In tile latter case, the Fburier series can be interpreted as an expansion of a function defined in the torus T ::: 1(2 I'Ll. We give tbe same inteI'pretat.ion for thE' discrete Radon transform.

1. RADON TRANSFORM

The square 0 < Xl, X2 < 1 can be viewed as an unfolding of the torus T = R2/Z2. In this case, for any k = (kl' k2) E A and pER, the union of the segments of the lines kl Xl + k2 X2 P + m. m E Z.

=

contained in the square corresponds to a closed geodesic "Y(k.p) on the torus T. Conversely, every closed geodesic on the torus T can be represented as the union of segments of this kind. Thus. for the restriction of a periodic function I to the square o < Xl. X2 < I, the discrete Radon transform F(kl' k2• p) can be interpreted as follows: F(klo k2' p) = (k l 2 + k 22 )- 1/2 I(k lt k 2: p) - 1(1). where j(kb k2 : p) stands for the integral of the function I on the torus T over the closed geodesic "Y(k,p) (taken with respect to the Euclidean length on T), and 1(1) for the integral of the function j over the family of parallel closed geodesics. Relation (5.10) implies the following assertion. THEOREM 5.4. Every smooth function I on the torus T = /R2/Z2 i.s reconstructed from the integrals j(k l • k 2; p) over the closed geodesics "Y(k.p) as lollows: (5.13) I(x)

= l(/)- +

~ L-

((k2 - l • k 2 : kl XI + k2 X2) - l(/) - ). l + k 2 2)- 1/2 I(k

("I."2)EA

REMARKS. 1°. The length ofthe geodesic "Y(k.p) is equal to (k 12 + k22)-1/2. Therefore. if one renormalizes the measures on the geodesics in such a way that the lengths of all geodesics become equal to one and denotes by (kl. k2; p) the integral over the geodesic "Y(k,p) with respect to the new measure. then (5.13) simplifies to

l'

(5.14)

lex)

L

= 1(1) +

(J(k l • k2:

kl XI

+ k2 x 2) -l(r»).

("I. k2)EA

2". Formula (5.13) takes even a simpler form for functions I with mean value zero on the torus, namely,

I(x) =

L

r(k l • k2; klzl +k2 X 2).

(kl,k,)EA

In this case, the inversion formula is local. To reconstruct the function I at a point on the torus, one must know only the integrals of this function over the closed geodesics passing through the point z. Z

6. Minkowski-Punk transform This section is devoted to an analog of the Radon transform in which the plane R2 is replaced by the unit sphere tfl c R3 and the lines in R2 by the great circles on tfl. Let I be an even smooth function on the unit sphere S c R3. By the Minkowski.fUnk translorm we mean the integral transform sending the function I to the integrals of this function over all great circles on the sphere S. The following problem arises: Reconstruct the function I from its Minkowski -F\mk transform. This problem had first been solved by Minkowski and Funk even before the Radon paper appeared. A similar problem can be formulated for functions on the three-dimensional unit sphere S3 C R4.

6. MINKOWSKI-FUNK TRANSFORM

23

REMARK. We consider only even functions on the sphere because the integral of an odd function over a great circle equal."! zero.

We represent the Minkowski--Funk transform by an explicit formula. Every great circle on the sphere S is the intersection of this sphere with a plane passing through the point 0, ({. x)

(6.1)

= O.

I{I = 1.

Thus. the manifold of great circles is parametrized by the points ( of the unit sphere S' in the dual space (R.3)" and the same great circle corresponds to antipodal points ~ and To write out the integral over the circle (6.1). we present this circle parametrically,

-e.

(6.2)

x=cos'P·o+sin'P·B.

0:5 !p <2'11",

<e.

where 0 and {3 are mutually orthogonal unit vectors such that et) The Minkowski-Funk transform is given by the following relation: (6.3)

MI(e) =

1 2

#

l(cos'P' 0

+ SinlP' {J)

dIP.

= <e, (3) = O.

lel = 1.

It is clear that the integral does not depend on the choice of vectors et and {J and that MI{ -(> = MI(e)· Thus, the Minkowski-Funk transform sends an C'\o"Cn function on the sphere Se R.3 into an even function on the sphere S' c (R3),. In order to identify functions on S and S'. we identify the spheres S and S'. To be precise, we assign to any point x e S the point e 5' corresponding to the equatorial great circle with the pole x. It follows from the de6nition that the Minkowski-Funk transform commutes with rotations of the sphere. In other words, for any rotation 9 w ...... w 9 of the sphere S, the Minkowski-Funk transform MI,«() of the function I,(e) = 1«(9) is equal to MI«(g). Wt> shall use this property to reconstruct a function on the sphere from the Minkowski Funk tTansform of this function. Consider the Hilbert space 'H of even functions on S whose absolute ,,'Blues are square-integrable (with respect to the invariant measure on S). We construct the spectral decomposition of the operator M on 'H. Introduce a filtration of 'H by subspaces invariant under rotations. Namely, let 'H2n C 'H be the space of spherical polynomials of degree 2n, i.e.• the subspace of functions on S obtained by restricting the homogeneous polynomials of degree 2n to 5 c R3. We obtain a filtration

e

(6.4)

'Ho C 'H2 C ... C 'H 2n C •..

of the spare 'H because the restriction to S of a homogeneous polynomial P(z) of degree 2n coincides with the restriction to S of the polynomial (X1 2 + X22+X3 2 ) P(x) of degree 2n+2. Moreover. the closure of the space U'H2ft coincides with 11.. Obviously, all spaces 'H2n arc finit.tHlimensional and rotation invariant, and the operator M is defined on each subspare 'H2 ... Note that the restriction to 5 of a nonzero homogeneous polynomial of degree 2n is a nonzero function on 5. This readily implies that (6.5)

dim 'H2n

= (n + 1)(2n + 1).

24

l. RADON TRANSFORM

We construct the grading of invariant subspaces of'H associated to the filtration (6.4): (6.6) where Ho = 'Ho is the subspace of constants, and H2n , n > O. is the orthogonal complement of 'H2n-2 in 'H2n. i.e.•

It follows from (6.5) that dim H2n = 4n + 1. The space H2n (n = 0,1,2, ... ) admits the following intrinsic description. It consists of the functions on the sphere S that are restrictions to S of the homogeneous harmonic polynomials of degree 2n (i.e., the polynomials P satisfying the Laplace equation lP P + lP P + lP P = 0). ~ lJi2f ~ The main fact is that each H2n is an eigenspoce 0/ the Minkowski-Funk operator M. This assertion immediately follows from the fact that the operator M commutes with rotations of the sphere S. whereas the sllbspaces H2n are irreducible (i.e., contain no proper invariant subspaces) and are pairwise nonequivalent. To describe M. it remains to compute the eigenvalue An of the operator M on each subspace H2n. To this end. we consider the function / E H2n obtained by restricting the polynomial (Z2 + i Z3)2n to S. Since (Z2 + i Z3)2n is obviously a harmonic polynomial, it follows that / E H 2n • Hence.

MI=An/· We write out the expression for 1 in the spherical coordinates lJ, loll

= cod,

W2

= sin 9 COSl{),

1013

= sin 9 sin", (0 < 9 < 7r.

I{) :

0 < 'P < 27r).

We have 1(9,1{) = e 2intp sin2n 9.

In particular, l(j.O) = 1, and therefore MI( j, 0) = An. On the other hand, MI(j. 0) is the integral of the function lover the great circle on which ..p = ±~, i.e.,

Hence, I-

(6.7)

An

= 2v7r(-1)

nr(n + ~) r(n + 1)'

It follows from the description of M that to reconstruct a function / on S from its image 'P = MI, it suffices to decompose 'P over the eigenspaces of the operator M, namely, 'P = 'Pn. where 'Pn E H2n. Then 1= A;; I !{In' In § 8 we shall find another inversion formula for the Minkowski-Funk transform. This formula will be based on a simple relation between this transform and the Radon transform for the projective plane.

E:=o

E:=o

7 RADON TRANSFORM OF DIFFERENTIAL "·OJU.IS

7. Radon transform oC differential forms The Radon transform can be defined Dot only for functions but also for other objects. In this section we consider thc Radon transform of difl'erential forms on the plarte Md in three-dimensional space. The main I)roblem still is to describe the image of the transform and t.o reconstruct the original form from its Radon t.ransform. More general discussion can he found in [31.

7.1. Radon transform of I-forms on the plane. Consider an arbitrary differt"lltial I-form on 22. W

= h{XI,X2)d:.tl

+ h(x"x2)dx2,

whose roet1icients belong to the Scbwartz space S(R2). One can take the int-egral of this form OW!I' any oriented line ill tilt> plane 1Il2. This defines 8 function OD the manifold H of all oriented lines, and \\"e refer to this function as t.he Radon transform of the original I-form w and denote it by "Rw. 'rhus. by definition,

"Rw(1)

(1.1)

= /, w.

I E H.

Wc write out an explicit expression for the function "Rw in coordinates on H. AB in § 1. we present tbe lille.s 011 th(' plane by parametric equatioos,

Xl = -tsill~ + peas"" X2 = teastp+psinrp.

(7.2)

For the positi\"e direction 011 the line (7.2) we take the direction in which t inc.rea5CS. Thus, the parameters (rp,p) define not only the line but aLw the orientation of this line, and hence they can be t.aken for the coordinates on the manifold H of all oricnted lines. Note that the same line with opposite orientation b88 the coordinates (tp + 1f, -plo In the coordinat.es (rp,p). the cxprollSion for "Rw becomes (7.3)

'Rw(t;?p) =

L~ (-sinrp !t(-tsint;?+peas'P, tC061?+psin~) _ + cos.p h( -t sin IP + pros",. t costp + psin",) ) dt.

We wish to roconstmct tbt> differential form w from the Radon transfonn "Rw. It is impossible to obtain a unique solution of this problem because of the following simple assert.ion. LEMM.A.

1/ the thJIerentiall-/orm "". is dosed., i.e., dI.AJ

= 0, then 'Rw == O.

Indood, every closed fnnIl W = !I d:r.l + h ~ with 11, 12 E S(JR2) is exact, i.e.. representable as wJ = du, where u E S(R2). Hence, the Integral of the c1o&ed form w over any line vanishes. We solve another geometric problem. Assuming that the integrals of a I-form dXI +h (Lx'l over an lines in R2 are known, i.e., for the given Radon t.ransform of w, find the exterior differential of 4.0.', i.e., the 2-form wJ

=h

dJ.J =

8h ( ax l

-

8ft) dx l CJ:r 2

1\ dZ2

== F(x I, X2) dXl " h

or. brieHy. rccon.ortnlct tbt> function F from "Rw.

2•

26

I. RADON TRANSFORM

PROPOSITION 7.1. To reconstnJct F, one must apply the inversion lormula lor the Radon tmnslorm on the plane to the /unction :" 'Rw{'I',p).

The proposition results from the following lemma. LEMMA. The &don translorm 'Rw 01 the lorm..,; = h dx l + hdx2 is related to the Radon tmru/orm 'R F 01 the /unction F = ~ - ~ by the lormula

(7.4)

'RF{",.p)

I)

= (Jp'Rw{",.p).

PROOF OF THE LEMMA. Obviously. for a fixed coordinate 'I' = '1'0. the integral of the function 'R F with respect to p from PI to 1'2 coincides with the integral of the 2-form diN over the strip 11' C R2 between the lines II = ('Po. PI) and = (lPo, 1'2). Hence. by the Stokes formula,

'2

1

1>2 PI

'R F (lPo, p) dp =

f. =1 -1 diN

..

w

12

w = 'Rw ("'0. 1'2) - 'Rw ('1'0. PI).

11

This implies (7.4). COROLLARY. The Radon tmnslorm 'Rw 01 a differentiall-Iorm w with coefficients in the space S(R2) is identically zero il and only il this lorm i.. dosed.

7.2. Radon transform of 2-forms on the plane. In analysis, it is customary to integrate differential k-forms over k-dimensional submanifolds. This corresponds to the contents of 7.1, where differential I-forms were integrated over lines on R2. However, if a family of submanifolds of dimension k is chosen, then one can integrate an m-form with m ~ k over this family. If m > k, then the result is a differential (m - k)-form OD the variety of submanifolds rather than 8 numerical function. In this subsection we consider the integration of a 2-form w over the lines on R2; the result of this integration is a 1-form 'Rw on the manifold of lines. The informal idea to evaluate 'R.w is as follows. Th find the value 'R.w(l. r), where I is a line and r is a tangent vector to the manifold of lines, one must infinitesimally shift the line I in the direction of r and integrate w over the corresponding infinitesimal 2-dimensional domain. The same idea works in the general situation as well. We proceed with the corresponding computations. First we define the Radon transform of a differential 2-form For that. we introduce the three-dimensional ftag manifold A whose elements are pairs fonned by an oriented line on the plane and a point on this line. Since points on the plane are given by coordinates (Xlo X2), and oriented lines by coordinates (""p), it follows that one can equip A with two coordinate systems. The first system is given by the coordinates (XloX2' "'). where (XI.X2) determine a point on the plane and '" gives the direction of a line passing through the point, and the other system is given by the triple (",.p,t), where (""p) detennine an oriented line on the plane and t gives a point on this line. The coordinates (x I , X2. "') are expressed in terms of the coordinates (",.p,t) by relations (7.2). The inverse relations look as follows: t = X2COS", - XI

sin""

21

1. RADON TRANSFORM OF DIFFERENTIAL FORMS

Let US \iew tht' 2-form w = 1dx l A.dx2 85 a differential form on the manifold A with coordinat~ (X 10 X2.1P). Th define the Radon traosform of w, we 6rst repretiC1lt the formw in the coordinates (rp,p, t) by using (7.2) for Xh X2 and the corresponding expressions for d.r.1t dx2, dXl dX2

= - sinlPdt - t cos",d", - P 8inlP~ + coslPdp, = cos ip dt - t sin IP dIP + P cos IP dIp + sin IP dp.

We obtain

w= -f(-t sinlP+PCOSip, t coscp+psiu'P)dpA.dt

- t/(-t sin 1,0 + p oosrp, t coscp + p sinrp}~ A. tit + ... where the dots stand for tht' terms that do not contain dt.

,

We denote by Wo the differential form obtained from w by deleting all terms that do not contain dt. For fixed values of lp, p. dip. and dp, the resulting form Wo is 8 differential I-fonn on the oriented line corresponding to (rp,p). Integrating the form over the line, we obtain a differentiall-form on the manifold of oriented lines; this form is caUed the Radon trall8fOml of the original 2-form w and is denoted by 'Rw. Thus, WP ohtain the following definition. DEFINITION. The Radon transform

w

'Rw of a differential 2-form

= I(xl, X2) dXIA. dX2

on the plane is the differential 1-form 'R.w=a(rp,p)dp+b(IP.p)~

(7.5)

on tlw manifold H of oriented

a(.p.p)

(7.6)

1 =- 1

=

lin~

on the plane, where

+01:.

-00

1(-tsin<;+pcoslP. tcos.p+psinrp)dt,

+00

b(cp.p)

-OD

t/(-tsinrp + pCOSIP, tcoscp+ psinrp)dt.

We establish some properties of the differentiall-form 'Rw. It foUows from the explicit expressions for tbe functiollS a and b that both of them arc infinitely differentiable and rapidly decreasing together v.ith all their derivatives on the manifold H of oriented lines, i.e., these functions belong to the Scbwartz space S(H). Further, t.he function a(f;?p) is the Radon transform of the function I. Q = 'R.I, and hence this function additionally satisfies Cavalieri's conditions (see § 4). Finally, one can immediately sec that

Ba

8",

and henre the differential form

8b

= (Jp'

'R.w is closed.

=

PROPOSITION 7.2. A differentuu I-form (l adp+bd

belongs to the image 01 the Radon translorm 01 the 2-lorms if and only if thu form is closed and the junction a satisfies Cavalieri's conditions. PROOF. In one direction the assertion has already been proved. Conversely, let {1 = a dp + b ~ b(' any form with the above-listed properties OD the manifold H = Si xRI. Then a = 'RI forsotuc function lE S(R2). Bysettingw = /dx1A.dx2,

28

1. RADON TRANSFORM

we obtain {} - 'Rw = c(cp.p)dcp. Since this I-form is closed. it follows that the function c does not depend on p. and it follows from the rapid dl.'crease that c = O. Thus. {} = 'Rw. REMARK. The fact that the form 'Rw is dosed can also be derived from the general fact that the operator 'R. commutes with the operator d of exterior differentiation. We only need to note that the form w itself is closed as a form of maximal degree.

To reconstruct the differential form w = I dx l /\ dx 2 from its Radon transform 'Rw = adp + bdcp, or. equivalently, to reconstruct the function I. one must apply the inversion formula for the Radon transform on the plane (8(!C § 1) to the function a(cp.p). We see that the Radon transform of 2-forms has trivial kernel. in contrast to the case of I-forms. We also note that the problem of reconstructing a differential 2-form w from its Radon transform 'R w is overdetermined because one must know only one coefficient of the form 'R w to find w completely. 7.3. Radon transform of2-forms in three-dimensional space. Consider an arbitrary differential 2-form in R3, (7.7) w =

h (X)dx2 /\ dx3 + h(x)dx3 /\ dx l + Ia(x)dx l

/\

dx 2.

X = (Xl. X2. X3),

with coefficients in the Schwartz space S(R3). Integrating the form w over all p0ssible orientable planes in R3, we obtain a function on the manifold of planes. We refer to this function as the Radon translorm of wand denote it by'Rw. We present the expression for 'R. w in the coordinates on the manifold of planes in R3. We define the planes by equations of the form (7.8) On the manifold of planes we take the coefficients al. 02. {3 of these equations for the local coordinates. Further, we take (Xl, %2) for the coordinates on each plane given by (7.8), and thus define an orientation on this plane. In this case. the restriction of the form w to the plane (7.8) can be obtained by substituting the expressions X3 = 0lXl + 02%2 + (J and dx 3 = 01 dx 1 + 02 dx 2 in (7.7). We finally obtain

'R.W(01. 02.{J)

= [:00 [:<>0 [(la _ 01 h -

02 h)(Xl. X2. 0lXl

+ 02.1'2 + j)] dx 1 dx2.

As in the similar problem for I-forolS on the plane (see 7.1). the form w can be reconstructed from its Radon transform 'R.w only IIp to a sllmmand which is a closed differential 2-form. Therefore, we formulate another problem: Reconstruct the exterior differential of the form w from the function 'R. w.

dw

= (~h + ~h + ~Ia) dx l VXl VX2 VX3

or, equivalently, reconstruct the function F

/\

dx 2 /\ dx 3•

s from'R.w. iJ + !!h = 8z';' lll~ + a £h. a %2

%3

7.3. In order to reconstruct the function F from the function 'R.w. one must apply the inversion 10matJla lor the Radon translorm 01 functions in the three-dimensional space to the function IB'R.W(Oloo2,.8). PROPOSITION

7. RADON TRANSFORM OF DIFFERENTIAL FORMS

29

The kernel of the Radorl tmn.~form of 2-forn18 in R3 with ropUlly decmu&ng coeffident.s Nnlsists 01 the closed lorms w. The proposition result... fronl the foUowing lemma. LEMMA.

The Radon trnnslonn of a 2-foml w

= h dx2 1\ Ua + h ha 1\ dZl + la dZ l 1\ dz2

i.., related to the Radon tronsfoml of the functum, F = ~ lormula

+~+~

by the

(7.9)

The proof of (7.9) essentially coincides with that of (7.4), with the strip between two parallel lines in a2 replaced by a domain in RS between two parallel planes. REMARK. Th(' remits of 7.1 and 7.3 can be

re.adUy extended

t~

differential

(n -1).forlOs in Rn.

7.4. Radon transform of 3-forms in three-dimensIonal space. The RadOll transfunn of a differential a-fOfDl in R3.

w = f(xlo X2. X3) dXJ 1\ dX2 1\ dxa.

f

E S(R3),

is defined similarly to that of a differential 2-fonn in R2; see 7.2. Namely, introduce the 5-dimensional Rag manifold A whose elements are the pairs given by a plane in RS and 8 point on this plane. Since points in RS are given by the coordi.na.tcs (Xh X2. Xs). and thc planes by the local coordinates (a!.
= I{Xb X2, OtZ'1 + a2X2 + tJ)(dtJ + X, dOl + X2 002) 1\ dXl 1\ dx2'

For chosen valucs of 01.°2. {j, d(J and for integration variables x}, X2. this 3-fonn can be regarded as a differential 2-form on the plane with the coordinates 010 O}. p. Integrating the 2-foml over this plane, we obtain a l-fonn at the point (01,02,11) of the manifold of planes in R3. We eaU this I-form the Radon transform of the original differential 3-fuml w Bud dCllott> it by 'R.w. Let us summaru.e. the Radon tron••form of a differential3-fonn w = I dz l Atix2A mean the following differential I-form 'R.I.J on the manifold of planes

DEFINITION. By

U3 on RS ill R3:

Wf.'

1. RADON TRA!IISFORM

30

where (7.11) a; (010 0 2 • .8)

= [:00 [:OO%i /(%.,%2. 0 1%1 + 02%2 +

6(01,02,.8)

=

1: 1:00 oc

.8) cU l cU2.

/(Xl,%2,01%1 + 02%2 +

i

= 1,2.

~)dxl dx2.

The next assertion immediately foUows from the definition of the I-form 'Rw. I) The coefficients of the form 'R, w satisfy the relations 1J0 1

1J0

2 -=. 1J02 1J0 1

IJb 1J0j -=-. 1J0i IJ{J

i=1.2.

Hence. the form 'R w is closed. 2) The coefficients 0" 02. 6 of the form 'R w are infinitely differentiable functions. which rapidly decrease together with all their derivatives as 181 ..... 00. 3) The function 6 is the Radon transform of the function / E S(R3). i.e., 6 = 'R/. One can readily see that conversely. if a differential I-form {l = 01 001 +02002+ 6d{J on the manifold of planes in R3 satisfies conditions I). 2). and 3). then this form is the Radon transform of a certain differential 3-form w = / dx 1 1\ cU2 1\ cU3. / E S(R3) (cf. a similar assertion in 7.2 in the case of R2). It also foUows from 3) that to reconstruct the form w from its Radon transform 'Rw = 01 001 + 02 d0 2 + 6d{J. one must apply the inversion formula for the Radon transform of functions in R3 to the coefficient 6 of the form 'R w.

8. RadOD tr8D8form for the projective plaue aud projective space We have started from the Euclidean definition of the Radon transform; later on, we have established that this transform is affine-invariant. In this section we show that the Radon transform is in fact projective-invariant [3]. In this respect. the Radon transform substantiaUy differs from the Fourier transform because the latter has no projective version. We consider the Radon transform for the real projective plane p2 and the real projective space p3. The definitions of the Radon transforms for p2 and p3 are similar; however. as in the case of affine spaces. the inversion formulas for these transforms are substantiaUy different: the formula is local for p3 and nonlocal for p2. Therefore. we begin with the case p3 because it is simpler. 8.1. Spaces p3 &Dd (p3)'. As usual. points in p3 are given by homogeneous coordinates Xl. %2, %3. %4. i.e.• by vectors % E R4 \ 0 defined up to a common factor. In other words. p3 is defined as the base of the natural bundle ~% '" %.

~

E lR \ O.

Since the preimages of points in p3 are one-dimensional subspaces in R4. it follows that p3 can be interpreted as the manifold of one-dimensional subspaces in R4. In the homogeneous coordinates %1.%2.%3.%4. each plane in p3 is given by an equation of the form

:u

8. RADON TRANSFORM FOR p2 AND p3

where the coefficients ~i are defined up to a common factor. Thus, the manif'old of planes in p1 is it~ a three-dimensional projective space with homogeneous coordinates el,{2,';3,{'" This space is said to be dual to pS and is deooted by (p3)'. 8.2. Radon transform for p3. Denote by F(m) the space of infinitely differentiable funct,ions on R" \ 0 satisfying the homogeneity condition I{~z)

(8.1)

= 1~lm I(x)

for

any

~:F

o.

The first task in passing from the affine case to the projective one is to select an appropriate function space on which the Radon transfonn can he constructed. As such a function space \\-e take the space F( -3). The reason why one should take the space F( -3) rather than of a space of functions on the projective spact> p3 itself is related to the fact that there is no projective-invariant volume fonn on pS. i.e., there is no differential ~form invariant with respet:t to the action ()f the group SL(4,R) of projective transformations on pS. On the other band. such a 3-form exists on the one-dimensional bundle R4 \ 0 over pS. This is the Leray differelltial form (8.2)

w(x):;

Xl

th2 "tix3 1\ tU4

+ (fonDS obtained by DOntrivial cyclic permutations of 1,2,3,4). which can com.-ewently be written 88 the determinant with comnmting and anticommuting ~'8I'iables, Xl

w(x)

1

X2

= 6 X3 x"

dx t dZ2

dxs

dx"

In what follows. we writt-, for brevity, w(x)

= [x, dx, th, dx).

Let .; E (R"), \ 0 and let hI.:

<e. x) =

e.

0

be the subspacc in R4 corresponding to Denote by O'dz) the interior product the Leray form w(z) and the fonn d({(. x» on R;I. O'E(z)

or

= d«{, z}) j w(x).

By definition, O'((x) is a differeutial 2-fonn on J!t4 satisfying the relation (8.3)

It is known that such a form on the entire space R4 is

not unique; however, the restriction of such a fonn to the subspace hE is uniquely defined. We can set, for instal\ce, () Ix. u, dx, hI t7E x = (~, u) ,

where U E R4 is an arbitrary (but fixed) vector such that (~, u) Let 1 E F(-3). Thm the differential 2-fonn I(x) O'((x)

:F O.

32

I. RADON TRANSFORM

on the space h( is homogeneous of degree zero, and this form is orthogonal to the fihers of the bundle h( -. Phf-. where Ph( is the projectivization of h(. Hence. this differential form can be pushed down from h( to Ph(. i.e.. to a plane in pl. DEFINITION. By the Radon transform of a function function 'R,I on (R")' \ 0 given by the formula

(8.4)

I

E F( -3) we mean the

I(~) = f

I{x) lTdx). J«(,zl=O It follo\\'8 from the definition that .p = 'R, I is an infinitely differentiable function 'R,

on (R")' \ 0 satisfying the homogeneity condition ,,(~~) .1'(-1). Thus. the Radon transform defines a mapping

= I~I-I .p(~). Hence • .p E

'R,: .1'(-3) -. F(-I).

It is convenient to represent the expression for 'R, I in the following form:

'R,I(~) =

(8.5)

L

l(x)6«e, x)w(x).

where 6(t) is the delta function on R. The integral must he understood as follows. First we represent the form in the integrand in an arhitrary system of variables that includes t = ({. x) as one of the variables. and then apply the distribution 6(t) to the variable t (i.e., take only terms containing dt. delete dt in these terms. and then set t = 0). It is remarkable that the differential fonn thus obtained on the plane (e. x) = 0 does not depend on the choice of the original variables. 8.3. Relation to the affine Radon transform for R3 and to the Minkowski-F\mk transform for the three-dimensional sphere. Both the Radon transform for R3 and the Minkowski-Funk transform for the three-dimensional sphere arc ,,'pedal cases of the projective Radon transform. Namely. one must replace homogeneous functions I E F( -3) by their restrictions to appropriate sections of the bundle R' \ 0 -. pl. To obtain the affine Radon transform. we take the hyperplane x" = 1 for the section and set 10(Xl, X2. X3) = I(x), X2. X3. 1). Since w(x)1

z.=l

= -dxl A dx2 A dx 3• formula (8.5) becomes

'R, I(el. e2. e3. e4)

= - JR3 f 10(Xl. X2. X3) 6(elXl + e2 X2 + e3 X3 + e4) dx1d.r2 d.r3 •

i.e .• 'R, I is the affine Radon transform of the function 10 on R3. To obtain the Minkowski-FUnk transform, one mlL~t replace the functions I and 'P = 'R, I by their restrictions to the sphere Ixl = 1 in R4 and to the sphere lel = 1 in (R')', respectively. In this case, the transform (8.5) coincides with the Minkowski-Funk transform M for the functions 00 the sphere Ixl = 1 in R4. oamely. MI(e) = 'R, I(e) for lel = 1. 8.4. Inversion fonnula for the Radon transform on p3. Our problem is to reconstruct a function I E F( -3) from its Radon transform 'R, f. Let us immediately present the answer, (8.6)

I(x)

= -(27r)-i 'R,I(~) cS"«~. x» w(~).

8. RADON TRANSFORM FOR p3 AND

,a

33

where 6"(t) is the second derivative of the delta function on R; the integral is taken over an arbitrary section r of the bundle R4 \ 0 ..... p3. Formally, the differential 3-form in the integrand,

0:.: = 'R./(~) 6"«~, x» w(~), is homogeneous of degree zero and is orthogonal to the fibers of the bundle R4 \ 0 ..... p3. Therefore, the integral does not depend on the choice of the section r. We must make sense of the integral (8.6). To this end, we set ~

= '1+p{,

where '1 E (R4)' \ 0 satisfies the condition ('1, x) = 0, p belongs to R, and { E (R4)' \ 0 is an arbitrary (but fixed) vector such that ({, x) ::F O. In the expression for 0:.: we pass &om variables ~ E R4 to variables '1 E R4 and pER and then delete the terms that do not contain dp. Since in the new variables we have w(~)

= ('1 + p{, d'1 + (dp, d'1 + (dp, d'1 + {dp), 6"«~, x» = I({, x)l-36"(p),

after deleting the terms without dp we obtain O~

= 1«, x)I- 3 'R. 1('1 + p() 6"(P)('1, (, d'1, d'1) Adp.

Integrating the form 0':.: with respect to p yields a differential2-form on the subspace h( : ('1, x) = 0 in (R4)', namely,

I<{. x)r 3 (L 2",) ('1) ['1, {,d'1, d'1), where

L=<~=E(i~'

This form is homogeneous of degree zero and is orthogonal to the fibers of the bundle h( ..... P(h(), where P(hd is the projectivization of h(. Thus, we can regard L 88 a form on P(hd. Hence, we have reduced the inversion formula (8.6) to the following formula. THEOREM

(8.7)

I(x)

8.1. 1/ '"

= 'R. I, where I E F( -3), then

= -(211")-21{{, x)l-t (fI, :.:)=0

(L

1~iJ~4

\i(;

~"'~~ )

['1,{,m"m,J,

V'II V'IJ

where ( E R4 U an amtnary (but fized) vector such that

<(, x) ::F O.

REMARK. In contrast to (8.5), the restriction of the differential form 0:.:

«(.

=

'R./«() 6"«(, z» w«() to the subspace x) = 0 (this restriction was introduced in (8.7» is not unique because it depends on the choice of the vector (. However, the difference of any two forms obtained in this way is an exact form. PROOF OF THE INVERSION FORMULA (8.7). Denote by Fi, i = 1,2,3,4, the subspace of functions I E .r( -3) that vanish on the subspace Xi 0 together with all their derivatives. Since the sum of the subspaces Fa is the entire space .r( -3), it suffices to prove the validity of (8.7) only for functions belonging to these subspaces. To be definite, assume that I E .r4' Then the function on R3 given by the formula

=

10(x.,x2.x3) = /(XltX2,X3, 1)

1. RADON TRANSFORM

34

belongs to the Schwartz space S(RS), and it follows from 8.3 that I{J = 'R. I is the affine Radon transform of the function 10' Thus. by the inversion formula for the affine Radon transform, 10(Xlo X2,XS)

=

-(27r)-21(~1 )WO(11), 'Y Vll4 'k=-'1I Z I-'I2Z2-'I3Z3

= 111 d'12 1\ d'13 - '12 d111 1\ d'13 + '13 d'h 1\ d'l2. Hence. I(Xlo X2,XS,X4) = IX 41- s 10(X1 , X2, XS) = _(27r)-2I x41- s {

where Wo(11)

X4 X4 X4

a:..1{J Wo(11).

)('1,z)=o ..,rH

The equation thus obtained coincides with (8.7) in the case of (= (0,0,0.1). Thus, the inversion formula is proved for (0,0,0,1). The case of arbitrary vector x) :F 0 can be reduced to the above case by a transformation of { for which coordinates in R4 and (R4)'.

<=

«,

REMARK. One could drop the assumption that ( is constant. Then a more general inversion formula than (8.7) is obtained; the integration in this new formula is over a two-dimensional cycle in the manifold of pairs 11, (.

8.5. OD the iDversloD formulas for the afIlDe RadOD tr8D8form OD RS and the Minkowski-Funk tr8D8form for Ss. We have already seen in 8.4 that the inversion formula proved in § 2 for the affine Radon transform is obtained from the projective inversion formula (8.7) for ( = (0,0.0,1). H for ( we take an arbitrary (but fixed) vector in R4 such that (1X1

+ (2X2 + (sxs +" = c > O.

then, using (8.7), we obtain a new inversion formula for the affine Radon transform. which differs from the inversion formula in § 2. Namely, let 7rz C (R4)' be the subspace orthogonal to (Xl,X2,X3, 1) E R4, P(7rz ) the projectivization of this subspace, and 1z an arbitrary section of the bundle 7rz - P(7rz ). If lE S(R3) and if I{J = 'R. I is the affine Radon transform of the function I e S(R3). then I(x)

= -(27r)-2 c- 31

t

'Y~ i,j-1

(i(j

C::~~ 111,(,d11,d111· J

REMARK. A more general class of inversion formulas for the affine Radon transform can be obtained by taking a smooth vector field 11 e (R4)' \ 0 transversal to 7rz \ 0 (i.e., such that (x, «11» > 0 for any 11 E 7rz \ 0) at the points 11 E 7rz \ o. To every such field there corresponds an inversion formula for the affine Radon transform.

<

Let us now turn to the Minkowski-F\mk transform for the three-dimensional sphere. Interpreting the elements of :F( -3) as even functions on the sphere Ixl = 1 in R4, we see that the projective Radon transform 'R. I and the Minkowski-F\mk transform MI of a function I E :F( -3) are related as follows: 'R./«()

= 1(1- 1 MI(I~I)'

Hence, the inversion formula for the Minkowski-F\mk transform M is obtained from the inversion formula (8.7) for the Radon transform 'R. by replacing 'R./({) with 1(1- 1 MI(

Jfr).

tl. RADON TRANSFORM FOR p2 AND

1"

35

8.0. Description or the image or the Radon transform tOr p3. THEOREM

8.2. The Radon transform 'R defines a vector space isomorphism

'R. : F( -3)

-+

F( -1).

PROOF. Introduce the operator R. on F( -1) by the fonnula

(R.!p) (x)

= -(2'11") -2 f

wee}

'P({) 6"«{, x»

J(p3)'

This definition implies that ~ E F( -3) for 'P E F( -1). It fonov.."S from what was proved in 8.4 that the operator 'R, is ~ective, and the composition R'R. is the identity operator on F( -3). Therefore, to prove the theorem, it suffices to show that the operator R. has zero kernel. Thus, let 'P E F( -1) and R.'P = O. Note that all derivatives or the function !p belong to F( -3). and hence the Radon transform of each of these derivatives is well defined. 6({{, x» = XiXj 6"({{, x), integration by parts shows that Since

a

-&

L:~~g

6«({, x»w({) =

XiX)

L

'P(e)6"«{, x»w({)

= O.

Since the Radon transform is injective, ~ 0, i. j 1,2,3,4, i.e., IP is a polynomial. On the other hand. VJ is of homogeneity degree -1, which is possible for VJ = 0 only.

=

=

8.7. Radon transform for the projective plane p2. The description or the Radon transfonn for p2 is similar to that in the above case of the projective space pS; so we describe the transform for p2 more briefly. The only substantial difference between the cases of pS and p2 is that the in\-ersion formula is local for pS and nonlocaJ for ]p2. By analogy with the case of projective space P', denote by F(m) the space of Coo functions on R3 \ 0 satisfying the homogeneity condition (8.1). The Radon transform for p2 is defined on the function space F( -2). Let w(x) be the Leray differential form on R3, w{x)

= Xl h2 1\ ha -

X2

or, briefly, w(x) = (x, dr, u). On every

h( : in 2

3

h1

1\ dX3

+ X3 UI 1\ U2,

8Ubspace

<e, x) == elXI + e2X2 + aX3 = 0

we introduce the differential l-form O"(z) by the rule O"(z)

= d«e, x» Jw(x),

i.e.,

d«{. x» 1\ O"dx)

= w(x).

H f E F(-2), then the differential form f(x}O"(x) is of homogeneity degree zero, and this form is orthogonal to the 6bers of the bwldle h( -. P~, where Ph( is the projectivization of h(. Hence, this form can be pushed down to Ph(, i.e., to a line in p2.

1. RADON TRANSFORM

36

DEFINITION. By the Radon traosform of a function function 'R. I on R3 \ 0 given by the relation

(8.8)

'R./«()= (

I e F( -2)

we mean a

I(X)C1dx).

1«(,20)=0

It follows from the definition that the function thus, the Radon traosform yields a mapping

tp

= 'R. I

belongs to F( -1);

'R.: F(-2) - F(-l). Similarly to the case of projective space p3, it is convenient to represent the expression for 'R. I in the form

'R./(~) =

(8.9)

L

I(x) 6«(, x»w(x),

where 6(t) is the delta function on R. We now proceed with the construction of the inversion formula for 'R.. As in the case of p3, we first present the formal answer, I(x)

(8.10)

= - 4!2

i 'R./(~) (~,

X)-2 wee),

where r is an arbitrary section of the bundle R3 \ 0 - p2. To make sense of this integral, we repeat the arguments used in 8.4 for p3. Namely, set = fI + p(, where fI e (R3), \ 0 satisfies the condition (fI, x) = 0, p belongs to R, and ( e (R3), \ 0 is an arbitrary (but fixed) vector such that {(, x) ;: O. We pass from the variables ~ e R3 to the variables fI e R3 and peR in the expression for the differential form

e

020

= 'R.I(~){(,x)-2w(e)

and then delete the terms that do not contain the differential dp. We obtain

= {(, X)-2 'R./(fI + ",)p-2 (fI,(,dfll" dp.

~

We interpret the integral of this form as foUows. One first takes the integral over p in accordance with the definition of the distribution p-2, i.e.,

[:00

F(P)p- 2 dp=

[:00

F'(P)p- 1 dp,

where the integral on the right-band side must be understood in the principal value sense. We obtain a differential I-form on the line and integrate the form over the line. Hence, the inversion formula (8.10) is reduced to the foUowing formula. THEOREM

(8.11)

8.3. 11 tp = 'R.I, where I I(x)

= - 4 12 «, x)-2 1r

where (8.12)

e F( -2), ( 1('1,20)=0

then

"'(fI,() (",(,dfll,

8. RADON TRANSFORM FOR p2 AND p2

and, E R3 \ 0 is an arbitrary (but fixed) vector satisfying the condition (C, z) The integral (8.12) mwt be understood in the principal value sense.

37

#: O.

As in the case of projective space (see 8.4), the proof of the inversion formula (8.11), (8.12) is reduced to the inversion formula in § 1 for the affine Radon tra.nsform on the plane. THEOREM

8.4. The Radon translorm 'R lor the projective plane p2 defines a

vector space isomorphism 'R: .1"(-2) - .1"(-1). PROOF.

Let

~: .1"(-1) - .1"(-2) be the operator defined by (8.10). It follows from what was proved above that the composition ~ 'R is the identity operator on .1"(-2). Therefore, to prove the theorem, it remains to show that the operator ~ is injective. Let us use the fact that the function Repl can be represented in the form ~:5=1

(8.13)

-

1

'Rlp(Zlt Z2, 1) = - 411"2

lrf 1tI«() cS,«(IZI + (2Z2 + es)w(e),

where (8.14) Indeed, substituting (8.14) in (8.13) and integrating with respect to (a, we obtain the original expression for Repl . %:5=1

Equation (8.14) defines the Hilbert transform of the function lP(el, e2, ea) with respect to Q. This transform is known to be injective. Further, if IP E F( -1), then 1tI is an odd homogeneous Coo function on (Ra)' \ 0 whose homogeneity degree -1 is equal to that of IP. Therefore, every derivative belongs to F( -2). Suppose that Rep = o. Then, using the integration by parts, we obtain

I£-

k::

cS«(I ZI + e2 Z2 + e3 Z3) wee)

k1tl(e) 6'«(lZl + e2Z2 + e3Za) wee) = 0, i = 1,2,3, i.e., the Radon transform of the function It; is zero. Since the Radon transform = -(i

It

is injective, it follows that = 0, i = 1, 2, 3, and hence .p is constant. Since the function 1tI is of homogeneity degree -1, we have 1tI = O. Since the transform (8.14) is injective, IP = O. As in the three-dimensional case, the inversion formulas (8.11), (8.12) for the projective Radon transform give new inversion formulas for the Radon transform related to the affine plane and for the Minkowski-Funk transform related to the t~ensional sphere. 1b obtain an inversion formula for the Minkowski-Funk transform, it su8ices to interpret elements of F(-2) and F(-I) as functions on the ~eosional sphere 52. If IP = MI is the Minkowsld-Funk transform of a function I on 52,

I. RADON TRANSFORM

38

then the inversion formula for with l~rl'P(1fr).

I

is obtained from (8.11). (8.12) by replacing

'P(~)

8.8. Radon transform for the projective space of an arbitrary dimension. The Radon transforms for p2 and p3 introduced above can be extended to projective spaces of arbitrary dimension. The Radon transform for P" is defined on the space F( -n) of Coo functions I(x) on Rn+! \ 0 satisfying the homogeneity condition

= I~I-n I(x).

I(>.x)

Similarly to the cases of p2 and p3. this transform is defined by the formula

(8.15)

'RI(~) = JP" I(x) cS«~. x» w(x).

~ E Rn + 1 \ O.

'L:::l1

where cS(t) stands for the delta function on R, (~,x) = ~.Xi' and w(~) = E:':ll( -l)i-l~i AJ~i d{j. The integral over P" should be understood as the integral over an arbitrary section ofthe bundle Rn+I\O -- P"; this integral does not depend on the choice of the section. By this definition. 'P = 'RI is a Coo function on Rn+! \ 0 satisfying the homogeneity condition 'P(~) = I~I-I 'P(~). Let us present without proof the inversion formulas for the projective Radon tra.ns. form. If 'P = 'RI, then l!jl (8.16) I(x) = (-1) f 'P(~) cS(n-l)«~ x» w(~) 2(211')n-l for odd n, and

ir

'

(8.17) for even n. The meaning of these expressions was discussed in detail for n = 2 and n = 3. Similarly to these cases, the inversion fonnula is local for odd n and nonlocal for even n. As in the case of affine Radon transform, these inversion formulas can be combined by using the distribution ( l)n-1 . (t - iO)-n = t- n + 11'1 cS(n-l)(t). (n - I)! The following inversion formula holds for any n:

(8.18)

I(x)

= (n -

1)1 (2~) n

L'P(~) «~,

x) - iO)-n w(~).

9. Radon transform on the complex afBne space The definitions and results of this chapter can be extended from the real case to the complex case. In this section we briefty present the definition of the Radon transform for the complex affine space, the relation between this transform and the Fourier transform, and the inversion formula for the Radon transform. In essence, the only distinction between the Radon transforms in the real and complex spaces is that the inversion formula for the Radon transform in the complex

9. RAOOS TRANSFORM ON THE COMPLEX AFFlNB SPACE

39

case is always local and has the same structure for the space of arbitrary dimeDSion. Therefore, we present tbe results for spaces of arbitrary dimen&i.on n > 1. The statements of t.he theorems are given without proofS.

9.1. Deflnltlon of the Radon traasfonn. We represent hyperplanes In CA by equations of the form (~. x)

-p=O,

where {~, x} ::; ~1 Xl + ~X2 + ... +~n X ... The Radon transform of a function / on C" is defined by the following formula:

'R/({,p)

(9.1) where

17

the form

= (2~r-l f

J«(.7:)=P

l(x)l7(x)/\I7(X).

is a holomorphic differential (n - I)-form on the hyperp1ane ({, x) 17 is defined by the relation

In the coordinates x., ... ,x.. , the form 17 ::;

(-1 )k-l ( ; 1 dXl

/\ ••• /\

17

= Pt

is given as follows:

dZi. /\ ... /\ dx"

It is also convenient to use the expression for 'R / of the form

(9.2)

'R/({,p) =

ar If:"

/(x)6«({, x) - p)dxdf,

where 6(·) is the delta function on C". The function 'R / satisfies the homogeneity condition ~EC\O.

(9.3)

9.2. Relation to the Fourler transform. The Fourier transform F 1 of a function / on en is defined as follows:

This transform is related to tbe Radon transform by the formula (9.4) Using the homogeneity (9.3) of 'R I. one can represent the relation between 'R. / and F / in the form

(9.5)

F/{>'{)::; (2nr" ~

I

X/(F., p)eiRe'\Pdpdj).

Hence, by the inversion formula (or the one-dimensional Fourier transform we obtain

X/CF..

p) = (2n"),,-2

~

I

F/(>.{)e- iRA!1." tU

tU.

I. RADON TRANSFORM

40

9.3. Inversion formula for the Radon transform. It is known that the operator :F- I inverse to F is

(9.6)

F-I
= (211")-n f

le..


Using relations (9.6) and (9.4), we obtain the following inversion formula for the Radon transform 'R.:

(9.7) I(z)

= (-I)n- 1 1I"-2n+2ar-1

i apna::;-l

'R./(e. ({. z»w(e) I\w(e),

wee)

where = E:=l( _1)"-1 e" d(1 1\ ••• 1\ di; A··· "d(n, and the integral is taken over an arbitrary surface r c en \ 0 intersecting once almost every complex line passing through the point O. 9.4. Case n becomes

(9.8)

I(zl. Z2)

= 2.

For n

= _11"-2 ~

i

= 2. the inversion formula for the Radon transform ('R.f):,,({lt {2; (I Zl

+ e2 Z2)

x ({I d(2 - e2 d(d 1\ (el d(2 - e2 d(d· We represent this formula in another system of coordinates on the manifold of lines. Let us define lines in 2 by equations of the form

e

(9.9)

Z2

= (Ut + {J

and take the coefficients Q, {J of these equations as local coordinates on the manifold ofUDes. Define the integral of a function I on C2 over the line (9.9) by the formula (9.10)

R./(Q. {J)

Obviously, the functions

=~

f

I(t,

Q

t + {J) dt tIl.

R./(Q, {J) and 'R./({I, {2; p) are related as follows:

'R./(el. (2; p)

= l{r2 R.1( - ~:. ~).

Substituting this expression for 'R.I into (9.8) and taking the line {2 obtain the inversion formula

(9.11)

l(zlo Z2) = _11"-2

~

fc

= 1 for r, we

(R.f):.,,(Q. Z2 - QZI) dodO.

9.&. Relation to Paley-Wlener theorems for the affine Radon transform In R2 and R3. In 4.2 we described the image of the space S(R2) of rapidly decreasing functions on the plane under the affine Radon transform (the PaleyWiener theorem). The proof was based on the relation between the Radon transform and the Fourier transform on the plane. A similar description of the image holds for the three-dimensional Radon transform. We obtain this description as a consequence of the description of the image of the projective Radon transform; Bee Theorem 8.4 in the case of projective plane p2 and Theorem 8.2 in the case of projective space p3. For simplicity. we restrict ourselves to the case of projective space. First we establish a relation between homogeneous functions I e F( -3) on R4 \ 0 and functions I e S(R3).

9. RADON TRANSFORM ON THE COMPLEX AFFINE SPACE

41

PROPOSITION 9.1. A function 10(x1.x2.x3) on R3 belong, to the BfNJC6 S(R3) i/ and only if it is obtained Q8 the restriction to the hyperp14ne x, = 1 0/ a function

I(x •• X2. X3. x,)

e .1'(-3) satisfying the lollowing condition: I vanishes lor X4 = 0 together with

Condition A. The function

all its deriva-

tives. PROOF.

tion

Let l(xl.x2.x3,x4)

I to the hyperplane X4 =

e .1'( -3) and let 10 be the restriction of the func-

1, i.e.,

10(x1o X2. X3)

= /(X1o X2, X3. 1).

Then lo(r{•• r(2. res)

= r- 3 /({l'~'es, l/r)

for any r

>0

and

lel =

1.

I satisfies Condition A, then it follows from this relation that 10 E S(R3). Conversely. let /0 E S(R3). We define the function / on 1t4 \ 0 by

If the function

= IX41-3/0(xlix4.x2/x4.x3/x4) /(Xl.X2,X3,0) = O.

/(Xt.X2,X3,X4)

Then I E .1'(-3), and the function 10 is the restriction of =0.

x,

x, -:F 0;

for

I

to the hyperplane

Note that the projective Radon transform 'RI or a function I E .1'( -3) and the affine Radon transform of the restriction 10 of f to the byperplane x, = 1 coincide. Thus, by Proposition 9.1, the description of the image, under the affine Radon transform. of the space S(R3) reduces to the description of the image, under the projective Radon transform. of the subspace of functiODs f E .1'( -3) satisfying Condition A. The answer is given by the following proposition. PROPOSITION 9.2. A function f its projective Radon transform

e .1'(-3)

,atisfies Condition A

il o.n.d only if

'R/({h~.e3,p) = Jpa ( l(x)6«{,x) + F4)W(X),

<e.x)

where = {.X1 + {2X2 + e3X3. rtJpidly decretl8e8 with rupect to P uniformly tJJith respect to { and the Cavolieri cond1tions hold, i.e., [k({) =

L:

oo

'RICe,p) pk dp,

k

Q8

P-

00

= O. 1,2•... ,

is a homogeneous polynomial 01 degree kin et.{2,{a. PROOF. Suppose that a function le .1'( -3) satisfies Condltion A. For any k. k = 0, 1,2•...• represent the product 'R/(e.p) pk in the form (9.12)

'R/(f..p)pk = (

Jpa I(z)

(_

(~'X})k cS({~,x) + p:E.)w(x). 3:,

Note that by Condition A, the function I(x) (- ~:})Ie belongs to the space .1'(-3) for any chosen Hence, the expression on the right-hand side of (9.12) is bounded with respect to p; thus. the function 'Rf(~,p) rapidly decreases with respect to P asp ..... 00.

e.

42

I. RADON TRANSFORM

Integrating (9.12) with respect to P. we obtain

1

+00 'R/(F..p)pkdp= f I(x) (_ (F.'X»)k IX4r I W(X). _~ ~3 ~

Hence. the integral r~: 'R1(F..p) ~ dp is a homogeneous polynomial of degree k in F.. Thus, the function 'R/(F..p) satisfies the hypothesis of the proposition. Conversely. assume that the Radon transform of the function 1 e F( -3) satisfies the hypothesis of the proposition. We show that in this case the function 1 satisfies Condition A, i.e.• the function and all its derivatives vanish for X4 = o. To this end, we use the inversion formula (8.6) for the projective Radon transform. where for r we take the cylindrical surface "Y : F.l + F.~ + F.l = 1 in R4. I(x)

= -(2nr21

1:

00

'R/(F.,p) 6"«F..x)

+ PX4) dp 1\ wo(F.)

for Wo(F.) = F.l df.2 A df.3 - F.2 df.1 1\ df.3 + F.a df.1 1\ df.2. Differentiating this relation with respect to XIoX2,X3,X4 and then substituting X4 = 0, we obtain

n 4 (

IJ ) •• IJXi l(x)Ia-,=o

= -(27r)-21 where k that

00

'R/(F..p) pk'F.~IF.~2F.:3 6(k+2) «F.. x» dp 1\ wo(F.),

= E ki • Integrating with respect to p, we see by the Cavalieri conditions

IT (olJ i=1

1:

(iXi

)k'/(X)Ia-,=o =

-(27r)-21pk,(F.)F.~IF.~2F.;36(k+2)«F.,x})Wo(F.). ..,

where p.. (F.) is a homogeneous polynomial of degree ~. Since the degree of the polynomial Pk,(~)~~I~:2~r is equal to k, it follows that this integral vanishes for any (Xl,X2,X3) ::F O. Thus, the function 1 satisfies Condition A.

CHAPTER 2

John Transform In this chapter we study the John transform (also called the X-ray transform).J of functions on R3 and analogs of this trBDSform for functions on the real projective space p3 and the complex space C3 • To each compactly supported or rapidly decreasing function f on R3 , the John transform assigns the integrals of this function over all possible lines in 1Il3. This gives a function 'P = .J/ on the manifold of lines. The transform .J was first studied by F. John [28), who established a close relation between this transform and the solutions of the ultrahyperboUc differential equation. As in the case of Radon transform X, we consider the following two main problems: Describe the image of.J and reconstruct a function f from its image .J/. However, the solution of these problems for .J is substantially different. The main distinction betllrren .J and 'R. (which also distinguishes .J among the transforms treated in the subsequent chapters) is that the John transform sends a function of three \1Uiables to a function of four variables (because the manifold of lines in 1R3 is of dimension four). First, this implies that a function belonging to the image of the John transform satisfies an additional diJferential relation (the Fritz John equation). Second, the problem to reconstruct a function / on R3 from the John transform .J/ is overdetennined. Namely, in general, to reconstruct the function /. it suffices to define .J/ on a tl1re&dimensional submanifold K of the four-dimensional manifold of lines H rather than on the entire manifold H. Certainly, both the existence of an inversion formula and the explicit form of this formula may substantially depend on the structure of the submanifold K. We now briefly describe the contenUi of this chapter. In § 1, the definition of the John transform of a. function on R3 is given. It is shown that one of the main special functions in analysis, namely, the Gauss hypergeometric function, naturally arises as the John transform of a. certain function on R3. The main problem of this section is to describe the image of the Schwartz space S(R3) under the John transform, and the solution of t.his problem is given in 1.3-1.6. We do not consider here the nonlocal problem of the inversion of the John trBDSform. However, we will collsider the romplcx vemoll of the problem. Following the scheme of Chapter 1. we then consider the John transform of differential forms OD as in § 2. This transform is well defined not only for 1-forms but also for the forms of degrees 2 and 3 (cf. the case of Radon transform). In each of these cases we give the description of the image of the John transform and obtain a formula reconstructing the original differential form from its John transform. In § 3, the main definitions and resulUi of § 1 are extended to the case of pr0jective space p3. The passage from the affine space to the projective one has two important advantages. First. the main definitions and results have a simpler and more invariant form in the projective case as compared to those in the affine case. 013

2. JOHN TRANSFORM

44

Second, since both the projective space p3 itself and the manifold of lines in p3 are compact, with these manifolds one can associate more natural function spaces as compared with those in the affine case. In § 3, the description of the image of the John transform is given in a projective-invariant form. A simple group-theoretic interpretation of the John transform is also presented. The last two sections of Chapter 2 are devoted to the analog of the John transform for the complex space C 3 . The merit of the complex version is that the complex case admits local inversion formulas; namely, to reconstruct a function I at a point x e C 3 from the John transform IP = .71, it suffices to know the function IP only on a set of lines infinitesimally close to x. A fundamental role here is played by the operator IC introduced by I. M. Gelfand, M. I. Graev, and Z. Ya. Shapiro (7) to solve a more general problem; to every function IP on H this operator assigns a differential (l,l)-form ICIP on the manifold of lines H. In terms of the operator IC we give a description of the image of S(C3 ) under the John transform and construct an explicit local inversion formula for this transform in §4. It turns out that the differential form ICIP, where IP = .71. is closed on the submanifold HZ C H of all lines passing through a given point x for any x E C3; the function I can be reconstructed by the foUowing inversion formula:

J.,. ICIP = c(-yZ) I(x), where r c HZ is an arbitrary cycle of real dimension two, and the factor c(r) depends on the homology class of the cycle only. Note that an analog of the operator It was introduced in § I for the real case; however, in that case the operator IC could be used only to describe the image of the John transform. In § 5 we first consider the submanifolds K of lines in C 3 that intersect an arbitrary algebraic curve A C C3. For these submanifolds we give an explicit local inversion formula reconstructing a function on C 3 from the integrals of this function over the lines in K. This example naturally leads to the foUawing problem: Describe all "good" three-dimensional submanifolds K (admissible complexes in the manifold H of all lines) for which there is an explicit local inversion formula reconstructing a function on C3 from the integrals of this function over the lines in K. The geometric structure of these admissible complexes is studied; it is shown that each admissible complex is either the manifold of lines intersecting a complex curve or the manifold of lines tangent to a complex surface.

r

1. John trBDSform in the real afBne space 1.1. John trBDSform iD R3. We present lines in R3 by parametric equations

x = ot + p,

t E R,

where {J = ({J1o f32,fJ3) is a point on the line in question and 0 is the direction vector of the line. DEFINITION. We define the integral of a function

line x

= at + (J by the formula

1

I

+00

(1.1)

1P(0,fJ)

=

-00

I(ot + {J)dt;

= (01002,03). 0 #: o.

E S(R3) over an arbitrary

I. JOHN TRAl"/SFORM IN THE REAL AFFINE SPACE

the function ';'(0, tJ) is called the John tronsform of the fuoct.1on f and is denoted by

3/.

According to thl' definition, <{) is a function on the manifold E of all pairs (a,p) e (Rs \ 0) x RS. Obviously, the function", does not depend on the choice of a point fJ on the line, i.e., 'P(a.fJ + ato) = ",(a,p)

(1.2)

for any

to e

R.

Thus, for any given 0, '" is a function on the quot.ient space R'I/{a}, where {a} is the subspace spanned by the vect-or a. It is also dear that the function '" satisfies the homogeneity condition "'(~Q,tn

(1.3)

= I~I-l ",(a, ~).

~ ~ O.

Therefore, the function 'P depends only on the line and on the admissible measure on the line defined by the vector 0, i.e., this is a function on the one-dimensional bundle over the manifold of lines. Moreover, if RS is also equipped with the Euc1idean metric, then in the definition (1.1) one can assume that 101 1; in this case, 'P bec:omes a function on the manifold of lines ibielC.

=

REMARK. Conditions

(1.2) and (1.3) are equivalent to the following system of

equations:

",(-0,13) = ",(0,13).

(1.4) Sometimes

\\'C

shall present lines in R3 by equations Xl

= al x 3 + iJ1.

1:

and tbe John traosfonn by the formula (1.5)

"'(al,a2.~,iJ:a)=

00

f(alX3+fJl,a2X3+.82,Xa)dxa.

It is clear that the functions t/J and 'P, where the latter is defined by (1.1), are related as follows:

1.2. John tnmsform and the Gausa bypergeometric function. The integrals (1.1) and (1.5) define t.he John transform not only for rapidly decreasing functions / e S(RlS) but also for any functions / on R3 for whicb these lutegraIs

converge or can be understood in some 8eose. In particular, the John traosform tP" of the following function on 1t3 is well defined: /~(Xl,Z2,X3)

= (Xl)~I-l (Z2)~2-1 (X3)~-1.

where t~ = t" for t > 0 and t~ = 0 for t < 0; the ~,'s stand for arbitrary complex numbers. For Re ~ > 0 the function ~ is defined 88 a classical funttion; it extends to a distribution meromorpbic in ~ (see (10]). We establish a relation between this Jobn transform and the Gauss bypergeometric function F(a.b,c; z).

46

2. JOHN TRANSFORM

By definition, the function "'). is given by the integral "').(010 02, Plo.82) =

100 (01X3 + Pd!I-1 (02X3 + ~)!2-1 X~3-1 dx

3•

The integral converges if Re ~i > O. i = 1,2,3, and Re (~1 + ~2 + ~3) < 2. and in this domain the integral defines an analytic function of ~1' ~2' ~3' Therefore, the integral can be defined for any values of the ~i 's 88 the analytic continuation of this function with respect to the ~i 's. Note that the interval on which the integrand is nonzero depends on the signs of the numbers 0i and Pi, i = 1,2. To be definite, consider the domain on which 0i < 0 < Pi, i = 1,2, in the manifold of lines H. In this domain. changing the variable X3 = t, we can reduce the integral to the form

-!!

1/)).(01002,P.. .82)

= p~I-1 P:2+).:s-11 0 2r).:s

1

1(1- xt)!I-l (1 - t»).2- 1 t).:s-1 dt,

:!t.

where x = Compare this expression with the weU-known representation of the Gauss hypergeometric function F( a, b, e; x) for Ixl < 1 88 the EuJer integral. F(a,b,c; x)

= r(b)~~~ _ b)

11

t"-1 (1- W-"-1 (1- xt)-a dt;

this integral converges for Re c > Re b > 0, and for the other values of b and e it must be treated 88 the analytic continuation with respect to b and e. We conclude that, for 0i < 0 < Pi, i = 1,2, and < 1. the function 1/)). can be expressed in terms of the hypergeometric ftmction,

:!ta

"').(0., 02, PI,.82)

= f(~2) f(~3) r(~2 +~)

a).I-1 a).2+).,-11 1-).3 F (_~ + 1 ~ ~ + ~ . 0 1.82) "'2 02 I , 3· 2 3. 02Pl .

"'I

This implies that F(a.b,e;x)

f(e)

= f(b)r(c-b) 1/)).(-x,-I,I,I)

for O<x< I,

where ~ = (1 - a, c - b, b). Similar expressions for the function 1/)). in terms of the hypergeometric hypergeometric function can be obtained for other combinations of the signs of 0i and Pi, i = 1.2.

1.3. Theorem OD the image of the operator.1. This subsection, together with the next three, is devoted to the description of the image of the space S(R3) of Schwartz functions under the John transform .1. First, it is clear that the functions in the image of the space S(R3) satisfy conditions of smoothness and rapid decreasing. FUrther, since the John transform acts from a space of functions of three variables to a space of functions of four variables, one can naturally expect that the functions in the image satisfy additional conditions.

1. JOHN TRANSFORM IN THE REAL AFFINE SPACE

47

We prove that each function I;' in the image of the John transform satisfies the foUowing system of differential equations:

( 6) 1.

CP'(J 80" 8/Jj

CPrp

..

= 80j 1}{3. '

I,}

123

= , ..

Indeed, if I;' = .1/. then it follows from (1.1) that 1}2rp(a, P) =

aa. apj

j+«> t I" -00

(at + IJ) dt

z • .zj

•

a!.'&fJ,

Thus. the function is symmetric with respect to the transposition of t.he indices i and j. We claim that relations (1.5) are not only necessary but also suftlcient for a function '" to belong to the image of the space S(J!l3). Let us proceed with detailed definitions and statements. Denote by H the manifold of lines in a 3 and by G ~ p2 the manifold of one-dimensional subspaces in 3 , i.e., the submanifold of lines passing through the point O. The manifold H is equipped with tJre structure of a canonical vector bundle over G. Namely, the projection 11' : H - G takes every line h eH to the parallel subspace a e G. Thus, the fiOO of the bundle 11' over a is the quotient space Ha =R3/a . Since H is a vector bundle with compact base space, it follows that the notion of rapidly decreasing function OIl S is \\-eJl defined, namely, a function on H is cal1ed mpidly decreasing if this function rapidly decreases on each liOO of the bundle

a.

H-G. DEFINITION. Denote by S(H) the space of functions rp(a, (1) on E satis£ying conditioDS (1.2) and (1.3) and such that the function ",(a,p), 10'1 = I, regarded as a function 00 H, is infinitely differentiable and rapidly decrea5ing together with aD its derivatives.

It immediately follows from the definition of the operator .1 that the image of the space S(J!l3) under the John transform belongs to S(H),

.1: S(R3) - S(H). THEOREM 1.1. A /unction r.p e S(H) belongs to the image of the space S(23) under the John tmnsfcmn .1. I.e., is Tl!p1'esentable in the fcmn 'P = .1/, where / e S(IR3), if and only if this function 8atisfies system (1.6).

The proof is given in 1.4-1.6. REMARK. It is interesting to compare this desttiptioo of the image of the John translform with the description of the image of the Radon tnmsform given in Cbapter 1. In the latter case, the main component in the description of the image is the nonloc:al Cavalieri conditions. whereas in the former case, there are only local conditions (1.6). Of coun;e. it is p06Sible to write down analogs of Cavalieri's conditions for thE' John transifrom, but they wiU be consequences of the differential equations (1.6).

We now present another equivalellt formulation of the image theorem. Introduce the operator" which takes e\.'ery smooth function ",(a, {3) on E to the following

2. JOHN TRANSFORM

differential I-form on E: ~

8~

It~ = 8{J1 do.

(1.7)

8~

+ 8fJ2 do2 + 8133 do 3 •

LEMMA. If ~ e S(H), then It~ con be pushed down from E to the manifold ii of oriented lines in R3.

Indeed, it follows from (1.2) and (1.3) that the form It~ is invariant with respect to the transformations {J - {J + ato and a - .\a, .\ > o. Note that the form ~ changes the sign if the orientation of the line alternates (a- -a). For any point x e R3, denote by HZ the submanifold of lines p88Sing through x, and by Itz~ the restriction of the form Itz~ to HZ. PROPOSITION 1.1. Condition (1.5) is equivalent to the condition that the differential form Itz~ on the submanifold HZ is closed for any x e R3, i.e., to the condition dttz~ = O.

PROOF. We have

_

Itz~ -

~ 8~(a,{J) IJla.

L-

~.

i=1

Hence, the exterior differential of the form

dltz~

~ 1P~(a,{J) = L80.8/3. iJ I,

I_

l3-z

=~(IP~(a,{J) _

f<1

Ba; BP,

I_

..1_. UUI'

/J-z

Itz~

is given by

doi A do)

IP~(Q,{J») 80., BPi

I

doi Ado,.

(J-z

This immediately implies the assertion. By Proposition 1.1, Theorem 1.1 admits the following equivalent formulation. THEOREM 1.2. A function ~ e S(H) belongs to the image of the space S(R3) under the John transform :J if and only if the differential form Itz~ on the submanifold HZ c H is closed for any point x e R3.

If the John transform is given in the local coordinates a), a2, by (1.5), then (1.6) is equivalent to a single equation for the function

REMARK.

fJ2 on H

1Pt/J

(1.8)

8o.8fJ2

P•• t/J,

IP!/J = 8a28{J••

and the operator It is given in the local coordinates by the following formula: (1.9)

8tP

8tP

K!/J = 8{J. do. + BfJ2 do2.

1.4. Space S(H'). This subsection, together with the next two, is devoted to the proof of the main theorem concerning the image of the space S(R3). The reader may omit them and immediately proceed to 1.7. To describe the image of the space S(R3) under the John transform, we first p88S (by using the Fourier transform) from the functions ~ e S(H) to functions on the bundle H' - G dual to the bundle H - G.

1. JOHN TRANSFORM IN THE REAL AFFINE SPACE

49

Note that the fiOOr of the bundle H' - G over a E G, i.e., the space H~ dual to R3/a , is canonically embedded in (R3), as the subspace of vectors orthogonal to a. Thus, the elements of H' are the pairs (a,e), where a E G and is a vector in (R3), orthogonal to a.

e

DEFINITION. Denote by S(H') the space of Schwartz functions on H', where H' - G is the bundle dual to the bundle H - G.

Introduce the Fourier transform Fo of a function ",(0, P) E S(H) along the fibers of the bundle H - G,

=..!.. (

",(0, P) ei (II.() daP, (E (R3/{0})', 211" JRa/{a} where {o} is the one-dimensional subspace spanned by the vector 0, and the volume element daP on R3! {o} is determined by the volume element dx = dx1 dx2 dx3 on R3 and the vector 0, i.e., (Fo"')(o,()

( 1+

00

JRa/{a}

I(ot + P)dtdaP = ( I(z)d:& JRa

-00

for any function I E S(R3). It follows from the definition of Fo and from (1.3) that Fo"'(~o,() = Fo"'(o,() for any ~ '" 0, and hence Fo'P is a function on H'. Recall that the Fourier transform F (of the functions on R3) given by

(FJ)(e)

= (211")-3/2

( l(z)ei«(·:Z:) d:&

JR

3

defines an isomorphism of the Scbwartz spaces

F: S(R3) _ S«R3)'). Similarly, the next assertion readily follows from the properties of the ordinary Fourier transform. PROPOSITION 1.2. II'P E S(H), then Fo'P E S(H'), and the mapping 'P - Fo'" defines an isomorphism 01 the Schwartz spares

Fo : S(H) - S(H'). We state two important properties of the space S(H') in the form of separate propositions. The first property is obvious, whereas the second needs a proof, which is omitted here. PROPOSITION 1.3.

11 F

E S(R3), then the function '" on H' given fly the

relation 'P(a,~) = F(~) lor any (a,~) e H' belongs to S(H'). Thus, we have a natural embedding S«R3)') ...... S(H').

PROPOSITION 1.4.

Let a function 'P E S(H') be such that ",(a, e)

lor a function

= F(e)

for any e::f: 0,

F E S(R3). Then the function ",(0,0) does not depend on the suba, and if we set F(O) = ",(a,O), then the resulting function F on the entire space (R3 ), belongs to S«R3)').

8J1(I.(%

2.

50

JOHN TRANSFORM

1.S. Description of the image of S(R3) in the space S(H'). We now find the image of S(R3) in S(H') under the composition mapping

L

Fo.7: S(R 3 )

S(H) ~ S(H').

PROPOSITION 1.5. For any function I E S(R3) we have (1.10)

Fo.71

= (21r)1/2 FI.

where FI is the Fourier translorm 01 the function I E S(R3). and thus FI is an element 01 the subspoce S«R3)') C S(H').

PROOF. Let (a.~) E H' and let

0

:f: 0 be an arbitrary vector

in the subspace

aE G. Then

(Fo.7f)(o.~) = 21 f

1r i R3/G

= -1

la

.71(o.p)ei(fj·~) daP

2'11" Ra/{a}

1+

00

I(ot

+ P) ei{fj.~) dtdaP

-00

=..!... f I(x) ei{z.~) dx = (21r)1/2(F/He). 21r iRS COROLLARY. The image 01 S(R3) under the mapping Fo.7 is the subspace S«R 3 )') C S(H'). Indeed, it follows from Proposition 1.5 that the image of S(1ll3) under the mapping Fo.7 is contained in S«R3)'). Conversely. if E S«R3),). then = FI. where I E S(R3). Therefore. Fo.71 (2'11")1/2FI (2'11")1/2j.

=

=

1

1

REMARK. Relation (1.10) implies an inversion formula for the John transform. Namely, to reconstruct I. one must apply the inverse Fourier transform on R3 to the function Fo.7I, i.e., 1= (F- 1Fo).1f.

1.8. Proof of Theorem 1.1 on the image of the John transform. Let ~ = Fol.{). To prove Theorem 1.1. is suffices to establish the following

IP E S(H) and assertion.

PROPOSITION 1.6. 11 a function I.{) satisfies (1.6), then ~(a,e) = F(~) lor any vector :f: 0, i. e., the function ~(a, does not depend on the subspace a orthogonal to il :f: o.

e

e e

e)

Indeed, it follows from Proposition 1.4 that, in this case. the function ~ = .7otp belongs to the subspace S«R3)') C S(H'). Then ~ = Fo.71 for some function f E S(R3). Thus, tp = .71. PROOF OF PROPOSITION 1.6. Let a be a one-dimensional subspace and let be a nonzero vector in a. Denote by~, i = 1,2,3, the family of subspaces a for which 0i :f: O. Since the ~ form a covering of G. it suffices to prove the assertion for each ~ separately. To be definite, consider V3 • We present every subspace a E V3 by a vector of the form 0 = (01,02. 1) and take the coordinates 01.02 of this vector as the coordinates on V3 • Moreover. it is clear that on every line in R3 parallel to a there is a point P = (PI, P2, 13:J) such that I3:J = O. Take the coordinates PI, tJ.z of this point as the coordinates on R3 / a. 0 = (01.02,03)

51

I. JOHN TRANSFORM IN THE REAL AFFINE SPACE

In this system of coordinates the measure on R3/0 becomes daP therefore

~(O,~) =

1: 1: 00

00

= dl3t d/J2. and

t/J(01o 02. PI./J2) e i (1J1El+83E2) dPI di32.

where t/J(0J,02,PI.fJ:z) stands for cp(01,02.I.PI,.82,0) and the pair 01,02 and the triple ~ = (~lt~2'~) are related as follows: 01~1

(1.11)

+ 02~2 + ~3 = 0.

By virtue of (1.11) we can take~" ~2 for the coordinates on H~ = (R3/a )'. Mol'& (~1'~2) -F 0. By the assumption of the theorem, the function t/J satisfies the equation

over, if ~ -F 0. then

lP t/J _ (flt/J 80. 18132 - 8028P1 . Hence, the Fourier transform of t/J with respect to PI, /J2 satisfies the equation I)p

~2 801 For

(~1.~2)

8rp

= ~180.2 .

:F 0, the general solution of this equation is rp(0,,02'~"~2) = F(~lt~2. -(01~1

+ 02~2»'

or, by (1.11), rp(01.02'~)

Thus. for any

chosen~, ~

= F(~1o~2,~3).

-F 0, the function ~(a.~) does not depend on a.

1.T. Analogs of the operator K. In 1.3 we have introduced a linear operator from the space 5(H) to the space {l1(H) of differentiall-fonns on the manifold of lines H, and this operator has the following important property. K

CONDITION A. If tp = .1/, then the differential form KIP is closed on the submanifolds HZ of the lines passing through an arbitrarily chosen point z e R3.

We pose the following problem: Describe all linear operators K having this property. In the statement of Proposition 1.7 we refine the setting of the problem by restricting the class of operators under consideration. Suppose that the lines in R3 are given by the equations

and the John transform 1/1 = .1/ is defined in the local coordinates Oh 02, PI, fJ:z on H by formula (1.5). By 1.3, the function t/J satisfies the equation

1P1/I 8018132

lP1/I = 8028PI ,

and the operator " is given by the formula (1.12)

2. JOHN TRANSFORM

52

One can readily see that, along with satisfy condition A:

I =

K

= K1, the following three operators also

=

Indeed, since d{3i Ha -X3 OOi, i 1,2, on HZ, it follows that the forms K2'" and K4'" are proportional on HZ to Kl'" and K3"', respectively. Moreover, K2tP + K3tP = d",. Hence, each ofthe forms Ki'" is cohomology equivalent on HZ (up to a factor) to the form Kl"" Consider the operators K of the form

8t/J

(1.13)

K'"

8t/J

8t/J

8t/J

= 001 WI + 002 W2 + O{JI W3 + O{Jl W4,

where Wi is an arbitrary I-form on H. PROPOSITION 1.7. Every opemtor KO/ the /ann (1.13) satisfying condition A is a linear combination with constant coefficients 0/ the opemtors "i, i = 1.2,3,4. PROOF. For convenience, we shall write spectively. Then (1.13) becomes

8t/J

4

KW =

Ea: i=l

Wi

A.

=

E

04

instead of (J1 and /32, re-

4iJ(0)

dot j

i

,

= 1.2,3,4.

j=l

I

Wi

(1.14) Wl

and

4

where

Wi,

It follows from the condition d(K"') H.= differential forms Wj satisfy the relations

(1.15)

03

1\

OOi = 0,

= 0, dwi = 0,

i = 1,2,3,4,

W3 1\004 +W4 1\003 = 0,

1\002 +W2 1\001

(1.16)

°that on each submanifold HZ the

i

= 1,2.3,4.

It follows from (1.14) that W1

= 4n 001

W3 =

a31 001

+ a13 003,

W2 = a22 002

+ a33 003,

W4

+ a24 do4,

= a42 002 + a44 004.

Substituting the resulting expressions for Wi in (1.15), we obtain all

= a22,

This implies that K

= all IC4 + a33 K2 + a31 K1 + a13K3'

Finally, substituting the last expression into (1.16), we conclude that all coefficients aij in this expression are constant.

2. JOHN TRA.~SFORM OF DIFFERENTIAL FOR..'dS ON It'

2. John transform of differential forms OD RS 2.1. DeflDitiOD of the JOhD trBDSform of di8'erential forms. Here we coWlider integral tra.osforms that take a differential form OD Jt3 to the integrals of this form over the lines in RS. These transforms are defined Dot only for diJferentlal forms of degree 1 but also for difFerential forms of degrees 2 and 3. In this ca&e we obtain differential forms (00 the manifold of lines) whose degree is less by one. Similar transforms of differential forms on R3 related to planes in RS were treated in § 7 of Chapter 1. Although the definition of these transforms can be given in an iovariant form, here it is more convenient to connect them with the choice of a specific local system of coordinates on the manifold of lines in RS. Let us present lines in RS by the equations (2.1)

and take the coefficients 0;, Pi of these equations for the local coordinates OD the manifold of lines. Introduce the four-dimensional flag manifold A whose elements are pairs (h,x), where h is an arbitrary line in RS and % is an arbitrary point of Jl3 belonging to the line h. We equip the manifold A with two systems of coordinates. The first system is (Xl, %2. Xs, 01. 02), where (Xl. X2. XS) are coordinates of a point in R3 aDd (al. 02) define the direction of a line passing through this point. The other system is (Oh02.Pl.P2.XS). where (01,Q2,{31,P2) are the local coordinates of a line aDd X3 is the coordinate of a point on this line. These two systems of coordinates are related by (2.1). We now define the John transform of a differential Corm w on RS of degree r ~ 1. Wc regard w as a differential form on A given in the coordinate system (Xl' X2. X3. 01, 02), then pass to the other system (OhQ'2.~lt.82,X3) and omit the terms not containing dx3 • The resulting expression is a differential I-form on the line with coordinates oi,A. i = 1,2, for any chosen wines 0 ... A. do•• df3•. lntegrating this I-form along the line, we obtain a differential (r - I)-form on the manifold of lines. We denote this form by .1w and call it the John trrm8/orm of the original differential form w. The above definition is convenient for calculations, but it is related to a specific coordinate system in R3 and in the manifold of lines H. There is an invariant definition of the John transform of differential forms in terms of operations of pullback and pushdowo. Namely, two operations on differential forms are defined for any smooth fibration 'If: A .... B, the pullback 'If* and the pU8hdown 'If... The fonner takes tilt> differential forms on B to dift'erential forms of the same degree on A; the latter (the i.ntegration over the fibers of the bundle) takes the di8immtiaI forms of degree r on A to forms of degree r - l on B, where I is the dimension of the fiber of the bundle A .... B (it is assumed that r ~ l). U a pair of librations A - B and A .... C (double 6bration) is given. then one can assign to this pair an integral transform

.1 : O(B) - O(C) from the space of differential forms on B to the space of differential forms on C. This form is defined 8S the composition :J = 'If.. 'If. of the puUback 'If. taking the forms 00 B to those OD A and the pushdown 'If'* taking the forms on A to tba6e on

2. JOHN TRANSFORM

C. It turns out that the operator .1 thus defined commutes with the operator of exterior differentiation d:

d.1 = :Id. The John transform .1 : n(R3) - n(H) is an example of such an operation related to the double fibration A - R3 and A - H, where A is the flag space. The property d.1 = .1 d of this operation can be verified by the immediate calculation. 2.2. John transform of 3-forms on R3. Let a differential 3-form w be given, w

= f(x) UI A U2 A U3,

f E S(R3 ).

In order to construct the John transform of w, one must first represent this form in the coordinates (01,02,PI,Pl,X3) on A: w

= f(ol X3 + PI, 02 X3 + 132, X3) (X3 001 + dPl) A (X3002 + dP2) A dz3.

According to the definition in 2.1, the John transform of the form w is defined as follows: (2.2)

.1w

= !P2(0, P) 001 A 002 + !PI (a, P) (001 A dP2 -

where (2.3)

!p,(o, P)

=

1:

00

f(ol X3

002 A dPI)

+ "'0 ( a, P) dPI A dP2,

+ p" 02 X3 + Pl. X3) (X3)' U3.

i

= 0, 1,2.

We study the properties ofthe resulting form. First. since f E S(R3 ). it follows that the coefficients 'P, of .1w are rapidly decreasing together with aD their derivatives. PRoposmON

2.1. The differential form

.1w is closed.

Indeed, it follows from the expressions (2.3) for the coefficients 'Pi of the form .1w that (2.4)

a!Pl

ap.

= a!po

ao,'

.=1 2

t

••

These relations are equivalent to the condition that the form .1w is closed. REMARK. There is another way to prove that the form.1w is closed. Namely. the original form w is closed as a form of maximum degree on R3. and the transform .1 commutes with the operator of exterior differentiation. PROPOSITION 2.2. The restriction of the differential form .1w to the submanifold of lines belonging to an arbitrarily fixed plane vanishes. PROOF. The submanifold P of lines belonging to a given plane is defined by the following system of equations in the local coordinates (a. P):

.\1 (01 - o~) + .\2 (02 - o~) = 0, .\1 (PI - /Il) + .\2 (132 - fil) = O. o Here the pair (a • {1J) stands for the coordinates of an arbitrarily chosen line on the plane and .\1 and .\2 are constant. Hence, on this submanifold P we have 002 = .\001 and dPl = .\dPt. where.\ = -.\1/.\2, and thus it follows from (2.2) that :lw Ip= o.

2. JOHN TRANSFORM OF DIFFERENTIAL FORMS ON

aa

THEOREM 2.1. A differential2-form n with rapidly decreasing coefficients on the manifold of lines H is the John transform of a 3-form w f(z) dz 1"dz2" dz 3, f e S(R3), if and only if the form n satisfies the conditions of Propositions 2.1 and 2.2.

=

PROOF. The necessity of the conditions is already established. We prove the sufficiency. Suppose a differential 2-form n satisfies the conditions of Propositions 2.1 and 2.2. We first prove that this form can be represented as in (2.2), i.e., that n contains no terms with OOi "d/3i, i = 1,2, and the coefficients of 001" d/J:z and 002 "dPl differ in sign only. Indeed, let

n = aldo:l "dPI + a2 002" d/J:z + hi 001 "d/J:z + b:z 002" dPI + ... , where the dots stand for the remaining terms. If hO is an arbitrary line and P is the manifold of lines in an arbitrary plane passing through hO, then

n Ip= (al + >..2 a2 + >"(hl +~» 001" dPl (see the proof of Proposition 2.2). Since n Ip= 0 by assumption and >.. is arbitrary, it follows that al = a2 = 0 and hi = -~. Thus, we have proved that the form n can be represented as

n = W:i(o, P) 001 "002 + "'I (0, P) (do:l "dl12 Since

0021\ dpt) + t/Jo(o, P) dPl 1\ dl12.

n is closed, the functions "'i are related as follows: i = 1,2.

(2.5) Therefore, the function t/Jo satisfies the equation

fPt/Jo 801112

=8

fPt/Jo 2 8 PI .

Q

Hence, by the theorem on the image of the John transform (see § 1), this function is the John transform of some function f e S(R3): 7/10(0, P)

=

i:

f(ol Z3

+ Plo 02 Z3 + /J:z, Z3) dz3.

We now prove that n = .1w, where w = f(z)dzl 1\ dz2 1\ dz 3 • Indeed, it immediately follows from formulas (2.3) defining the coefficients l{Ji of .1w that t/Jo = 1{Jo. Further, these formulas, together with (2.4) and (2.5), imply that ~

Ii,

i = 1,2. Hence, since "'I and 'PI are rapidly decreasing, it follows that "'1 We can similarly see that W:i = 1{J2, and hence n = .1w.

=

= I{Jl'

Note that the form w = f(z) dz1 1\ dz2 1\ dz3 can be reconstructed from only one coefficient I{Jo of the form .1w because the function 'Po is the John transform of the function f.

2. JOHN TRANSFORM

2.3. John tr8DSfonn of 2-forms OD Ra. Consider an arbitrary differential 2-form w

= hex) dx2" dx3 + hex) dxa "dx. + h(x) dx. "dx2

with coefficients in SeRa). In the coordinates 0.,02, fJh 132, X3 on the flag space this form can be represented as

w = h (X3 do2 + dfJ2)" dxa + 12 dxa " (X3 do. + dlh) + fa (01 dx3 + Xa dol + dl3d" (02 dxa + X3 do2 + dfJ2). According to the definition in 2.1, the John transform of the 2-form w is the I-form

:Iw = al dol + a2 do2 + bl d131 + b, dfJ2

(2.6)

with the coefficients al = 021/Js -

(2.7)

= -0. 1/Js + '!/Jl' b, = -01 'Pa + 'Ph

t/I-z.

a2

61 = 02 'Pa - 'P2,

where

i = 1,2,3,

(2.8) (2.9)

i

= 1,2,3.

Consider first the problem of reconstructing the form w from its John transform

:lw. This problem can be reduced to the corresponding problem for functions

by

using the following assertion. PROPOSITION

2.3. The functions 'Pi are e:rpretlsed in terms of the coefficients

01 the lorm :Iw as lollows: (2.10)

lJ~ lJo l 'Pa = - lJ0 2 lJfJ2 '

'PI

= 01'P3 + b"

'P2

= 02'P3 -

6•.

Relations (2.10) immediately follow from (2.7)-(2.9). Since the functions 'Pi are the John transforms of the coefficients I. of w. it suffices to apply the inversion formula for the John transform of functions to the 'Pi'S to reconstruct the coefficients li. We now proceed with the description of the image of the John transform :I. To this end, we first establish relations for the coefficients of the differential form

:lw. PROPOSITION 2.4. The coefficients ai, lollowing relations:

~

01 the differentiallorm :Iw satisfy the

(2.11) and

(2.12)

l _ a(lJa lJfJ2

Dbl ) _

lJ02 -

0

,

where

a = lJo)lJ{3, lJ2 -

lJ2

lJ02 lJfJ) .

Indeed, relations (2.11) immediately follow from (2.7), (2.8), and (2.9). Relation (2.12) follows from the above equation 'Ps = ~ - ~ because a'P3 = O.

,. JOHN TRANSFORM OF DIFFERENTIAL FORMS ON

57

R3

PROPOSITION 2.5. Relations (2.11) are equivalent to the condition that the restriction .1w Ip 0/ the differential/onn .1w to the sulnnani/old o/line8 that all belong to a plane P is a closed/onn on P, i.e., d.1w Ip= o. PROOF. We have

d.1w = (8b 1 _ 8a 1) do 1 1\ d/31 + (8~ _ 8a l ) dol 1\ df32 8al

8/31 8a l 8132 bt _ 8a,) do, 1\ d/Jl + (~ _ 8a,) do, 1\ df32 + .... 8a, 8/31 8a, 8132

+ (8

Therefore, relations (2.11) are equivalent to the condition that the form d.1w contains no terms with do i I\d/3i' i 1.2. and the coefficients of do l l\dfJ2 and do,l\dPl differ in the sign only. In turn, according to 2.2, this condition is equivalent to the one formulated in the proposition.

=

THEOREM 2.2. Suppose 0 is a differentiall-/onn with infinitely differentiable rapidly decreasing coefficients on the manifold o/line8 in R3,

o = a1 do 1 + a2 do, + b1d/31 + b, df32. For the fonn 0 to be the John tmnsfonn 0/ a differential 2-/onn w on R3 with coefficients in S(R3). it is necasary and sufficient that its restriction Olp to submanifolds Po/lines that belong to the same plane be closed /onns for any plane and that the coefficients % satisfy the additional condition (2.12). PROOF. We have already proved the necessity. Let us prove the sufficiency. Suppose a differential form 0 satisfies the hypothesis of the theorem. Then the coefficients of this form satisfy conditions (2.11) and (2.12). Define functions !Pi. i 1,2,3, by (2.10). It follows from (2.12) that 61{)3 O. We prove that also 6IPl = 0 and 6!p2 o. Indeed,

=

6!p2

=

=

= a, 6IP3 - :~ -

6bl

8 ( 8b1 80 1 ) Blb1 Blbl 8a2 - 8132 - 8a18f32 + 8a,8/31

= - 8PI

!!.... (8a 1 _ 8b1 ) = 8132 8/31 8al One can similarly prove that 6IPl = O. =

o.

This implies (see § 1) that the functions IPi are the John transforms of some functions /i E S(R3), i.e., the functions !Pi are given by (2.8). Introduce the differential form w = h(x)dx2 Adx3

We prove that 0

+ h(x)dx3 Adx1 + la(x)dx 1 I\dx,.

= .1w. Indeed, .1w = 41 dol + 4, do, + b1 dPl + ~ df32,

where

bl

= a2 IP3 -

and the functions !Pi and w by (2.8) and (2.9).

tPi

!p2,

b, = -al !P3 + IPI'

are expressed in terms of the coefficients of the form

2. JOHN TRANSFORM

58

By Proposition 2.3,

861 - -Bal = -80 2 B/J2'

= 02 'P3 - b- I . Comparing (2.8) and (2.13), we conclude that bi = bi, i = 1,2. Hence, compar-

(2.13)

'1'3

'1'1

= 01 '1'3 +~.

'P2

ing relation (2.11) for 0i and bi with the similar relations for lit and bi , we see that = i,j = 1,2. Since the functions a. and ~ are rapidly decreasing, we have Oi, i = 1,2, and therefore {} = .1w.

I;-

Bt,

a. =

2.4. John tr8D8form of I-forms on R3. Consider a differential I-form on

By definition, the John transform 'I' manifold of lines in R3:

;p(o,P)

= .1w of "" is the following function on the

= L:OQ [h(ol %3 + PI. 02%3 + /J2. %3)01 + 12(01 %3 + Plo 02 %3 + fh, %3) 02 + 1a(01 %3 + PI. 02 %3 + /J2. %3)J dx 3 •

We find the kernel and the image of the transform .1. PROPOSITION 2.6. The kernel Ker.1 of the John transform coincides with the subspace of all exact differentiGll-forms on R3 with coefficients in S(R3). PROOF. Obviously, each exact I-form W on R3 belongs to Ker.1. Conversely, let w e Ker.1, i.e., .1w = O. Then .1 dw = d.1w = O. Since the John transform .1 has zero kernel on the differential 2-fornlS. it follows from the relation .1 dw = 0 that dw = 0, i.e., the form w is closed. Since the form w is closed, it is exact. PROPOSITION 2.7. A function 'I' in the Schwartz space on the manifold of lines H belongs to the image Im.1 of the John transfoma of l-fofT1&8 if and only if the differential dip belongs to the image of the John transform of differential 2-fofT1&8 onH.

PROOF. Let w e Im.1, i.e., 'I' = .1wl, where WI is a I-form with coefficients in S(R3). Then dip = d.1Wl = .1dwlo i.e., dip e Im.1. Conversely. let dip e Im.1, i.e., dip = .1W2, where W2 is a 2-form on R3 with coefficients in S(R3). Then .1~ = d..11M2 = 0, and since the John transform has zero kernel on the a-forms in R3, it follows that ~ = 0, i.e., the form W2 is closed. Since each closed 2-form on R3 with coefficients in S(R3) is exact, it follows that W2 = dwlo where Wl is a I-form on R3 with coefficients in S(R3). Thus, dip = .1dw I = d.1WI, and hence d ('I' - .1WI) = o. Therefore, 'I' = .1WI • THEOREM 2.3. A function 'I' in the Schwarlz space on H is the John transform of a I-foma on R3 if and only if this function satisfies the equation

(2.14)

2

~ cp = 0,

where ~

IJ2

= IJolfh -

IJ2

1Jo2PI·

PROOF. It follows from Proposition 2.7 and the description of the image for the John transform of differential 2-fomlS (see 2.3) that 'I' e Im.1 if and only if the coefficients aj = and bi = i = 1,2, of the differential form dip

8£

Gf"

69

3. JOHN TRANSFORM IN apS

satisfy relations (2.11) and (2.12). Obviously. relations (2.11) hold identically for the differential form dJ.p. and then (2.12) coincides with (2.14).

3. John transform in the three-dimensional real projective space In this section. the main definitions and results of § 1 are extended to t.he case of projective space p3. The passage from the affine space to the projective space has two important advantages. First, the main definitions and results in the projective case have a simpler and more invariant form as compared with those in the affine (''&se. Second, since the projective space p3 itself and the manifold of lines in ps are compact, the function spaces related to these manifolds are more uatW'8l as compared with those in the affine case. We also give a group-theoretic interpretation of the John transform.

3.1. Manifold of lines in PS. Let us define points in pS by homogeneous coordinates X1>X2.X3,X4. i.e., by vectors:E E R" \ 0 defined up to a oonzero f~ tor. In other words, the points of the space p3 are interpreted as one-dimensional subspaces of R4. To the set of points on a line there corresponds the set of onedimensional subspaces in R4 belonging to some ~imensional 8Ubspace. Thus, the lines in p3 are interpreted as two-dimensional subspaces of 4 , and hence the manifold of lines in ps is identified with the Grassmannian G2,4 of two-dimensional subspaces in 1Il4. We introduce a coordinate system on the Grassmannian G2,4. Let h E G2,4 and let. % = (%I,%2,X3,X4). Y = (Yt.Y2,1I3,Y4) be an arbitrary basis in the subspace h.

a

DEFINITION. By the Plucker COOrdifUltes of a two-dimensional subspace h in R" one means the coordinates of the bivector :E A 'g, i.e., the follcn\ing number set:

Pi;

= Xi Y; -

X; 'U.,

i,j = 1,2,3,4,

i ::f:.j.

Under any change of a basis in h, in which case the vectors % and '11 are replaced by some linear combinations of them, all PlUcker coordinates are multiplied by the same number. Thus, the Pliicker coordinates of a subspace h are defined. uniquely up to a factor. Conversely. if the Pliicker coordinates of two subspaces are proportional, then these subspaces coincide. Thus, each two-dimensional subspace of R4 is uniquely determined by its PlUcker coordinates, and they can be taken as homogeneous coordinates on tbe Grassmanni8D G2,4' Since Pli = -Pji. it foUows that wc can restrict ounlelves to the six coordinates Pi;. i < j. The Pliicker coordinates Pij satisfy the quadratic relation

(3.1) PI2P3-1- P13P24 + P14P23 = O. This relation can be obtained, for instance, by expanding the following determJnaot of order four with respect to the first two rows: Xl

X2

X3

:E4

111

'Y2

113

11..

%1

:E2

%3

:E4

= O.

111 112 Y3 Y4 We prove that, conversely, if some numbers Pi;. i::f:. j, are not simnltaneously zero and satisfy (3.1), then these numbers are the PlUcker coordinates of a subspace h E G2,4. Indeed, to be definite, let P12 ::f:. 0; then one can assume that P12 = 1.

2. JOHN TRANSFORM

60

Consider the subspace spanned by the vectors x = (1.0,-P23.-P24) and y = (0,I,P13,P14)' Using relation (3.1), one can readily see that the Plucker coordinates of this subspace are equal to Pij' Thus, there is an embedding G2,4 ....

pS

that assigns, to each two-dimensional subspace with Plucker coordinates Pij' i < i, a point of the projective space pS with the homogeneous coordinates Pi]' The image of G2,4 under this embedding is the quadratic surface (3.1) in pS. 3.2. John transform in p3. The first problem that arises when passing from the affine space to the projective one is to choose a function space on which the John transform :I must be constructed. As in the case of Radon transform, spaces of functions on p3 are not convenient for this purpose. The natural object on which the John transform will be defined is the space F( -2) of infinitely differentiable functions on R4 \ 0, i.e., on a one-dimensionaJ bundle over p3, that satisfy the following homogeneity condition: (3.2)

J(>.x)

= >.-2 J(x)

for each

>. #: O.

Let us define the John transform of an arbitrary function J E F( -2). Take he G2 ,4. Choose an arbitrary basis u, v in h and denote by s and t the coordinates of points on h with respect to this basis, x=su+tv. To a function

J E F( -2) we assign the following differential J(su + tv) (sdt - tds).

I-form on h;

It follows from the homogeneity condition (3.2) that this form is homogeneous of degree zero, and the integral of the form over a contour S in h enclosing the point does not depend on the choice of this contour.

o

DEFINITION. By the John transform of a function J E F we mean the following function of a pair of linearly independent vectors u, v E R4;

(3.3)

(:If) (u, v) =

~

Is

J(su

+ tv)(sdt -

tds);

the integral is taken over an arbitrary contour S surrounding the point 0 in the subspace h spanned by u and tI. It has already been mentioned that the integral does not depend on the choice of the contour S. The function", = :IJ thus defined depends not only on the subspace h E G2 ,4 but also OD the choice of the basis in h. It follows from (3.3) that, when passing from one basis u, v to any other basis u' = 0 u + {J v, v' = "Y u + 6 v, the function '" is transformed by the following law: ",(ou + (Jv, "YU + 6v)

= 106 -

(J"Y1- 1 ",(u, v).

In particular, if the area elements in the bases u, v and u' , v' coincide, i.e., if

106 - {J"YI = I, then ",(0 u + (Jv, "Y u + 6v) = ",(u, v). Thus, the John transform of "'(u, v) depends only on the Plucker coordinates Pi; = utV; - u]v.. i.e., this transform is a function on the quadratic cone Kc R6

3 JOHN TRANSFORM L~ Rp3

given by (3.1). Obviously,

K\ 0

8a

is a one-dimeusional bundle over the Grassmanniao

G2,4. In what follows, \\'e often write :p(p) instead of tp(u, v) having in mind that P is a point of the cone K. i.e., a point of R6 with coordinates Pij, i < j, satisfying (3.1). DEFINITION. Denote by 'H( -1) the space of COO functions tp on K\ 0 satisfying the homogeneity condition

(3.4) It follows from the definition of the John transform .:I that .:I

tp E F( -2}. Thus, the John tr8ll8form is a mapping .:I: F(-2} - 'H(-I).

3.3. Relation to the John transform in the aftlne space. For

~

nience. in this subsection wc denote the John transforms in the affine and projecth'e spaces by .:10. and .:lp, respectively. Recall that the John transform .:10 of a function F on a3 is given in local coordinates (Ot. 02, PJ, I~) OD the manifold of lines in R3 as follows: (.:IoF)(Oh02,/3,.;'J.z) =

(3.5)

[:OQ F(01 X3 + PI, 02X3 + .132, Z3)Ulj.

Suppose a3 is embedded in R4 as the hyperplane rot = 1, and assign to every function / E F( -2) the restriction F of / to this hyperp1ane, i.e., put

F(XltZ2.X3)

= /(ra, x 2,z3,l).

By the homogeneity condition (3.2), the function / can be reconstructed uniquely from the function F, and convenely, any COO function F OD Jl3 which appropriately decreases at infinity is a restriction to R3 of a certain function! E :F( -2}. It follows from (3.5) that (.1.. F)(ol. 02.

/Jt.~) = [:00 J(t31 + Q1 t, ~ + 02 t, t, 1) dt = (.:11'/)(11. v),

where u = (/3)'112.0,1) and v = (01,0:2,1,0). (For the contour S used in the definition of the function .:I,,! ill (3.3), one takes the pair of lines 8 ±l.) Thus, passing from the vect0r8 U, tI to the PlUcker coordinates, we obtain tlle following assertion.

=

PROPOSITION 3.1. The JolIOtDing relation holds between the John tmnsjorm .Jp! oJ a jrmction ! E :F( -2) and the aJfine John tramform .J.F, V1he~ F stands Jor the restriction 0/ / to the hyperplane x" = 1:

where

= (.:I,./)(p), = -010 P23 = /32, P24 = -02,

(.JoF)(01t02.~,.82)

(3.6) P12

= .131 0 2 -

112 0 1, PI3 = Pl. Plot

P34 =-l. In turn, it follows from (3.6) and the homogeneity condition (3.2) that (.:I,,/)(P) =

1P341- 1 (.:IoF)(P14/P34. P24/PM, -Pl3/P34, -P23/Pad·

and

2. JOHN TRANSFORM

62

3.4. Description of the image of the John transform. AB in the affine case, any function 'P( u, v) in the image of the John transform .J satisfies additional relations. Namely, it immediately follows from the definition (3.3) of the John transform that i.j

(3.7)

= 1.2,3,4

(i .pj).

where Ui, v. are the coordinates of the vectors U and v. We represent these conditions on the functions 'P = .Jf in an invariant form by viewing 'P 88 a function of the Pliicker coordinates Pi]. Le., 88 a homogeneous function on the cone K given by the equation P12P34 - P13P24 + P14 P23 = 0 in R6. Let H( -1) be the space of Coo functions on R6 \ 0 satisfying the homogeneity condition (3.4). Introduce the following differential operator on H(-I):

~=

(3.8)

IP

_

iJp128p:w

IP

iJp13 ~4

+

IP

iJp14 ~3

.

PROPOSITION 3.2. The operator~ can be pushed doumfrom H(-I) to 'H(-I) under the restriction operation H(-I) - 'H(-I). In other words, ifif' IK= O. then

~'P

IK= O. PROOF.

Write r

= P12P34 -

P13P24 + P14P23

and pass to new coordinates on R6 in a neighborhood of an arbitrary point pO e K by replacing one of the coordinates Pi] (i < j), say, 1'34, with r (this change of coordinates is admissible if P12 -:/: 0). We must show that the expression ~'P IK= il.'P Ir=o represented in the Dew coordinates contains DO derivatives with respect to r. Denote by f/J the function 'P in the new coordinates, i.e., 'P(P) = 1/!(P12,P13,P14.P23.P24. r ). Immediate calculations show that (3.9)

IP f/J dl f/J ,,' 1Pf/J lJ1/J IP f/J ~'P = - 8p13~ + iJp14~3 + L.J Pij {Jpi]lJr + 3 lJr + r lJr2'

where the prime means that the sum is taken over all indices i and j (i < j) except for (i.j) = (3,4). Further, it follows from the homogeneity condition for 'P that Pi] ~ + 2 r ~ = -f/J, and hence

E'

L'PiJ

:'' 'tw =

-3 ":: - 2r

:~.

Substituting this expression into (3.9), we obtain (3.10)

dlf/J

~'P = - 8p13~4

dl f/J

+ 8p14DP2:s -

dl '"

r lJr2'

The assertion follows from (3.10). COROLLARY.

The operator ~ is well defined on the function space 'H(-I).

3. JOHN TRANSFORM IN RP'

63

PROPOSITION 3.3. Each of the equations (3.7) for the function 11' E 'H(-I) is equivalent to the equation

_ fPrp _ fPrp 1111' = 8p12Op:u 8p138JJ24

(3.11)

+

fPrp

{Jp14~

= o.

PROOF. Let us prove, for instance, that the relation .Jf!.L. = ~ is equivBul8V2 lJUilJVa alent to (3.11). For the local coordinates on the cone K we take the PlUcker coordinates Pij, i < i, (i,i) -F (1,2). By (3.10), in these coordinates we have

1111' = _

= "iV, - ",Vi. fPrp = _ L

Since Pij

8u1~

+

fPrp 8p131JJJ24 it follows that Vi"j

iJ=3.4

fPrp

fPrp 8p14~

,

{JplilJJJ2j

This implies that

fPrp

Iflrp

-- - = (113"4 8u IOv2 8u2OvI

V4"3) I1rp,

which proves our assertion. THEOREM 3.1. A function 11' E 'H(-I) is representable in the form rp = ~f, where f E :F( -2), if and only if this function satisfies the equation 1111' = O.

The "only if" part is already proven. For the proof of the "if" part, see 3.6. 3.5. Another way to deftne the John tr8D8form. Let us define dimensional subspaces of R4 by systems of independent equations (3.12)

«(, x) = 0,

~

('I, x) = 0,

i.e., by pairs (, '1 of linearly independent vectors of the dual space (R4)' rather than by pairs ", V of linearly independent vectors in R4. Denote by hE" the subspace of R4 given by (3.12) and denote by PhE" the projectivization of this subspace. Introduce a differential I-form O'E,,(x) on hE" as follows: (3.13)

«(. dx) 1\ ('I, dx) 1\ O'E,,(x) = w(x),

where w(x) = Xl dx2 1\ dx3 1\ dx4 + ... (cyclic permutations). If f E :F( -2), then the differential I-form on hE" given by

f(X)O'E"(X) is homogeneous of degree zero, and this form is orthogonal to the fibers of the bundle h(." \ 0 - Ph E." (i.e., it vanishes on vectors tangent to the fibers). DEFINITION. By the John tronsform of a function I E :F we mean the following function of a pair of linearly independent vectors (, '1 E (24 )':

(3.14)

(~' 1)«(, '1) =

1

"''''

I(x) O'E"(X)'

where 'YE" is any section of the bundle hE." \ 0 depend on the choice of 'YE".

Ph E.,,; the integral does not

2. JOHN TRANSFORM

64

88

Using the symbolism of delta functions, one can represent the integral (3.14) follows:

(':1' I)(e, 71)

(3.15)

=

i

I(x) 6( (~, x}) 6( (", x» w(x),

where r is an 81'bitrary section of the bundle R4 \ 0 - p3. Simil8l'ly to 'P .11. the function t/J .1'1 satisfies the following conditions. 10. It depends only on the bivector q = ~"'" i.e., on the minors qiJ = ~i", _~i"i. 1fl. When viewed 88 a function of q, the function t/J is a Coo function on the cone q12tf4_ q13q24+q14q23 = 0 in (Rs),. and t/J satisfies the homogeneity condition

=

=

t/J('\q)

(3.16)

= l,\r 1 t/J(q).

ao. The following relations hold: lPt/J

(3.17)

"i

lPt/J

lJei 8qJ = lJei/Jrti'

i.j

= 1.2.3,4

(i ~ j).

e

where ~i and are the coordinates of the vectors and "I. As in the case of functions. 'P = .11, one can show that each of relations (3.17) is equivalent to the relation

/It/J

(3.18)

lP '" 24 + IJ IJ2 '" =IJqlPt/J 121Jq34 - lJq13IJ q q148q23 = o.

Let us find a relation between the operators .1 and .1'. We say that bivectors P = {Pii} in R4 and q = {tt J } in (R")' 8I'e dual and write P '" q if Pi)

= sgn(i,j.k,l),f'

for any permutation (i.j,k,l) ofthe indices (1,2,3,4). One can readily see that the same subspace h C R4 corresponds to dual bivectors p and q, and the differential I-forms on h related to these bivectors coincide. This implies the following assertion. PROPOSITION

3.4.

1/ q '" p, then :I1(P) = :I' J(q).

Denote by 1{,' ( -1) the space of Coo functions on K' \ 0 satisfying the homogeneity condition (3.16), where K' stands for the cone q12tf4 - q13q24 + ql"q23 = 0 in (Rs),. By Proposition 3.4, Theorem 3.1 can be restated in terms of functions in H'(-I).

t/J e 1{,'(-1) is representable in the lorm t/J = :1'1, e F( -2), il and only il this function satisfies the equation /It/J = O.

THEOREM 3.2. A function

where I

3.6. Proof of the theorem OD the image of the John transform. The "only if" part of Theorems 3.1 and 3.2 was already established. We prove the "if" part in Theorem 3.2. Let t/J(~,,,) be a Coo function of the bivector 71. and let t/J satisfy the homogeneity conditions (3.16) and equations (3.17). We prove that t/J = :I'J for some

e"

le .1"(-2).

Introduce the manifold A of pairs (x. "I), where x e R4 \ 0, " e (R4)' \ O. and (", x) = O. Let (x,,,) e A. Denote by L" C R4 the subspace orthogonal to ". Note that x e L". For any e (R4)' \ {"I}. where {,,} is the subspace spanned by ", we set

e

3 JOHN TRANSFORM IN Rp3

Since

i.e.,

1/l depends on ~ "11 only. it follows that 1P,,(e + ;\11) =

~«() for

each A E R,

w" is a function on «R")'/{,,}) \ O. It is clear tbat this function is infinitely

differentiablt' Ob «)R4)'It,,}) \ 0 and satisfies the homogeneity condition ~,,(~) = IAI-l1/1,,(~). Hence. by the theorem on the image of the Radon tJaDSform for the projective plane, the function 1/1" is the Radon transform of a er» function I,,(z) dcli.ncd on thesubspace L., C 4 dual to (R4)' I{,,}, and 171 satisfies tbehomogeneity condition I,,().z) ,\-2/,,(x). Thus. the following function F is well defined on the manifold A:

a

=

F(z. ,,) = I,,(x). LEMMA.

For any x

e R4 \

0, the junction F doe8 not depmd on ".

It suffices to prove the 858Crtion for some special point z, say, for In this case, the condition that the vectors '1 and z are ortbogonal becomes 111 = 0, i.e., " = (0, fil, ~ ,,,"). We must show that 8P~.,,) = 0, . 2 3 4 For .Instance, 1et us prove that 8F~.!f) 0 ."" =. We write out an explicit expression for F(z,11) in terms of I,,(z) by using the inversion formula for the Radon transform on the projective plane. Let L be the coordinate subspace e;4 = 0 in (R")'. If'14 :p 0, then L is transversal to the vector '1, and we can assume that ttJ" is a function on the subapace L. Since (~, ZO) = ~l, the inversion formula for the Radon transform implies that PROOF.

z = Xo

= (1,0,0,0).

,= . ,.

F(zo.,,) = c

j ~,,«() I

(s=l

=cjaw('1,() 8(1

(e l ) -2 ~l

I

(e;1)-1dF. 1 "dF.2.

= ~, we obtain 8F(zo.'1)

lJq2

=cj()21/1('1·(> 1Jf.2 8'11

dF. 2

('=1

Differentiating the resulting relation "ith respect to

~

"

I

fil

and applying the formula

({l)-1~1"dF.2=O.

(s =1

We now return to the proof of the theorem. Using the above lemma, we define a function I(x) on R' by the formula

I(z) = I,,(z), 1] is all arbitrary vector in ()R4)' orthogonal to z. The function I is h0mogeneous, I(;\z) = ;\-2/(z). and infinitely differentiable on L" \ 0, where L" is a 3-dimensional subspace of 1Il4, and hence I is coo-dift'erentiable 011 Rot \ O. Therefore, I E F( -2). MoteO\"el'. it is clear that 3'1 because this relation holds for the restriction of '" to the set of subspaces belonging to an arbitmry three-dimensional space.

where

= '"

3.7. John transform as an Intertwining operator. The John transform has a simple grol1~theoretic interpretation. Consider the group G = GL(4,R) of nondegenerate matrices of order four. Any element 9 E G defines a linear transform z ...... X9 of Rnding a point x = (Zt,Z2,Z3,X,,) to the point with coordinates

a"

zj

= E:.1 Xi !}.J'

r

2. JOHN TRANSFORM

66

To any element 9 E G we assign linear operators TJ1) and TJ2) on the spaces F(-2) and 1t(-1), respectively, (TJ1) f) (x)

= I(xg),

(TJ2) rp) (u, v) = rp(ug, vg).

Obviously,

T91(') .....ti) 1.92

= T(i) 9192'

• •

= 1,2.

r,1)

for any 91,92 E G, i.e., the operators and T~2) form representations of the group G in the corresponding function spaces. The next assertion immediately follows from the definition of the operator :J. PROPOSITION

3.5. The operator:J T~2):J

(3.19)

= :JTJ1)

01 John translorm satisfies the relation lor each 9 E G.

In terms of representation theory, relation (3.19) means that the operator :J : F(-2) -1t(-1) is an intertwining operator for the representations T(l) and T(2) ofthe group G. One can prove that an intertwining operator F( - 2) - 'H( -I) is unique up to a factor. Thus, the condition that :J is an intertwining operator uniquely determines the John transform up to a factor. We now establish the group-theoretic meaning of the additional relations (3.7) for a function cp = :Jf. Let 9 be the Lie algebra of the group G and let {ei,} be the standard basis in 9 (i.e.• etj is the matrix with only one nonzero entry which is equal to 1 and stands at the (ij)th place). The representations 7'<1) and T(2) of the group G induce the corresponding representations of the Lie algebra 9 of G. Denote by E~) and E!;) the operators of these representations corresponding to eiJ E g. The formulas for the operators TJ1) and T~2) immediately imply the following expressions for E~) and Eg): (1)

(3.20)

E i,

(3.21)

(2) _ E'j rp -

01

1 = Xi 8x. ' 1

8rp Ut Du.

Orp

+ Vi Dv ..

1

1

The next assertion immediately follows from (3.20). PROPOSITION

(3.22) i,j,k,l

3.6. Each /unction

(EW Ei~) - 6j lc E~I»

= 1,2,3,4, where 6ij

I

E F( -2) satisfies the lollowing relations:

1 = (E!,I) Ei~) -

6,Ic E~J»

I,

is the Kronecker delta.

Since :J is an intertwining operator, this implies the following corollary.

= :J1 satisfies the relations 6j lc E~2» rp = (E~,2) Ek~) - 6"" E~;»

COROLLARY. The /unction rp

(3.23)

(E!]) Ei~) -

where the operators E!]) are given by (3.21).

rp.

4. JOHN TRANSFORM IN THE COMPLEX AFFINE SPACE

67

Relations (3.23) obtained from group-theoretic considerations are equivalent to (3.7). Indeed, if we substitute the explicit expressions for the operators E~:) into (3.23), then (3.23) becomes

~~ auJOv,

~~ OvjOu,

UiVk-- +V.Uk--

~~ ~~ = U , Vau,Ovj k - - +V.Uk vu.""J

A...,A.. •• '

i.e., (Ui Vk - Uk Vi) (

~I{J

~~

aujOv, - Ou,Ovj

)

= O.

4. John tr8D8form in the complex afIlne space The definition of John transform for the complex affine space and the description of the image of this transform are similar to those given in § 1 for the real case. We briefly present this material in 4.1 and 4.2. The main point at which the complex John transform differs from the real one is that the former admits local inversion formulas. Subsections 4.3 and 4.4 are devoted to the construction of these formulas.

4.1. John tr8D8form in C3 • Similarly to the case of real affine space, we present complex lines in C3 by equations of the form

aecJ\O, /3ecJ, and the integral of a function I on C3 over a line x = at + /3 x=at+/3,

relation

~(a, /3) = ~

(4.1)

fc I(at + /3)

is defined by the

dt" di.

The function ~(a, /3) thus defined on the manifold E of pairs (a, /3) is denoted by :JI and referred to as the John trans/mm 0/ the function I. In what follows, it is assumed that all functions I belong to the Schwartz space S(C3) of infinitely differentiable functions rapidly decreasing, together with all their derivatives, on

C3 , Similarly to the real case, the functions ~(a, /3

(4.2)

i.e., the function (4.3)

I{J

~

= :JI

satisfy the relations

+ ato) = ",(a, /3) for each

to

e C,

does not depend on the choice of a point

~(.\a, /3)

= l.\r

2 ~(a, /3)

/3 on the line, and

for each .\ # O.

Thus, ~ depends only on the line itself and on the choice of the direction vector a. If the space C3 is equipped with Euclidean metric, then in the definition of the John transform in (4.1) one can assume that 101 = 1. In this case, condition (4.3) is reduced to a simpler one: 1{J(~a, /3) = ",(a, (J) and I~I = I, and '" can be regarded as a function on the manifold of complex lines H. By analogy with the real case, denote by S(H) the space of all infinitely differentiable functions l{J(a, /3) satisfying condition (4.2), (4.3) and such that, for 101 = I, these functions (regarded as functions on H) rapidly decrease together with all their derivatives. One can readily see that the image of any function I e S(C3 ) under the John transform:J belongs to the space S(H). Sometimes it is convenient to present lines on C3 by equations of the form

2. JOHN TRANSFORM

68

and define the John transform of 1/J on H by the formula (4.4)

1/J(0102, 13r. fJ.z)

=~

fc

= .7f

f(olxa

in the local coordinates 01, 02. 131,

fJ.z

+ 131, 02Xa + fJ.z, xa) dxa A tUa.

4.2. Differential form Ki{J and the theorem on the image of the John transform. The complex analog of the differential form "tp introduced in § 1 is the following differential (1, I)-form on the manifold E of pairs (0,13): a fP (4.5) "tp = .~ ofj doi II.dOj . 1.)=1

1

:8. ,

Here tp stands for an arbitrary function on E. H tp satisfies (4.2) and (4.3), then the differential form "'P can be pushed down from the manifold E to the manifold of lines H. Correspondingly, in the local coordinates 0102, 13r. fJ.z the form "1/J becomes fP1/J

2

(4.6)

,,'"

= .L

_

{Jfj. (JP doi

1.)=1

1

" dOjo

,

Similarly to the real case, one can describe the image of the space 8(C3 ) under the John transform by using the form "tp. Namely, denote by HZ the submanifold of lines in C3 passing through an arbitrary point x E Ca and by "ztp the restriction of the form "tp to HZ. THEOREM 4.1. The function tp E 8(H) belongs to the image of the Schwarlz space 8(Ca) under the John transform .7 if and only if the differential form "ztp on the sulnnanifold HZ is closed for any point x E C3. The condition that the differential form K.zl{) is closed for any point x E Cl is equivalent to the following conditions on the function tp: fPtp {JOi {J13j

fPtp

= OOj {J{3i ,

fPtp = {Jlii {J{3,

fPtp

= {Jlij (J13i- .

The proof of Theorem 4.1 is the same as in the real case.

4.3. Inversion formula. In the complex case, in contrast to the real case, the operator" can be used not only to describe the image of the John transform .7 but also to obtain an inversion formula reconstructing a function f E S(Cl) from its John transform tp = .7f. THEOREM 4.2. If tp = .7 f, f E 8(C3 ) and "ztp is the restriction of the differential (1, I)-form "I{) to the submanifold HZ C H of lines passing through a point

x

E

Ca, then

(4.7)

for each cycle.., c HZ of real dimension 2, where c(..,) depends on the homology class of the cycle..,. Moreover, if the cycle.., is not homologous to zero, then cb) ::f:. O. PROOF. Since the differential form "zI{J is closed, it follows that, for a fixed f, the integral (4.7) depends only on the homology class of the cycle ..,. It is known that the second homology group of a two-dimensional manifold HZ is a group with a single generator i, and for a representative of the homology class i one can take

4. JOH."J TRANSFORM IN TilE COMPLEX AFFINE SPACE

69

an Eu1er cycle, i.e., the manifold of lines passing through the point x and belooging to a chosen plane. Therefore, it suf6.ces to evaluate the integral (4.7) over some Euler cycle. We show that, if.., is an Euler cy("le, then t.be inversion formula (4.7) coincides with the inversion formula obtained in Chapter 1 for the Radon transform in C2. We express the John transform in the local coordinates by (4.4). Consider the Euler cycle "1 formed by the lbw passing through a point zO = (xy,xg,xg) and belonging to the plane h given by the equation XI = Let 10 be the restriction of the function I to the plane h and let 1/>0 be the restriction of the function '" = .11 to the submanifold of lines lying in the plane h, i.e.,

xY.

lo(x2.x3)

= l(x~,x2.xs),

~'o(Q,~) = "'(O,Q;X~,~) = ~ We have

1

fc lo(QXa +

(lJ2l/1o(a. fj)

I -

le

')' K z 1S' -

IJ/JIJP

I

i3.. ~-et4

t3,Xa)dz3I\.dZa. do I\. dii

.

On the other hand, the function VJo is the Radon transfunn of t.be flmction 10 on the plane h. Hence, by the inversion formula for the Radon transform on the complex plane (see (9.11) in §9 of Chapter 1) we obtain

L c

()2~'o(Q,P) JliQ Jli.Q VI-' vI-'

I

n

'- (0 0) ° uO' I\. uO' = 2 7r2·I 10 X2- xa . .3.3-

~Z2-QZ3

Thus, it is proved that

for an Ewer cycle ") C HZu • Obviously, the inversion formula (4.7) is local; namely, to reconstruct a function an arbitrary point of x E C3 by this formula, it suffices to know only the integrals of the £m1ction I over the lines infinitt>iy close to x. We empbasize that there is DO analog of this inversion formula for the John transCorm in the real space because in the real case we have K-z'P = 0 for each one-dimensional cycle '"Y c HZ.

I at

J.,.

4.4. Aoalogs of the operator K. In this subscctioD \\'C construct a family of linear operators K from the space S(H) to the space (}
xeCa. Each of these operators can be used to construct an inversion Formula in the same way in which we used the original operator K above. Coosider tht> John transform in local coordinates (4.4). In these coordinates, the conditions for 8 function t/J to belong to the image or the John transform become (4.8)

~~ 8a1 lJ{3z

~'"

= 1Jo2IJfjl '

~~

001 aiJ2

~~

= 002 apl .

2. JOHN TRANSFORM

70

We introduce the following four linear mappings Ob',9) denotes the space of differential (p,q)-forms on H: 1

lJ

2

IJ

-+ 0(11+ 1,9),

where 0(",9)

IJ

" = IJPl dol + 1JfJ, do2, " = IJPl dPl ,,3

IJ

+ IJ{J, dfJ"

= 80 .!.....- dol + 1J0I2 .!.....- do 2 • 1

,," = vuI A~ dPl + "012 AIJ dfJ,. Further, denote by ;Ci, i obtained from ,,' by replacing respectively. Write

= 1,2,3,4, the linear mapping O(p,q) -+ O(p,q+l) 1/;;. JIr-, doJ , and dl3j with i/r:. =-88 ,OOj, and dPj' ,

UP,

,,'J = "i ;ci,

,

UP,

= 1,2,3,4.

i,j

According to this notation, if t/J is a function on H, then "ijt/J is a differential (1,1)form on H. In particular. ,,11t/J coincides with the above form "t/J. Denote by "ijt/J the restriction of the form "ij t/J to the submanifold H:r: C H of lines passing through a given point Z e C 3 • PROPOSITION 4.1. For any function t/J = :JI and any point z e C3, the diJferentiallorms ,,~t/J are related oslollows:

(4.9)

(4.10)

,,!It/J = _Z;1 ,,!1t/J = _Z;1 ,,!2t/1 = IZ31-2 ,,~t/I. ,,!3t/J = _Z;1 ,,-:t/J = _Z;1 ,,!"t/J = IZ31-2 ,,!"t/J. ,,!It/J = -z3'l ,,!1t/J = -z3'1 ,,!2t/J = IZ31- 2,,!2t/J, ,,~t/J = _Z;1 ,,~t/I = _Z;1 ,,~t/J = IZ31-2 "!"t/I, ,,!'t/J + ,,!'t/J = d(;c't/J), ,,~t/J + ,,':t/J = -d(;Cit/J). i = 1.2,3.4.

=

PROOF. Relations (4.9) are immediate because dA -Z3 do. and dA -Z300i, i = 1,2, on the submanifold HZ. Relations (4.10) follow from (4.8). COROLLARY 1. The diJferentiallorms "ijr.p, i,j molDgy equivalmt up to lactora. COROLLARY 2.

=

= 1,2,3,4, are painlJise coho-

I1 t/J = :JI, then all diJferentiallorms ,," r.p satisfy Condition A.

By Proposition 4.1, to construct an inversion formula. one can use not only the original operator" = "ll but also any other operator of the form (and any nonzero linear combination of these operators). For instance. for the operators ,,"" and ,,33, the inversion formulas are

,,'j

l = l ,,~t/J ch) I

cb) IZ 31"/(z),

,,!"t/J

(4.11)

=

Z 31 2

/(z).

5. PROBLE~LfJ OF INTEGRAL GEOMETRY FOR LINE COMPI..EXES IN

et

5. Problems of integral geometry for line complexes In

ca

11

It was already noted above that the problem of reconstructing a function 1 R3 or <;3 from its John transform tp = :J1 is owrdeternlined.; to reconstruct I, it suffices to know the function r.p OD some three-dimensional submaniCold K or the manifold of lillea H rather than OD the entire manifold H. We refer t.o threedimensional algebraic subvarietie8 of lines K C H as line complezes. We present examples of complexes for which the problem of reconstructing a function 1 has an elementary solution. These are the line complexes having the following property: For almost every point x E C 3 • there is a plane hz containing x and such that almost all lilles on this plane belong to the complex. In particular. this property holds for the complex formed by the lines that intersect a given line and for the complex of lines parallel to a given plane (the latter can be reganled as the complex of lines that intersect a line at infinity). It is clear that the problem of reconstructing the function at a point x from the integrals of this function over the lines of this complex is reduced to the inversion formtda for the Radon tnmsform

011

on the plane /tz. In this section, the problem of reconstructing a function 1 is soh,red for various line complexes in C3 . A specific feature of the resulting inversion formulas is that they are local, i.e., the function at a point x E is reconstructed from the integrals of this function over the lines of the complex that are infinitely close to x. We begin with the complex of lines in C 3 that intersect a chosen algebraic curve. We claim that the in\wsion formula for this complex is a simple consequence of the inversion formula obtained in § 4 for the John transform. After this we shall consider a broader class of line complexes ("admissible complexeS') admitting an explicit local inversion formula.

ca

5.1. Problem of integral geometry for a complex: of lines in C3 ID~ tersecting a curve. Consider the complex K of lines in C3 intersecting a chosen algebraic curve A C C 3 • We formulate the foUowing problem: Reconstruct a fuoo.. tion 1 E 5(C3 ) from tile integrals of 1 over the lines of the complex, i.e., from the function (5.1)

.(0,.\)

=~

k

I(ot +.\) dt dl,

(0,.\) E (cl \ 0) x A.

Denote by fh.. and 8A the (1,0)- and (O.l)-component& respectively, of the operator d of exterior diffenmtiation on the manifold A. Formula (5.1) defines tbe integral transform .(0,.\) of a function J e 5(ca). To any point xD E C 3 we assign the following differential (l,l)-form on A:

(lJA 1\ OA) .(0,.\) Io=zD-.\ . THEOREM 5.1. The lollounng inversion lormula lor the integral translorm (5.1) holds for almost every point XO E C 3 :

(5.2)

where k i8 the

l(xO) ~

= (2k1f2i)-1

i

(Dh A8A).(Q.'\)

IQ=z"-'\,

01 the algebrau: curve A.

PROOF. Wl' apply the inversion fomllda (4.7) for the John traosfonn '() = .11 (see §4). when> for rD we take the submaoifold of lines passing through the point

2. JOHN TRANSFORM

72

XO

and intersecting the curve A, i.e., the manifold of lines

x = (xo - ~) t By the definition of the operator (5.3)

"'I'

I..,.0=

+ ~,

~E

A.

le,

~ lP'{J(o, (J) ~ 8fJ,.8i).

iJ=l

I

J

In.O_A d~i"d~J""-A

I

Since the restriction '1'(0, (J) (j=A is equal to +(0, ~), the right-hand side of (5.3) is equal to (8A A8A)+(Q,~) lo=zO-A' Hence, by the inversion formula (4.7) for the John transform,

i

(8A

A8A)+(0,~) lo=zO-A = c(-yzo)/(xo).

Let us find the coefficient c(-yzo). H the order of the algebraic curve A is equal to k, then the cycle "'"(zo is homology equivalent to k -y, where "'"( is an Euler cycle. Since c(-y) 211" 2 i by §4, it follows that c(ro) 2k1l"2 i.

=

=

REMARK. If the curve A is given parametricaUy, ~

lII(o,s)

= u(s), then we set

= +(o,u(s».

In this case, the inversion formula takes the form

I(x)

= (2k1l"2i)-1 lef IIIZ,(x-u(s),s)d8Adi. .

REMARK. Real analogs of the inversion formula (5.2) play an important role in tomography. The real case is more complicated, and the reason is not only that real inversion formulas for line complexes are nonloca1. As a rule, it is impcl6Sible to reconstruct a functioin in R3 in terms of integrals along the lines intersecting a curve. The reconstruction is possible only for functions with strong restrictions on their supports. We do not consider such problems in this book.

5.2. Deflnition of admissible Une complexes in C 3 • Consider an arbitrary line complex K in C3. As above, the problem is to reconstruct a function I E S(C3 ) from the integrals of this function over the lines of the complex. We need to find conditions on K under which the reconstruction problem can be solved similarly to the case of a complex of lines intersecting a curve. using the inversion formula of § 4 for the John transform. Let HZ c H be a submanifold of lines passing through a chosen point x E C3 and let = K n HZ. The manifold is a cycle of real dimension two for almost every point x. Therefore, the following inversion formula holds for the John transform.1: if 'I' = .11, then

r

r

1..,.

(5.4)

"'I' = cf(x),

where " is the operator introduced in § 4. By the assumption, we know only the restriction • = 'I' IK of the function tp to the manifold of lines of the complex K. Thus, formula (5.4) gives a solution of the problem of integral geometry for the complex K if and only if the differential form can be expressed in terms of the function. and its derivatives along K only. An algebraic 3-dimensional subvariety of lines for which this condition holds

"'PI..,.

5. PROBLEMS OF INTEGRAL GEOMETRY FOR LINE COMPLEXES IN

CS

73

is called an admissible line complex. One can prove that this definition does not depend on the specific choice of the form K'P. An example of an admissible line complex is the complex of lines intersecting an algebraic curve in C 3 , which was treated in 5.1. If the complex K is admissible, then, substituting the explicit expression for KtPl'Y6 in terms of the function. into (5.4), we obtain an inversion formuJa for the integral transform related to this complex. REMARK. Our condition of admissibility is a purely local condition on a comis a cycle plex K. The requirement that K is an algebraic variety implies that for almost all x and the number of lines in K passing through a generic point x is constant. Combining these global geometric conditions with the local admissibility condition, we can write down the inversion formuJa for smooth complexes.

r

The next three subsections are devoted to the study of the analytic and geometric structure of admissible complexes. In particular, we claim that each admitmible complex is either the variety of lines intersecting an algebraic curve in C3 or a complex of lines tangent to some algebraic surface in C3.

5.3. Necessary and sufBcient conditions for a complex K to be admissible. Consider the John transform .1 in the local coordinates al. a2. Ph fJ'l on the manifold of lines H (see 4.1). If tP = .11 is the John transform of a function I e S(C3), then, according to § 4, the following inversion formuJa holds:

1

~

fPtP

-

L.J ~ 00; 1\ diij 'Y" i.j=I.2 a.a)

(5.5)

= C IX312 I(x),

where "{Z C HZ is an arbitrary cycle of real dimension two. Let K be an arbitrary complex. We set = HZ n K and, for each h e "(Z, denote by T"H, T"K, and T"r the tangent spaces at the point h to the manifold of lines H, the complex K C H, and the cycle respectively.

r r,

DEFINITION.

Introduce the endomorphism A" : T"H - T"H by setting

A,,(OOI,002,d{JI.dfJ'l) LEMMA.

= (0010002,0,0).

il and only il A"(T,,,,{Z) c T"K

A complex K is admissible

lor almost every point x e C3 and almost

every line her.

Indeed, the integrand in (5.5) is the derivative of tP taken in the complex direction A"(T,,r) for any h e Hence, the form in the integrand in (5.5) can be expressed in terms of. = '" IK if and only if this direction belongs to the 8Ubspace

,.z.

T"K. Let a complex K be given in the local coordinates (a, P) by the equation F(a, 13) = 0. PROPOSITION

5.1. The complex K is admissible

(5.6) at any point h

OF OF _ OF OF Oal OfJ'l 002 0131

= (a,l3)

01 K.

=

il and only if

°

2. JOHN TRANSFORM

74

= (001l002,d{3.,dlJ.z) be a vector in ThH. Then the condition

PROOF. Let ~

that

~ E

TIaK becomes

8F

8F

8 0 1 001

and the condition that

~ E

where al

=

~

-

x311;

8F

+ 81J.z djj.z = O.

d{J.

TIaHz is

+ d{Jl = 0,

X3oo.

Hence, the vector ~ E Tlar

8F

+ 8 0 2 002 + 8{31

X3 d0 2 + dlJ2

= TIa(HZ n K) is of the fonn ~ = (a.,a2, -X3aJ, -X3(2), and a2 = -(I!; - X3 :~).

= O.

It follows that AIa(~)

=

(al,a2,O,O). By the lemma, for the complex to be admissible, it is necessary and + = 0, which is equivalent to (5.6). sufficient that al

I!;

(12::'

We now present another proof of Proposition 5.1; this proof uses direct computations. First we assume that the complex K is given by an equation solved with respect to 1J.z:

IJ.z = U(01. 0 2,{J.). On the manifold of lines H we introduce new coordinates al. 02, {3J, S = IJ.z instead of the coordinates 011 02, {31,1J.z and denote by X(01.02, {JI, s) the function t/J in these new coordinates, i.e., U(01.02'~)

X(01,02, {311

IJ.z -

U(01l02,{3d)

= 1IJ(0.,02,{31,1J.z).

Note that in the new coordinates the complex K is given by the equation s = O. Let us represent the fonn ,,," in the new system of coordinates. We have i = 1,2.

Hence,

For the complex K to be admissible, it is necessary and sufficient that the restriction of this form to contain no term with 8x/8s. i.e., that

r

fJu- dOl 0

(5.7)

On the other hand, on hence (5.8)

fJu- 002 +0 V02

VO.

r

( -X3:;'

we have X3oo.

+ ::1)

001

I= ..,,,

+ d{3.

O.

= 0 and X3OO2 + dlJ.z = O. and

+ (X3 + ::2)

002

= O.

It follows from (5.7) and (5.8) that the function U(0},02,.8.) satisfies the equation

fJu (5.9)

80.

fJu fJu

+ 8 0 2 1J{3. = o.

If the complex K is given by an equation of the form F(o, {J) = O. then fJu/8~ = - Pa, / F's-;,. i = 1.2, and fJu/8{31 = - F~II F~. Substituting these expressions in (5.9). we obtain (5.6).

5. PROBLEMS OF INTEGRAL GEOMETRY FOR LINE COMPLEXES IN

CS

7&

5.4. Geometric structure of admissible complexes. The admissibility condition obtained in 5.3 for a line compll",x in (:3 has an ioteresting geometric interpretation. Let us first present some geometric notions and ract~ related to the manifold of lines H. To any line h E H Vo-e assign a three-dimeoslonal manifold B h or all lines intersecting h. The family of one-dimensional subspaces or Th H tangent to H h forms a tbrctHlimensional cone rh c TIIR. One can readily see that the manifold Hh, where h = (oP,po). is giveD in the local coordinates (0, fJ) by the equation (01 - Q~) (P-J -~) - (02 - Q~) (.81 - ~)

= O.

Hence, the equation of the cone rh is dol d~ - 002 dPt

= O.

The generatrices of the cone rh can be interpreted as isotropic directions with respect to the holomorphic metric

d82 = dot ~ - do2 dp'.

For this reason, it is natural to call rh c ThR t.he isotropic cone at the point h.

rh-

We state several simple facts concerning the isotropic cone Note first that the planes tangeot to rh are given by the equatiODs

al dol

+ ~ ~ + 01 <#11 + ~ diJ2 = 0

whose coefficients satisfy the relation al ~ - ~ lit = O. Further, it is clear from the equation of rlt. that this cone contains two onedimensional families nlo n2 of two-dimensional planes,

=

fi1 : dfJl Ado.. fi2: do, = Ado l ,

= Ado2; diJ2 = AdBt.

dJ32

Moreover, exactly one plane or each or the families passes through any given point (oftbe cone) that. differs from the vertex. We refer to these pIa.nes as a-pltmes and p-planes, l't!SpeCtively. These families of planes in ThH have a simple geometric meaning; namely, the a-planes are tangent to the submanifolds of lines intersecting h at a single point and the ~planes are tangent to the submanifolds of lines belonging to some c:bosen plane of passing through h. Any two planes in the same family intersect only at zero, and any two planes in different families intenect along a line. Com--enely. every genera.trix of the isotropic cone r" belongs to one of the planes of each of these families. If a tbree-dhnensioual subspace or ThH is tangent to the isotropic cone rh!

ca

then the iotersect.ion of this 811bspace with r" Is the unloo of a single a-plane and a single ~plane. Conversely, if a tbrecHlimensional subspace of ThB contains a plane of the isotropic cone rh, then this subspace is tangent t-o the cone. PROPOSITION

5.2. The admissible line compleze8 in

ca are uactly the line

comple:tes K th4t are tangent to the corn!8ponding isotropic cone rh at (lny smooth point h E K. Indeed, the admissibility condition proved in Proposition 5.1 is equi,,'alent to the condit.ion that the tangent subspace ThK to the complex K at any point h E K is tangent to the isotropic cone rh.

76

2. JOHN TRANSFORM

r

As before, let be the manifold of lines of the complex K that pass through = HZ n K. a point x, i.e.,

r

PROPOSITION 5.3. A line complex K is admissible if and only if, for any smooth point h E K, all stcbspaces Thr c ThK n fh. x E h, belong to the same {J-plane.

Indeed, if the complex is admissible, then. for any smooth point h E K, the subspace T",K is tangent to the cone f"" and hence the space T",Knf", is the union of an o-plane and a fJ-plane. In this case, all subspaces T",r. where x ranges over the line h, belong to the same {J-plane. Conversely. let a complex K be such that, for any smooth point h E K. the subspaces T",r form a two-dimensional plane if x ranges over the line h. Then the space T",K n f", contains this two-dimensional plane, and therefore the space T",K is tangent to the isotropic cone f",. Hence, the complex K is admissible by Proposition 5.2. Proposition 5.3 admits another (equivalent) formulation. For any complex of lines K. denote by rz c C 3 the cone formed by the lines of K that pass through a point x E C 3 . Further. if h is a smooth point of the complex K, then, for any point x of the line h. denote by 7r""z the plane in Cl tangent to fZ along the generatrix

h. A line complex K is admissible if and only if the planes where x mnges over a line h, coincide for any smooth point h E K.

PROPOSITION 5.4. 7r",.z.

5.5. Description of admissible complexes. It follows from Proposition 5.4 that, along with complexes of lines in Cl intersecting a chosen curve, the complexes of lines tangent to a chosen (algebraic) surface in C 3 are also admissible. Indeed, let K be a line complex tangent to a surface P. For any smooth point h E K. denote by 7r", the plane in C 3 which is tangent to the surface P at the tangency point of h and P. Obviously, if x is an arbitrary point of the line h and fZ is the cone formed by the lines of K passing through x, then the plane 7rh,z tangent to rz along the generatrix h coincides with 7rh. Hence, the complex K satisfies the conditions of Proposition 5.4, and thus it is admissible.

The admissible line complexes in C 3 are precisely those complex three-dimensional submanifolds of the manifold of lines H that consist either of the lines tangent to some complex surface in C3 or of the lines intersecting some complex cunte C3 (possibly at infinity). THEOREM 5.2.

To prove the theorem, we introduce the notion of a critical point on a line of an arbitrary line complex K. Consider a smooth point h of K (i.e., a line in C3). A point x E h is said to be critical if the a-plane (of the isotropic cone fZ) corresponding to this point belongs to T",K. Similarly. a two-dimensional plane 7r :::) h in C 3 is said to be critical if the corresponding fJ-plane belongs to T",K. PROPOSITION 5.5. If K is an admissible line complex in C3. then, for any smooth point h E K, there is a unique critical point x Eh (possibly at infinity) and a unique critical plane 7r :::) h.

Indeed, if the complex K is admissible, then T",Knr"" where r", is the isotropic cone, is the union of a single o-plane and a single fJ-plane. Corresponding to these planes are the critical point x E h and the critical plane 7r :::) h, respectively.

5. PROBLEMS OF INTEGRAL GEOMETRY FOR LINE COMPLEXES IN

CS

TT

COROLLARY. The manifold Kt of smooth points of K admits a standard embedding in the manifold of cOf71plete flags on Cp3.

Namely, to any It E Kt we assign a triple, a Bag (x, h, 11'), where x is the critical point and 11' is the critical plane Cor h. Let K be an admissible line complex given by an equation F(Q, p) == O. We find the coordinates ofthe critical point x = (Xl,X2.XS) of a line (o,P) E K assuming that this point is not at infinity. Tht~ equatiolls of the a-plane in r la corresponding to x are xsdo:j

+ d/31 ::; 0,

j

= 1.2.

The conditions that this plane belongs to the hyperplane F!l dOl ~ d{:Jl + PtJ.a dr~ = 0 tangent to K are F~l -

X3

F'13t ::; f~a -

x3

F'~

+ F~2 ~ +

= O.

This implies the following assertion. PROPOSmoN 5.6. The coordinates of the cnticol point of cm admissible complex K on a line (0,8) E K are given by the relations

F!.

F!,

FfJ.

F'.82

X3=-=-,

PROPOSITION 5.7. The dimension of the manifold of criticol points on the lines

of an admi8sWle complez K does not

~

two.

PROOF. Let K" c K be the subset of smooth points h E K whose critical points x are finite. To prove the proposition. it suffices to show that the mapping K" -. C3 taking every line (n,p) E Kit to its critical point x = (%10:1'<2,%3) is everywhere degenerate. It follows from tlte equations of the lines that dXi -

Q

1 dxa

= xada, + d/Ji,

i == 1.2.

Hence, it follows from the relations Xa = ~ = ~ and Pa. 001 + ~2 002 + it. Ba F'/,l diJl + p~ dll2 = 0 that the differentials of the coordinates of the critical points are related as fonows: (5.10)

F's,.

(dXl - Qt

dXa) + ~ (dx2

- Q2 dx3 )

= O.

This proves the assertion.

It follows from Proposition 5.6 that, for an admissible complex K, the manifold of critical points is either a complex Clln-e A or a complex surface S in C3. In too first CaRe the complex K consistM of the lines intersecting the curve A. In the other case it follo\\.-s from (5.10) that the lines (a, P) of the complex K are taDgent to the surface at their critical points. Thus, the proof of the theorem is complete. REMARK. There is a simplt' relation between the problem of recoDStrt1cting a function f E S(Ca) from the integrals of f over the lines of an admissible line complex K in C 3 and the following problem for functions tP(Ql,a2'Pt.~) on C": Find a solution ,p of the system of equations

(5.11)

dltb IJo.l

1J8J

= 1Jo.dlrJ! , 2 IJ{Jl

78

2. JOHN TRANSFORM

satisfying the boundary condition 1/IIK =~. Here K stands for a complex submanifold of et given by an equation F(Q, (3) = O. It is assumed that F satisfies (5.6), i.e., that K is a characteristic submanifold of system (5.6). In this way we obtain a Gomsat problem if derivatives do not enter the boundary data. Correspondingly, such a problem can be weD defined only on characteristic submaniftds. It follows from (5.11) that 1/1 is the John transform of a function / on C 3 , i.e., 1/1 = ~ /. Therefore, the solution of the boundary value problem is reduced to the application of the inversion formula for admissible complexes. Namely, using this formula we first find the function / from the function ~ = 1/1IK. Then the original function 1/1 is obtained as the John transform of the function /. i.e., 1/1 = ~/. Conversely, the problem of integral geometry to reconstruct a function / from the function ~ = 1/1IK is reduced to the solution of a boundary value problem. Namely, if the boundary value problem for the system of equations (5.11) is solved, i.e., the function 1/1 is reconstructed from the function ~ = 1/I1K, then the desired function / is determined by the inversion formula for the John transform.

CHAPTER 3

Integral Geometry and Harmonic Analysis on the Hyperbolic Plane and in the Hyperbolic Space The specific feature of transfonus in integral geometry as compared with other integral transforms is that the former often admit an immediate generalization from the flat case to the curved one. We illustrate this fact here by cousideriDg anaIogti of the Radon transform for tbe hyperbolic plaue and hyperbolic space. We shall see that the scheme of constnlcting these new transforms is just like that for t-he Radon transform in the affine space. and even the structure of inversion formulas is similar. However, the nonzero curvature of the hyperbolic space leads to a richer geometry. In particular. on the hyperbolic plane there are two analogs of lines, namely, linetl themselves (i.e., geodesics) and horoc:ycles (circles of infinitely large radius). Two different versions of the Radon t.ransfonn are relat..ec:l to these 3'0 analogs of lines. For the byperbolic plant> (and space), there is an analog of the Fourier transform using unitary representations of their motion groups. Since these groups are noncommutative. the theory of Fouricr integral for the Lobacbevsky plane (space) is much more complicated than that of its Euclidean analog. Seemingly, the simplest way to construct a non-Euclidean theory of the Fourier integral (representation theory) is to use the Radon transform for the horocycles (borospheres). The point is that in the case of hyperbolic space. as in the affine case, the analog of the Radon transform for horocycles (horospberes) and the analog of the Fourier transform are related by the one-dimensional (affine) Fourier transform.. For this reason, t-he horocycle (borospherical) Radon transform is quite important when constructing harmonic analysis on the hyperboli(' plane and space. This transform admits generalizations to other important classes of homogeneous manifolds. By the example of hyperbolic plane and space we simulate quit.e general constructions of harmonic analysis on homogeneous manifolds. When reading this chapter, it is very useful to keep in mind (and compare with) the corresponding constnlCtions of §§ 1 and 2 of Chapter 1. 1. Elements of the hyperbolic pJanimetryl

1.1. Models of the hyperbolic plane. The hyperbolic plane £.2 is the honwgeneous space of the group SL(2,R) with tbe isotropy subgroup 80(2) (the subgroup of orthogonal matrices). In the Poincare modelthis plane is represent-ed as the upper half-plane Im z > 0 of the plane of complex variable t = x + iy. where the motions corresponding to matrices 9 =

(~~)

E SL(2,1ll) are given as follows:

QZ+,

z 1-+ z 0 9

= fJz + 6 .

1Tbe main Facts lIhout hyperbolic geometry are presented without proofs. 79

3. INTEGRAL GEOMETRY IN 1:. 2 AND 1:.3

80

Obviously, the stabilizer of the point The Riemannian metric on £2 is

%=

i is the subgroup SO(2).2

(1.1)

The invariant distance d between points defined by the relation

%1

= Xl + iYI

and

Z2

= X2 + i1l2

is

(1.2)

The area element in the metric (1.1) is (1.3)

dv

= y- 2 dx dy.

Another way to define the hyperbolic plane is to consider the manifold of real symmetric matrices (1.4)

u

= (: :).

where a> 0 and

detu = 1.

with the motions g E SL(2, Ill), where g' stands for the transpa;ed matrix. Note that the manifold of matrices of the form (1.4) can be represented 88 a sheet of a hyperboloid of two sheets in R3: u ...... g'ug,

(1.5)

QC -

f12

= I.

a,e> O.

The correspondence between the points % of the upper halF-plane and the DUV trices u = u( %) is described 88 follows. The point % = i corresponds to the identity matrix e because the point i and the matrix e have the same isotropy subgroup. Hence, for each g = (~~) E SL(2, R), the matrix u = g'g corresponds to the point

. I . lor ~ g = (11112 = 8"+1 i+ . In partlCU ar,'log = X +'&y %11 2 11-0) 2 ,and we see that to -I( 2+ 2) -I ) a point % = X + iy corresponds the matrix ( 11 ~ 11 11 in other words. to a 1I:1t 11 matrix (~ :) corresponds the point z = ~ + ~. Geometrically, the passage from the

. log

1/

I

1/

_1%

;

sheet of the hyperboloid (of two sheets) to the half-plane is carried out as follows. We first project the hyperboloid (1.5) parallel to the axis a to the plane (b,c); this projection is the half-plane e > O. After this we make a projective transformation in the plane (b,e) by setting (b,e) ...... (~, ~), and this transformation sends the half-plane e > 0 into itself. It is convenient to pass in (1.5) to the new coordinates Xl = ~. %2 = G;C, X3 = b. Then we obtain a realization of the hyperbolic plane £2 as the upper sheet of the hyperboloid of two sheets in R3 given by (1.6) [x, x) == x~ - x~ - .r~ = 1. Xl > O. In this realization, the motions become linear transFormations in R3 that preserve the upper sheet of the hyperboloid, i.e., the elements of the group SOo(I,2) e! SL(2,R)/{±e}. The Riemannian metric in this model is (1.7)

ds 2

= -~ + dx~ + dx~.

2Note tbat tbe c:enter of the group 5£(2, R). which consists of two elements ±e, act.s ineffectively (preeerves any point). Therefore, it is convenient to view tbe quotient group 5L(2,R)/{±e} as tbe motion group.

1. ELEMENTS OF HYPERBOLIC PLANIMETRY

81

The invariant distance d between points x and , is (1.8)

coshd = [x, ,I.

The area element in the coordinates (1.9)

X2, X3

on £2 is

dv = dx2 dx3 • Xl

1.2. Horocycles. On the hyperbolic plane one can define circles of infinitely large radius (horocycles), which are the limits of non-Euclidean circles as the center and the radius of these circles consistently tend to infinity. In the Poincare model, the horocycles are represented either as (Euclidean) circles tangent to the real axis (the absolute) or as lines parallel to the real axis. The latter objects can naturally be interpreted as circles of infinitely large radius tangent to the real axis at the point X = 00. It follows from this description that every horocycle is given by an equation of the form

(1.10)

1(2z - (11 2 = y,

~

= (~1'~2) E R2 \ O.

Thus, the manifold of horocycles is parametrized by the points (~lt(2) E R2 \ 0, and the horocyc1es corresponding to the points (~1t~2) and (-~1'-~2) coincide. Denote by heel = h(elt ~2) the horocyc1e given by (1.10). REMARK. If a plane £2 is represented as the upper sheet of the hyperboloid given by (1.6), then the horocycles are the plane parabolic sections of this hyperboloid, namely, the plane sections of the form [~, x] E (1 Xl - (2 X2 - e3 x3 = 1, where ~ is a point of the upper nappe of the cone [~, ~I = 0, > 0, ~ E R3.

el

Obviously, the motion group 8L(2,R) acts on the manifold of horocycles H, and this action is transitivej the motion corresponding to a matrix 9

= (~ ~) sends

a horocycle h«(lt~2) to the horocycle h(ael + -r(2,P(1 + 6e2). In the language of motions, the horocycles are described as follows. They are the orbits3 of the unipotent subgroup Z of the matrices (~y) and all their translations. A group conjugate to Z acts on every horocycle, and the horocycle is a Euclidean line with respect to this action. Thus, the manifold H of horocycles is a homogeneous space of the group 8L(2, R), namely, the homogeneous space with the isotropy subgroup ±Z. Therefore, when passing from the hyperbolic plane £2 to the manifold of horocycles H, we obtain a new homogeneous space with the same motion group but with a different stabilizer. Constructions of this type play an important role in geometry. Every horocycle has a unique common point with the absolute, which is called the center of the horocyc1e. In the Poincare realization, this is the tangency point with the real line. Horocycles with common center are said to be pamlleL Note that a horocycle h(~lt~2) is tangent to the real axis at the point X = ~j hence, every pencil of parallel horocycles is of the form {h(~elt ~2) 1 0 < ~ < oo} for some chosen pair (el,e2). All points of a horocycle h(~el,~e2) are placed at the same distance 211og~1 from the parallel horocyc1e h(~1te2}' We refer to the expression p(Zje) -210g~

=

3In the Poincare model, these are the lines parallel to the real axis.

82

3. INTEGRAL GEOMETRY IN 1:. 2 AND 1:.3

as the oriented distance from a point z E h(~) to a horocycle heel. This distance is given by (1.11) (because ~ = (y-II{2Z - {112)-1/2). Every horocycle heel partitions the hyperbolic plane c,2 into two parts c,+({) and C,_({), where c,+({) abuts on the absolute and the boundary of c,-(e) has only one common point with the absolute, namely, the center of the horocycle heel. The domains c,+({) and c,-(e) consist of the points z E C,2 for which p(z;{) > 0 and p(z; e) < 0, respectively. REMARK. The homogeneous spaces c,2 (c,2 = SO(2) \ SL(2,R» and H (H = ±Z\SL(2, R» are dual to each other. By construction, to points { E H correspond curves heel in c,2 (horocycles). At the same time one can assign to a point z E c,2 the curve l( z) E H corresponding to the set of horocycles passing through z. Each of these curves is a trajectory of the subgroup SO(2) or a subgroup conjugate to SO(2). This trajectory partitions the manifold H into two parts, namely, the domain H+(z) in which p(z;e) > 0 and the domain H_(z) in which p(z;e) < O. In the realization H = R2 \ 0, the curves I(z) are all possible ellipses on R2 centered at the origin 0 and bounding a domain of unit area, and H+(z) and H_(z) are the exterior and interior domains of the ellipse 1(%), respectively. Note that the manifold of all orbits of the subgroup SO(2) and the subgroups conjugate to SO(2) is not homogeneous; in the realization at hand, this manifold consists of all ellipses on R2 centered at the origin O.

1.3. Geodesics. Consider now the "lines" (geodesics) on the hyperbolic plane c,2. In the Poincare model, these are the semicircles on the half-plane Im z > 0 with centers on the real line, and the half-lines orthogonal to the real line. Hence, every "line" is given by an equation of the form

(1.12)

(x - a)(x - b)

=

+ y2 = O.

=

a, bE R U (00),

a

-=F

b.

In particular, for a 00 (or b 00) we obtain the family of (Euclidean) half-lines orthogonal to the real line. It follows from (1.12) that the manifold of "lines" can be parametrized by the pairs (a, b), a -=F b, where a, b are the points of the projective line pI, and the "lines" corresponding to the pairs (a, b) and (b,a) coincide. In the language of motions, every "line" is an orbit of the subgroup D of diagonal matrices or of a subgroup conjugate to D, and all "lines" can be obtained from one another by motions. Thus, the manifold of "lines" is the homogeneous space D \ SL(2, R). We thus obtain another homogeneous space with the same motion group SL(2, R) and a different isotropy subgroup. Note that the manifold of all orbits of the subgroup D and subgroups conjugate to D is not homogeneous. In the Poincare model, this manifold coincides with the manifold of all half-lines on the half-plane Im z > 0 and of circular arcs having exactly two common points with the real line. IT the hyperbolic plane is represented as the upper sheet of the hyperboloid of two sheets given by (1.6), then, in this realization. the geodesics are the curves of intersection (one branch of a hyperbola) with planes in a3 passing through the origin 0, i.e., with the planes (1.13)

2. HOROCYCLE TRANSFORM

Denote by "Y«() the geodesic given by (1.13). Obviously, a plane (1.13) intersects the hyperboloid (x. xl = 1 if and only if < O. Moreover, since "Y(~) = "Y(e} for each

~.;:.

le,e]

0, we can set

le. e) = -1. Thus, the manifold of geodesics on tbe hyperbolic plane can be parametrized by the point-s of tbe byperboloid [~. () = -1 of w.'O sheets in R3, and the geodesics corresponding to the points and coincide.

e

e

-e

2. Horocycle trausform

2.1. Definition of the operator 'R,.h. In this section we study an analog of the Radon transform related to borocycles on the hyperbolic plane. DEFINITION. By the horocycle translorm we mean the integral tnmBfmm 'R,.h on £2 sending each smooth compactly supported function I OD £2 to the integrals of I over the horocycles.

(2.1)

'R,."

ICe) = j,,{O f I(Z)d8,

where h(el is tJle horocycle given by (1.10) and d8 the hyperbolic plane. The expression for 'R,." can also be written

88

= I¥ is the length element on

follows:

(2.2)

where p(z;{) is the distance from the point % to the borocycle h({) (see (1.11», and 6(·) is the delta function on R. It follows from the definition that the operator 'R,./a commutes with the arlion of tbe group SL(2,R). Thus, we have a pair of homogeneous spaces of the same group SL(2, R), namely. tbe hyperbolic plane and the manifold ofboroc:ycles, and an operator sending functions on one space to functions on the other and. commuting with the motions. Note that tbe integration is over the orbits of the isotropy subgroup of t.he second space and over the shifts of these orbits. 2.2. Inversion formula. Let us find t.he inversion formula for the integral transform 'R,.h following the same scheme as in tbe case of Euclidean plane (see § 1 of Chapter 1). First, take a radially symmetric function I depending only on the IlOIl-oEuclldean distance from a point z to the point i:

I(z}

= F(1I- 1/ 2 Iz - il}·

Since the operator "R" commutes with the motions of the plane £,2. it follows that "Rh I(e) depends only OD the distance from the point i to the borocycle h(~),

i.e., 'R,." I({)

= F(f')'

where r

= P(i;~) =

log({~ +~~).

3. INTEGRAL GEOMETRY IN t:. 2 AND 1:.3

84

Thus, the transform F( r) is equal to the integral of the function lover the horocycJe h(er / 2 ,O) for which 11 = er, i.e.,

F(r)

=

(2.3)

1 1

+00

F(e- r / 2.jx2 + (er - 1)2)e- r dx

-00

F(s)sds

+00

-2

21 s1nh il er / 2 Js2 - 4sinh2

-

~

(cf. (1.2) in § 1 of Chapter 1). Hence, the passage from F(s) to er / 2 F(r) reduces to the Abel transform. and it follows from (2.3) that

(2.4)

F(s)

11

(er / 2 F(r»' dr . 7r 2unb i>O J4sinh2 ~ _ s2

= --

In particular. I(i)

(2.5)

= F(O) = -1.1 (er / 2 !(r~)' dr. 00

27r

0

Slnh

2

Note that e r / 2 F(r) is a smooth even function by (2.3). Therefore. the integral (2.5) converges (because (er / 2 F(r»~=o = 0). and the expression for I(i) can also be represented in the form

1+

• = _1.

1(·)

(2.6)

00

47r

(e

r/

-00

2 F(r»' dr. sinh ~

Assume now that I is an arbitrary smooth compactly supported function on £2; we apply to I the averaging I(z) = F(y-l/2Iz - iD over the nOD-Euclidean circles centered at the point i,

F(

y

-1/2Iz _ il) = 1.1211" I 211"

0

( z~9 + sin 9 ) -zs1D9 + cos 9

dJJ:

the expression for F can also be represented in the form

F(s)

=~ f

7r It:.2

I(z) 6(,I- 1 Iz

- il2 -

s2) ~:y. 11

Note that I(i) = F(O). Let F(r) = 'Rh I«(), where r = log«(~ + ~~). Since the operator 'Rh commutes with the action of the group 8L(2, R), it follows that

F(r)

= 1.1211" 'Rh l(er/2 cos9,e-r/2 sin 9) d9 27r

= 21

(2.7)

11"

0

e- r

1

'Rh 1«()«(1 £1(2 - (2 £1(1)

p(i;()=r

= .!.e-r f 'Rh I«() 6(p(i; () - r) £1(1 £1(2, 7r la2 where p(i;() is the distance from the point z = i to the horocycle h(~). and (2.6), we obtain (2.8)

. _ - . _

1

I(t) - 1(1) - --2 11"

100 (e 0

r / 2 F(r»'

'oh r

SI

2

_ 1 dr - -47r

1+00 -00

Using (2.5)

(e r / 2 F(r»' 'oh r dr SI 2

2. HOROCYCLE TRANSFORM

85

(cf. (1.4) in § I of Chapter I). Substituting the expression in (2.7) for F(r} into (2.8), after elementary manipulations we obtain 1 [ e- p (l;()/2 ( { ) {) ) h (2.9) f(i)=-411"2JH±(i)Sinh~ ~1{){1 + {2{){2 +1 'R f({)d(1d(2,

where H±(i) are the domains of H2 in which p(i; {) > 0 and p(i;~) < 0, respectively. To obtain an expression for f at an arbitrary point z, one must apply the group translation taking the point i to the point z. We thus obtain the following assertion. THEOREM 2.1. A smooth compactly supported function I on t,2 can be expressed in terms of its horocyele transfonn 'Rh f by the following inversion fonnul4:

(2.10)

__ 2-1°C 2

f(z) -

11" 0

2-j+OC (e

(e r / 2 F(z,r»'r _ _ . hr dr 4 Stn 2 11"

-00

r / 2 F(z,r»'r 'oh r SI 2

dr,

where (2.11)

1

(2.12)

[

e-P(z;()/2 ( { )

{)

)

h

f(Z}=-411"2JH±(%)Sioh~ {1{)~1 + {2{){2 +1 'R f({)d(1d(2

(e/. (1.5) and (1.6) of Chapter I), where H±(z} are the domains of H in which p(x; {) > 0 and p(z; {) < 0, respectively.

it + {2i&

Note that {I parallel horocycles.

is the operator of differentiation along a pencil of

2.3. Asgeirsson relations. To invert the transform 'Rh, we used (2.4) for 8 f. 0, this formula contains an additional information on the relation between the averages of the function f and the horocycle transform 'Rh f of f. 8

= 0 only. For

THEOREM 2.2. The following Asgeirsson relations hold for the hyperbolic plane:

(2.13)

11

(e

F(Z;8} = - 11"

r / 2 F(z,

2sinbr/2>_ J4Sioh2

r»), r

i - 82

dr,

where F(r; s) is the mean 0/ the function / over the non-Euclidean circles centered at the point z,

(2.14) and

21

F(z,s}=-

11"

£2

2 dx'dy' f(z')6(" -1 y' -1 Iz'-zl 2 - s )--,

11'2

F( z, r)

from

is the mean of the function 'Rh f over the set of horocycles equidistant the point z; see (2.11). We also have

(2.15)

F-( z,r ) --

21

00

21ainb 51

er / 2

F(Z,S)8ds . 8 2 - 4 sinh2 i

J

3. INTEGRAL GEOMETRY IN 1:. 3 AND 1:. 3

86

Formulas (2.13) and (2.15) are obtained from (2.4) and (2.3) by a group translation; compare with the Asgeirsson relations (1.8) and (1.10) in § 1 of Chapter 1. 2.4. Symmetry relation. It follows from (2.15) that r ..... er / 2 F(z.r) is an even function of r, Le..

er / 2F(z, r) = e- r / 2F(z, -r) for any z and r. Substituting the expression (2.11) for F(z, r). we obtain the following symmetry relation for the function 'Rh I:

(2.16) e- r / 2

f

'Rh 1«()6(p(z;()-r)~1 ~2

= er / 2

f

'Rh 1«()6(p(z;{)+r)~1 ~2'

2.5. Inversion formula for the horocycle transform in another model of the hyperboUc plane. Let the hyperbolic plane {,2 be represented as the upper sheet of the hyperboloid in R3 given by

=

[x, x) x~ - x~ - x~ = 1. XI > 0. In this model, the horocycles are the sections of the hyperboloid by the planes

[(,xl

=eIXI-e2X2 -{3X3

= 1,

where { is a point of the upper nappe «(I > 0) of the cone K given by ((,{I The horocycle transform is then given by the following formula:

'Rhl«()

(2.17)

=f

= 0.

(eK,

l(x)6([{,xl-l)dx.

11:. 2

where dx

= x.l dx2 dx3 is an invariant measure on {,2.

THEOREM

2.3. A MTWOth COfIIpoctl,l8upportedjunction Ion {,2 can be expressed

in terms 01 its horocycle translorm 'Rh I by the lollowing inversion lormula: (2.18)

I(x)

= (2!)2 ['Rh I«() ([{,xl-l)-2~.

where

~ = {l' ~2 ~3.

The integral must be understood in the sense of regularized value [101. namely, as the value at I' = -2 of the analytic function of I' given for ReI' > by the convergent integral

°

(211')-2 ['Rh I«()

H~,xl- 111' cl(.

Note that formulas (2.17) and (2.18) are similar to the corresponding formulas for the Radon transform in Chapter 1. Similarly to the case of Radon transform, it suffices to prove the inversion formula (2.18) for functions on {,2 depending only on the distance from the point XO = (1,0,0). We can readily see that, for these functions I, the inversion formula (2.18) coincides with the one proven above.

3. Analog of the Fourler transform on the hyperbolic plane and the relation between this analog and the horocycle transform 3.1. Fourier transform on R2. At the naive level. the classical Fourier integral on R2 is the expansion of a function in L2(R2) in the eigenfunctions of the operators of parallel translation, Le., in the exponential functions ei(~IZ+~211). However, this integral is also closely related to a larger group, namely, the full motion group of the Euclidean plane. Let us indicate this relation.

3. ANALOG OF THE FOURIER TRANSFORM

Consider the Laplace operator fl. on the Euclideao plane, i.e., t.he second-order differential operator ~ = + commuting with the motions. It is known that this operat.or on L2(R2) has continuous Lebesgue spectrum coinciding "ith the half-line (-oo, 0). The problem is to construct sufficiently many distributional eigenf'unctions of tltis operator such that any function I E L2(22) can be expanded in these eigenfunctions. We sball argue at the naive level without specifying what object must be regarded as a distributional eigenfunction in the case of continuous spectrum and what is an expansion of a function / E L2(R2) in these distributions. Choose a one-parameter subgroup of translations in the motion group of the Euclidean plane; for instance. take the subgroup of translations parallel to the y axis. Each of the eigenspaces of the operator ~. say, the eigenspace corresponding to the eigenvalue _).2 (). E R \ 0), contains a pair of distributjons that are iDV8riant 'CIIritb respect to this subgroup, namely, e:t:·Az. Take one of these functions. for instance, eW:, and apply to it all possible motions. Obviously, the function e&A= is multiplied by a number under any translation and turns into e'~(z_9~aia') under the rotation through tJu.. angle IJ about the origin. Thus, for every eigenvalue _).2 of the operator ~ we obtain the family of eigenfunctions e i A.{zmatl+1/alnl) (in part.icular, the funct,ion e-j.\z). There are no linear relations among these functions, and every eigennmction 1>. of the operator A corresponding to the eigenvalue _).2 can be expanded in these functions,

£ lis

(3.1)

b.(x,y)

= 2...

f u~(IJ)e·~(zc:od+l/tdn9)

2w 0

d8

(). > 0).

Wc introduce the following norm in the space of eigenfunctions of the operator A that correspond to the eigenvalue _).2:

1I/l1l2 =

{3.2}

2~

1 lu,,(IJ)12 2 •

dB

and consider tht' related Hilbert spare H". In H" we obtain & unitary representation T" of the motion group of the Euclidean plane; namely, iD terms of the functions u~, to each translation (x. 7/) - (x + Xo,1I + 110) corresponds the operat.or of multiplication by the function eU(zo .-8+w aid), and to the rotat.ion through the aogIe a about the origin corresponds the operator of rotation u" (.) 1-1< u~ (. + a). It can readily be seen t.bat the representation T" is irreducible. The representation of the motion group in the space L2(R2) can be decomposed into the representations T". Let us describe this decomposition. Every function / E L2{R2) can be expanded in the integral

1= LOQ )./~ d>..,

(3.3)

where /l is given by (3.1) witb u~(8) = j(>.cos8,).sinIJ) (; stands for the Fourier transfonn of f). Moreover, the following Plancberel formula holds: (3.4) where

1l1A.1I is gi~-en by

(3.2).

3. INTEGRAL GEOMETRY IN

88

t:,2

AND

t:,3

3.2. Fourier transform on the hyperbolic plane. We now carry out similar considerations (at the same nonrigorous level) in the case of hyperbolic plane. Consider the Laplace-Beltrami operator A. on the hyperbolic plane 1:.2 , i.e., a second-order differential operator commuting with the motions of 1:.2 ; it is of the form

2(1fl Ifl) 8x2 +{Jy2'

A.=y

Let L2(.c2) be the space of measurable functions on 1:.2 with square-integrable absolute value (with respect to the invariant measure dv = On L2(I:.2) the operator A. has continuous Lebesgue spectrum coinciding with the interval (-00, -1/4). Let Z be the unipotent subgroup of matrices of the form ( : Y). By the Fourier integral on the hyperbolic plGne we mean the expansion of a function I e L2(.c2) in some special distributional eigenfunctions of the operator A., namely, in the distributional eigenfunctions that are invariant with respect to the subgroup Z, and the shifts of these functions. Just these functions are analogs of the exponential functions e i «(l Z+(211). Each eigenspace of the operator A. corresponding to some eigenvalue of the form (s e R) contains two functions invariant with respect to the subgroup Z, namely, the functions y~. Let us take one of them, say y~; under all possible translations, this function transforms to functions of the form

11").

_lit

(3.5)

+.(%;{)

= (yl{2% -

{lr 21¥ )

= exp (l+iS --2- P(%;{) ) , a point % = x + iy to the horocycle h({).

where p(%; {) is the distance from We call functions y~ %onal horocycle functions and translations of the form (3.5) are simply referred to as horocyde Junctions. We see that if { and {' are proportional, then the corresponding functions •• differ by a factor only. There are no other relations among the functions +. for S > 0, and if we take a family of functions +.(%;{) in which s ranges over the positive reals and { over a set having exactly one representative on each line in R2 passing through the origin 0, then we obtain a complete function set in which every function in L2(I:.2) can be expanded. Let us define the Fourier transform of a function on the hyperbolic plane 1:.2 by analogy with the classical Fourier transform. DEFINITION. By the Fourier translorm of a smooth compactly supported function I on .c2 we mean the inner product of I and the generalized horocycle function (3.5),

(3.6)

F I(£.,s)

=

1 t:,2

dxdy

1(%)·.(%; {)-2 .

Y

Since the functions •• are homogeneous with respect to {, it follows that the function F I is also homogeneous with respect to {,

(3.7)

F I(>.{;s)

= l,\ri . - 1 F I({;s).

3.3. Relation to the horocycle transform and the inversion formula.

The completeness of the family of horocycle functions +. means that an inversion formula for the Fourier transform F must exist. Let us construct it. First, we establish a relation between the Fourier transform F I and the horocycle transform 'Rh I. AB in the Euclidean case, the passage from a function I to

89

3. ANALOG OF THE FOURJER TRANSFORM

F / can be carried out in two steps. We first integrate the function / over the horocycles p(z;e- l / 2() == p(z;() - ~ = 0, and then integrate the resultiDg function multiplied by e-!!f!l with respect to ~. The first integration gives

Thus, the following assertion holds. THEOREM 3.1. The Fourier trans/onn on £,2 and the horocycle trans/onn are related as follows:

(3.8)

By the inversion formula for the one-dimensional Fourier transform, this implies that

1

+00

'Rh /«()

(3.9)

= (211r 1

-00

F /«(; 8) ds.

One can obtain the inversion formula for the Fourier transform :F from the lmown inversion formula (2.12) for the horocycle transform 'Rh. Note that in the Euclidean case we obtained the inversion formula for the Radon transform from the inversion formula for the Fourier transform; here we proceed in opposite direction. Substituting in (2.12) the expression for 'Rh / in terms of F / given in (3.9) we obtain (3.10)

f(z)

1 f 1+

= - 8r 1R2

00

-00

e-p(,::()/2

sinh ~

(8 + 8(28 + 1)F /«(; (18(1

(2

8) dsd(1 d&.

It remains to simplify the resulting expression. We first note that the homogeneity condition (3.7) for :F J implies

Further, we pass in (3.10) to the "polar" coordinates ( = erE-', where -00 < r < +00 and F.' belongs to a contour r c R2 \ 0 that intersects once every ray issuing from the point O. We obtain

Integrating over r (the integral over r must be understood in the principal value sense), we obtain the foUowing assertion.

3. INTEGRAL GEOMETRY IN t;2 AND t;3

90

THEOREM 3.2. A smooth compactly supported junction Ion (,'l can be expressed in terms 01 its Fourier translorm :F I by the lollowing inversion /ormvla: 4

I(z)

= S7r12

(3.11)

1+ (£ 00

x

=

S!2

7rS

stanh"2

-00

1:

00

:F I(~i s) exp ( - 1 ~ is P(z;~»)

stanh

~s

(£

(~1 d{2 -

{2 d{d) ds

:F 1(~;s)ct_.(Z:{)(~l d{'l - {'l d{d) ds.

In the proof of (3.11) we changed the order of integration; however, we did not prove rigorously that this change is indeed possible. 3.4. Symmetry relation. We show that the function I can be expressed in > 0 only. This is related to the following fact: the shifts of the functions,,~ belonging to the same eigenspace of the operator ~ can be expressed in terms of one another. Namely, terms of the values :F I(e; s) for s

,,¥ = c(s)

1:

00

(ylz -

ell- 2 )A¥ d{lo

where c(s) = 7r- 1/ 2 r ('~'.) Ir (¥). This readily implies the following symmetry relation for :F I: the integml

£:F f(e;s)ct-.(Z;eHel de'l- e2d{1) is an even /unction 01 s. Thus, by the symmetry relation, the inversion formula for the Fourier transform F can be represented as (3.12)

I(z)

=

4!21

OO

stanh ~s

(t:F f(e;s)ct-.(Z;~Hel d{2 -

e2d{1») ds.

REMARK. The symmetry relation for :F I can also be obtained from the symmetry relation (2.16) for the horocycle transform 'Rh I. Namely, substituting in (2.16) the expression for 'Rh I in terms of I and passing to the "polar" coordinates ~ = e'e', e' Er, we see that the integral

(3.13) e- r / 2

t 1: 1: 00

00

:F

e(l-i.)t:F I(e: s)6(2t+p(Zie)- r)dsdt(el d{2-{2d{d

is an even function of r. Integrating over t, we prove that the integral

{1+

lr

00

e-ier:F I({;s)e-¥p(z~)ds(el d{2 - {2d{d

-00

is an even function of r. Obviously, this condition is equivalent to the symmetry condition for :F I.

4nus (onaws from P.v.f+ao -OD 'o __

L". Te _l.p 2 •

-1f1WIIIU

.,-.rOdr olnh(a+l>

o dr = i e 6·P f+ ao alnr. dr = = e'·PP.V.f+ ao "-il ola r alDhr -DO

-00

4. RELATION TO THE REPRESENTATION THEORY

91

3.6. Plancherel formula. The definition of the Fourier transform F the inversion formula (3.12) imply the foUowing aasertion.

I

and

THEOREM 3.3. The lollowing Plancherel formula holds lor any smooth compactly supported function f on C2:

(3.14)

(lI{z) 12 d:c: y = 4 12 (00 ( Btanh 7r2BIF 1{(;s)1 2«(1 d(2 y 7r 10 lr

11:.2

~2d(1)ds.

Indeed, replacing the function fez) in the integrand of J l{z)/(z)~t'l by the expression for I(z) in terms of F I, we obtain

(1/{z)12d:c:Y=412 ( (00 (stanh7r2sl(z)F/{(;S) Y 7r 1/:.210 lr l+iS] d:cdy xexp [--2- P(z;() (~1d(2-(2d(1)ds7

11:.2

= 4!2 1000

£

stanh ~sl.r 1{(;s)12«(1d(2 -(2d(1)ds.

By the Plancherel formula, the Fourier transform can be extended from the smooth compactly supported functions to all functions I E L2(C2) according to the classical scheme.

4. Relation to the representation theory of the group SL(2, R) Let us show the relation between integral geometry and the representation theory of Lie groups by the example of the hyperbolic plane C2. We have the Hilbert space L2(C2) and a unitary representation T of the group SL(2, JR) in L2(C2),

Tgl(z)

= I(z 0 g),

9 E SL(2,R),

where % ..... % 0 g is the motion (corresponding to g) of the hyperbolic plane. The problem of representation theory (harmonic analysis for the hyperbolic plane) is to decompCl8e L2(C2) into irreducible invariant subspaces. This problem is similar to that of decomposing a function in L2(1R2) in the Fourier integral, and irreducible invariant subspaces can be treated as analogs of exponential functions. We show that the solution of this problem is given by (3.12) and (3.14) (see §3). We first describe the irreducible invariant subspaces themselves. Consider the eigenspace corresponding to an eigenvalue - 1~.2 of the Laplace-Beltrami operator~. As was already said before, this space is spanned by the distributions •• (z;() = (yl(2% - (21-2)~, where ~ ranges over a set containing exactly one representative of every line in R2 passing through the origin O. Thus, the elements of this eigenspace can be presented in the form

(4.1) where the function

f.(z) = U.

£

u.«() •• (z; ()«(1 d(2 - (2 d(l),

is homogeneous, u.(~) = 1~1'·-lU.«(),

and the integral is taken over any contour r c R2 \ 0 intersecting at one point every ray issuing from the origin O. (The integral does not depend on the choice of r because the differential form in the integrand is invariant with respect to the

3. INTEGRAL GEOMETRY IN r:~ AND

£,3

homothetic transformations ( ...... ~i cf. (1.14) in Chapter I.) In the space of functions I" we introduce the norm

i

11/,112 = lu,«()1 2«(1 ~2 -

(4.2)

(2 ~J)

and define the Hilbert space H, as the L2 space with respect to this norm. It follows from what was said in 3.4 that the spaces H, and H _. coincide. The motions on (.2 determine a unitary representation T. of SL(2,R) in H •. In terms of the functions u.«(), this representation looks 88 follows:

(T,,(g)U.)«(l,e2)=U.(Q(1+'Y~,,8(1+6(2)'

g=

(~ ~).

It is known (see, e.g., 18]) that this representation is irreducible. Introduce the projection operator p. : L2(£2) - H. by the rule

p.I = I., where I, is given by (4.1) in which u.«() = F I«(,s). Obviously, these operators p. commute with the action of the group SL(2, R) on the spaces L2(£2) and H •. Thus, the problem of decomposing the space L2(£2) into irreducible subspaces consists in expanding the functions I E L2(£2) in their projections I •. The solution of this problem is given by (3.12) and (3.14), since these formulas can be written 88 follows: (4.3)

(4.4) where

1

1= 4'1r2

1 £,a

[00 stanb"2/• 'lrS

0

1I(z ) 12dzd1l Jj2"

ds ,

I.=p./i 'lrsll 12

1 foo = 4'1r2 10 s tanb"2 1.1

ds,

11/.11 is given by (4.2).

Recall that the proof of the main formula, (3.12), used the inversion formula for the operator 'R" of horocycle transform. We clarify the role of this operator from the groul>'theoretic point of view. The operator 'R" sends a function I(z) on the homogeneous space £2 = SO(2)\ SL(2,R) to the function 'R" I(e) on the homogeneous space H ±Z \ SL(2,R), and this operator commutes with the action of the group SL(2, R) on these spaces. Moreover, the irreducible components transform into irreducible components; however, a priori this mapping may have a nonzero kernel. In fact, the kernel of the mapping is trivial because there is an inversion formula for and the problem of decomposing the representation of SL(2, R) in L2«(.2) is reduced to the problem of decomposing the representation in a function space on H = ±Z \ SL(2, R). The solution of the latter problem is very simple for the following reason. The homogeneous space H is endowed with an action of the commutative group of homothetic ~ commuting with the action of the group SL(2, R). Consider transformations the corresponding commutative group of the operators AA:

=

'R",

e. . .

(AA/)(e)

= I(~).

The common eigenspaces of AA consist of homogeneous functions. Since the operators AA commute with the operators of the representation of SL(2, R), it follows that their eigenspace8 are invariant with respect to this group. It is known that the representations of SL(2, R) acting on these subspaces are irreducible. Thus, the decomposition of the function space on H into irreducible components is reduced

5. INTEGRAL TRANSFORM

93

to the decomposition into the eigenspaces of the operators A~, and since the group of operators A~ is commutative and isomorphic to the additive group R, this decomposition is reduced to the one-dimensional Fourier transform. The composition of the horocycle transform 'Rh and the above one-dimensional Fourier transform is the Fourier transform :F on the hyperbolic plane. Thus, the role of the horocycle transform is that this transform reduces the problem of expanding functions on the homogeneous space £2 = SO(2) \SL(2, R) to the expansion problem for functions on another homogeneous space ±Z\SL(2,R), where this problem admits an elementary solution. Similar transforms reducing decomposition problems for group representations related to homogeneous spaces to a problem of integral geometry can be defined not only for £2 but also for other homogeneous spaces. This tool, known 88 the method 01 horospheres, was first introduced in [5]. The construction used in this section to introduce irreducible representations of the group SL(2, R) can be included in the general scheme of realization of irreducible representations in function spaces on homogeneous spacesj this construction originates &om Frobenius. A finite-dimensional irreducible unitary representation T of a group G can be realized in functions on a homogeneous space U \ G if and only if the representation space contains a nonzero vector invariant with respect to the subgroup U i the dimension of the space of these vectors is equal to the multiplicity of occurrence of the representation T in the space L2(U\G). The Frobenius duality theorem explicitly defines an operator that embeds T in L2(U \ G). In the infinite-dimensional case one needs some analytic refinements. These are the functions 1/~ considered above; they are invariant with respect to the subgroup Z, and the construction of irreducible representations of the group G = SL(2, R) that correspond to these functions follows the Frobenius scheme. Since every irreducible space contains two (distribution) vectors invariant with respect to Z, it follows that this space is embedded in L2(±Z \ G) with multiplicity two. However, every irreducible representation enters the decomposition of L2(£2) with multiplicity one, due to the symmetry relation for :F I.

5. Integral transform related to lines (geodesics) on the hyperbolic plane £2

5.1. Deftnltlon and the inversion formula in the Poincare model. Consider another analog of the Radon transform for the hyperbolic plane, namely, the integral transform related to "lines" (geodesics) on £2. According to § 1, every "line" on £2 is given in the Poincare realization by an equation on the half-plane lID % > 0 of the form (5.1)

(z - a)(z - b) + 1/2

= 0,

where a :F b and a, b e R U (oo)j we denote this "line" by G(a, b). It readily follows from (5.1) that under the motion z .... a line G(a, b) transforms into the line

i:!l

G(~,~). DEFINITION. Denote by 'R' the integral transform taking each smooth compactly supported function 1 on the hyperbolic plane £2 to the integrals of this

3. INTEGRAL GEOMETRY IN t:,2 AND t:,'

function over the lines in £,2 (i.e., over the geodesics with respect to the metric in £,2),

(5.2)

'R,9I(a,b) = (

ds

I(z)ds,

= /dz/. 1/

10(11,11)

Obviously, this transform commutes with the action of SL(2, R). Let us find the inversion formula reconstructing I from 'R,' I. In the proof of this formula we follow the scheme used in the case of Radon transform on the Euc1idean plane and the horocycle transform on £,2. We first consider a function I E D(£,2) that is coDStant on the non-Euc1idean circles centered at the point i, i.e., on the orbits of the subgroup SO(2) of matrices 9 (coe' - .iD'). this condition means that I has the form

= sin' coe' '

Since the transform 'R,I commutes with the action of the group SL(2, R), it follows that the function 'R,I 1(0, b) is also invariant with respect to the tnmsforms in 80(2). One can readily see that the fraction ~*-T is a full invariant of a pair of points a, b E pI with respect to 80(2). Thus, 'R,I 1(0, b) = from (5.2) that

F(r)

='R,I I(oo,r) =

(00 F

(r2

10

=1,(r2+l)1/2 F( s 1(r2 +l)1/2

It follows

+ (1/ -1)2) dy 21/

1/

ds

00

= (00

F (~"!1 ).

1) """"'i'9;==ri====:=t= S2 - (r 2 + 1)

J

(s-lF(s

-1» .,f82 -sds(r2 + 1) .

Thus, F(r) is an even function of r, and it follows from the inversion formula for the Abel transform that S-1 F(s

- 1) = _! 1r

In particular, for s (5.3)

fooH=i ../r2F'(r) dr; +1- 2

s

~ 1.

8

= 1 we obtain I(i) = F(O)

= _! 1r

(00

10

Per) dr r

(the integral converges because F'(r) = oCr) as r - 0). We now assume that I is an arbitrary smooth compactly supported function on £,2 and 7 is the result of averaging I over the DOn-Euc1idean circles centered at the point i,

5. INTEGRAL TRANSFORM

95

Since the transform 'R9 commutes with the motions of (,2, it follows that

'R9j(a, b)

=F =

(1

1..

+ab) a-b

f'hr'R 9

21r 10

I

(aCOS9 + sin 9 bcos9 + sinB ) -asinB+cos8' -bsin8 + COS 8

d(J

'

and hence

F(r)

= 'R9j(oo,r) =

(5.4)

21 f2fr 'R' I (-cot8, rC~89+sin(9) d8 1r 10 -rsm +008

=.!. f+oo 'R9 1 (t, rt -1) ~. 1rl_oo

r+t

l+t

In view of (5.3), we obtain

I(i)

(5.5)

= -I(i) = --1r2 LOO -F'(r) dr, 0 r

where F(r) is given by (5.4). To find an expression for I at an arbitrary point z, it suffices to apply a group translation 9 sending the point i to the point z, for instance, 111/2 9 = ( rg-1/2

0)

y-1/2

•

Under this translation, I(i) goes to I(z) and the transform 'R9 1(o.,b) goes to 'R9 1(x + o.y, x + by). We obtain the following result. THEOREM 5.1. A smooth compactl,lsupported /unction I on {,2 can be u;pre8sed in terms 01 its translorm 'R9 I related to the lines in (,2 by the lollowing inversion

lormula: I(z) where

~

F(z; r)

= _~ fOO F~(z;r) dr, 1r 10 r

1f+OO 'R I ( x + ty, x + rtr +- 1)y

= ;: Loo

9

t

dt 1 + t2·

5.2. Relation to the Radon transform on the projective plane. We now assume that the hyperbolic plane {,2 is represented as the upper sheet of the hyperboloid given by (x, x] 1, where (x, xl xY - xi - x~. In this realization, the geodesics on (,2 are the curves 'Y(~) which are the intersections of the hyperboloid with the planes

=

=

and the integral transform 'R9 is given by the formula

(5.6)

'R91(~) =

1

I(x)ds,

(~,~l = -1,

-r(E)

where ds is the length element on (,2. One can readily see that the expression for

'RI can also be represented in the form (5.7)

'R9I(~)= f l(x)6«(~,x})dx2dx3,

l£a

where 6(·) is the delta function on R.

Xl

3. INTEGRAL GEOMETRY IN £2 AND £3

96

In this model, one can obtain the inversion formula for the integral transform 'R' similarly to that for the Minkowski-Funk transform related to the unit two-dimensional sphere; in other words, the inversion formula can be found as a consequence of the inversion formula for the Radon transform 'R on the projective plane; see § 8 of Chapter 1. Namely, to any function I E (,2Coc we assign a function Fin R3 \ 0 satisfying the homogeneity condition F(.u) = ..\-2 F(x) and such that F 11~.~1=1= I and F = 0 for [x,x] < O. It follows from the definition of the operator 'R that 'R F(e) = 'R'I(e) for [e,e] = -1 and 'RF(e) = 0 for [e,e] > O. Therefore, the following assertion is an immediate corollary of the inversion formula given in Chapter 1 for the integral transform'R (see (8.10) on page 36). THEOREM 5.2. A function I on (,2 can be expressed in terms 01 its translorm 'R' I related to the geodesics by the lollowing inversion lormulo.:

(5.8)

I(x) = 41 1r

f 'R' I(e) (e, x) -2 w(e). J1E•EJ =-1

For the meaning ofthis integral, see § 8 of Chapter 1.

6. Horospherical transform in the tJuoe.dimensional hyperboUc space {,3 The definitions and results of §§ 1-5 for the hyperbolic plane can be extended to hyperbolic spaces of higher dimensions: however, the inversion formula has different forms for spaces of odd and even dimensions. For this reason, we give an independent exposition for the three-dimensional hyperbolic space {,3. Here we put an emphasis on explicit formulas.

6.1. Models of the hyperbolic space. Let us briefly present the elements of geometry in {,3. The hyperbolic space (,3 is the homogeneous space of the group SL(2, C) with the isotropy subgroup SU(2) (the subgroup of unitary matrices). The following quatemion model of the space {,3 is convenient for our purposes: a point of {,3 is a quatemion with three components, w

= x + iy + zj,

Z

> 0,

where x. y. Z E R, ;2 = j2 = -1, and ;j = - ji; the motion of (,3 corresponding to a matrix (~~) E SL(2,C) is given as follows:

= (wiJ + 6)-I(WQ + 1). Obviously, the isotropy subgroup of the point w = j is SU(2). w ...... w'

The Riemaonian metric on (,3 is (6.1) ds 2 = z-2(dz2 + dy The distance d between points by the relation (6.2)

where (6.3)

Wl

+ dz 2 ).

= Xl + Yl; + zd and w, = X, + 1I2i + z,j is given

97

6. HOROSPHERlCAL TRANSFORM

and I . I stands for the norm of a quaternion t'lernent in the metric (6.1) is ofthe fonD (6.4)

dv

(lwl' = z2 + If + Z2).

The vo)wne

= z- 3 dxdydz.

In a di1ferent way, the hyperbolic space can be given by the manifold of Hermitian-symmetric matrices (6.5)

u=(: :),

O.C

> 0,

detu = 1,

with motions u ...... g"ug, 9 e SL(2,q. where gO stands for the Hermitian--conjugate matrix. Note that the manifold of matrices (6.5) can be interpreted as a sheet of the following byperboloid of two sheets in R":

(6.6)

QC -

~

-

~

= 1,

a,e> 0 (bJ + i~ = b).

The correspondence ~-een the quaternions w and the matrices u follows:

.

.

x+yt+zJ......

(z-I(X 2 +y2+.:2) -1 (') If: X -111

= u( w) is as

») ;

Z-l(X+iY z-1

!

in other words. to a matrix (5!) corresponds the ql1atemion 1V = + ~j (cf. the case of the hyperbolic plane). In particular, the identity matrix corresponds to the point w:::: j. It Is convenient to pass in the equation of the hyperboloid given by (6.6) to the new coordinates Xl = ~, X2 = ~. X3 = bt, X4 = b:z. Then we obtain a realization of the hyperbolic space as the upper sheet of the following hyperboloid:

(6.7)

Ix,x] == x~ - x~ - ~ - x~ = 1,

Xl

> O.

In tbis realization. the motions arc given by linear transforms in R4 presening the upper sheet of the hyperboloid. i.c.. by elements of the group 800(1,3) 9!! 8L(2, C)/{±e}. The Riemannian metric in this model is ds 2 = -~ + dz~ + dzi + dz~i the in,'lUiant distance d between points X and 11 is

ooshd = (X, 11); the volume element in the coordinates dv

x2. X3

on

1:,2

is

= dz, tlz3 U4 . XI

6.2. Horospheres. In the hyperboUc SpaL'e one can define spheres of in6nit.ely largt' radius (horospheres), which arc limits of non-EucHdean spheres as the eent« and the radius of these spheres simultaneously tend to infinity. A horosphere can be represented either as a (Euclidean) sphere (in the balf-spac:e .z > 0 of R3) that is tangent to the plane z = 0 (the absolute). or 88 a plane (in this haIf-space) parallel to the plane % = O. The latter can be naturally interpreted 88 a sphere of infinitely large radius tangent to the plane If: = 0 at a point at infinity. In the quatemion model, every horosphere is given by an equation of the form (6.8)

IW{2 - {112

= Z.

W

= x + ui + zj.

3. INTEGRAL GEOMETRY IN C 3 AND

98

,3

where ~ = ({1'~2) E C 2 \ O. Thus, the manifold of horospheres is parametrized by the points ~ E C2 \ 0, and the horospheres corresponding to the points { and ~, IAI = 1, coincide (cf. the equation of horocyc1es on the hyperbolic plane). Denote by h({) the horosphere given by (6.8). REMARK. H the space £3 is represented as the upper sheet of the hyperboloid (6.7), then the horospheres are the plane parabolic sections of this hyperboloid, namely, the sections by the planes of the form [{,x] = {I Xl +{2 X2+{3X3+{4 X4 = 1, where ~ is a point of the upper nappe of the cone [{,~I = 0, and thus {I > 0 ({ E R4 \ O).

The motion group 5L(2, C) acts on the manifold of horospheres, and this action is transitive; the motion corresponding to a matrix 9 = (~~) takes a horosphere !(~.. {2) to the horosphere !(O{l

+ "'~2, .B~l + 6{2). In the language of motions, the horospheres are either orbits of the subgroup Z of the matrices of the form (: Y), tEe (these are the planes parallel to the plane z = 0) or shifts of these orbits. Thus, the manifold of horospheres is the homogeneous space of the group 5L(2, C) with the isotropy subgroup ±Z. Each horosphere has a unique point of tangency with the absolute; this point is called the cemer of the horosphere. In our interpretation, these are the points of tangency with the plane z = O. Horospheres with a common center are said to be paralleL Note that the horosphere h(~} is tangent to the plane z = 0 at the point ~; hence, every pencil of parallel horospheres is of the form {h( A{) I 0 < A < oo}, where { is fixed. The (oriented) distance p(w,{) from a point w E £3 to a horosphere h(~) is defined in the same way as the distance from a point to a horocycle on the hyperbolic plane, and is given by a similar formula w =x+yi+ zj.

(6.9)

6.3. Horospherical transform. DEFINITION. By the horospherical transform in £3 we mean the integral transform 'Rh taking a smooth compactly supported function f on £3 to the integrals of this function over the horospheres,

(6.10)

'Rh f(~)

=[

f( w) M,

lh(f.)

where h({) stands for the horosphere given by (6.8) and M for the (non-Euclidean) area element on h(~). The expression for 'Rh can also be written as (6.11)

'Rh f({) = [

lC3

f(w)6(p(w;~»dxd~dz, z

where p( w; {) is the distance from the point w = X + yi + zj to the horosphere h({); see (6.9). It follows from the definition that the operator 'Rh commutes with the action of the group 5L(2, C).

6. HOROSPHERlCAL TRANSFORM

99

6.4. Inversion formula. We now proceed with the coDStruction of an inversion formula for the integral transform Rh by repeating the arguments used in the case of hyperbolic plane. We first 88SUlDe that f is a radial function dependiDg only on the distance from the point w to the point j,

few)

= F(Z-1/2Iw -

jl).

In this case, Rh f({) depends only on the distance from the point j to the horosphere h(~), i.e., Thus, the value F( r) is equal to the integral of the function . h( e r/2 , 0)· given by z = e r , I.e.,

. . = f.

F(r) (6.12)

R2

F

f

over the horosphere

(JX2 + y2 +/2(er - 1)2) -:;dxdy r er

e-

= 271"100 F (Jt2 + (er

er / 2

o

-1)2)

tdt

= 27r

e2r

(00 121uab 11

F(s)sds. er

First, this implies that er F( r) is an even function of r,

er F(r)

= e- r F( -r),

and second, (6.13)

(erF(r»'

= -27r( coshrF(2sinh i) + sinhrcosh iF' (2 sinh i))·

It follows from (6.13) for r = 0 that

IU> =

(6.14)

F(O)

1

....

= - 27r (er F(r»'lr=o

(d (2.5) in this chapter and (2.4) in §2 of Chapter 1). Assume now that arbitrary smooth compactly supported function on £3 and J(w)

= F(Z-1/2Iw -

I

is an

jl)

is the function obtained by averaging I over the non-Euclidean spheres c:entered at the point j. Since these spheres are orbits of the subgroup of unitary matrices 9

= ( ~tt :) , lal2 + IPI2 = 1, it follows that f(w)

= F(z-1/2Iw -

jl)

=/

l«wP + O)-l(wa

-P» dp.(a,p),

where dp.(a,p} is the invariant measure on the group of unitary matrices, i.e., on the three-dimensional unit sphere lal 2 + IPI 2 = 1, normalized by the condition

/ d,,(a, P)

= 1.

We can readily see that the expression for F can be written in the following convenient form:

(6.15)

3. INTEGRAL GEOMETRY IN £.2 AND £.3

100

=

=

=

Let F(r) 'Rh f(e), where r p(j;e) log(leil 2 + le21 2). Since the transform 'Rh commutes with the action of the group 5L(2, C), it follows that

F(r) (6.16)

=

1.

'Rh l(er / 2Q, er/ 2fJ) d/J(Q, fJ)

101 2 +1111 2 =1

= 11"2 (~r

12

'Rh l(e r/ 2el,e r / 2e2) 6(p(j;e» del del de2de2,

where pU; e) is the distance from j to the horosphere heel. By (6.14),

IU)

~ = -IU) = - 211"1 (er F(r»"lr=o.

Substituting here the expression for F(r) given in (6.16), we obtain

I(j)

r

= - 81

11" JI~112+1~212=1

(6.17)

=-

8~

L2 'Rh I(e) d/J(e)

L

L2'Rh I(e) 6(p(j;e» dv,

where (6.18)

B L = el Bel

B

-B

-B

+ e2 ae2 + el ~1 + e2 ae2 + 2,

dv

= (i)2 2 del del de2 de2.

This gives an expression for IU) in terms of its horospherical transform 'Rh I. Hence, applying a translation that takes j to an arbitrarily chosen point w, we obtain the following assertion. THEOREM 6.1. A function I on £3 can be expressed in tenns 01 its horospherical translonn 'Rh I by the lollowing inversion lormula:

(6.19)

I(w)

where (6.20)

F(w,r)

= 11"-2

~ " = - 211"1 (er F(w, r)) Ir=o,

L

'Rh l(er/ 2e" er / 2e2) 6(p(w.e))dv;

hence,

(6.21)

I(w)

= - 8~ /(;2 L2 'Rh I(e) 6(p(w;e»

dv,

where the operator L is given by (6.18). Since 'Rh I(e i 'el. e i'e2) where L~ = el + e2~ + 1 and L~ REMARK.

ii

4L:'Rh 1= 4L:'Rh I

= 'Rh I(el, e2), we have L~ 'Rh I = l~ 'Rh I, = el ~ + e2at; + 1. Therefore, L2 'Rh 1=

= 4L~L~'Rh I.

Note that this inversion formula is local (in contrast to the case of hyperbolic plane); namely, to reconstruct a function I at a point w, one must know the integrals of I only over the horospheres infinitely close to w.

101

6. HOROSPHBRICAL TRANSPORM

6.5. Symmetry relation. We obtain the fobing result as a by-product, applying a group translation to (6.12). THEOREM 6.2. The following

Asgeirsson relation holdJJ for the hyperbolic space:

211'12111lnh il F(w; s)sds. 00

eTF(w:r) =

(6.22)

Here F( w, s) standa for the result of averaging the function f over the centered at the point w, i.e., (6.23)

F(w.s)

= 11'-1

La

=(211')-1

%1/2%,1/2110 -

LJsinh

sphere~

w'r 1f(w')6(z-l z ,-1 Iw - w'12 _ 82)dr'~dz'

d(w;w')

1-1 6(4sinh2 d(w~w') _ 82 )

f(w'}dz':!d.t,

uthere d(w.w') 18 the non·Euclide4n dUltance between w and 'Ill, a:nd P(W,T) is the result of averaging the junction 'R" J over the horospheres equidistant from the

point w; see (6.20). COROLLARY.

The junction eT F( w. r) is

etre1I

as a juncticm oJ r,

eTF(w,r) =e-rF(w,-r). Substituting bere the expression for the following symmetry relation:

Fgiven by (6.20), we see that 'Rh I satisfies

er J'Rh f(e r / 2{l,e r / 2{2)6(p(w;{»du

(6.24)

= e- r f'Rh J(e- r / 2(1,e- r / 2(2)cS(P(W;(»dv.

,3

6.6. Inversion formula for the horospherical transform In another model of the hyperbonc space. Let us represent the byperbolic space as the upper sheet, Xl > 0, of the hyperboloid Ix. x} == ~ - ~ - ~ - ~ = 1 in R4. In this model, the horospheres are the sectiODS of the ~ by the byperplalle8

le. x] ==e1 Xl -

(2%2

-(aX3 -{.X4

= 1.

when> { is a point of the upper uappe K, Xl > 0, of the cone giwn by [(,{j The borospberical transform is given by the following formula:

'Rh J({)

(6.25) where U

= ( J(x) 6«e. x] lc.s

1) U,

= xII U2u3 dx4 is an invariant measure on

=

o.

~ E K,

'3.

THEOREM 6.3. A smooth compactly .rupported function f on ,3 am be ~ed in tenns of the horospherical transform 'Rh J by the following inversion formula:

(6.26)

lex) =

-s!,2 L'Rh J«()&H([(,X] -1)~,

~ ={11c1Qd(3d(4,

where 6" (.) is the second denvative 0/ the delta function cm R.

3. INTEGRAL GEOMETRY IN

102

(;2

AND

(;s

Formulas (6.25) and (6.26) are similar to the corresponding formulas of Chapter 1 for the Radon transform and, as in the case of Radon transform, it suffices to prove the inversion formula (6.26) for functions on C3 depending only on the distance from the point xO (1,0,0,0). One can readily see that, for these functions I, the inversion formula (6.26) coincides with that proven above.

=

8.'1. Integral tr8DSform related to completely geodesic surfaces in C3 • Here we consider another integral transform on C3; it takes functions on C3 to their integrals over the completely geodesic surfaces in C3. We denote this transform by

'R' .

e

We represent the hyperbolic space 3 as the upper sheet of the hyperboloid (6.7) in R4. In this model, the complete geodesics are the sections of this hyperboloid by the planes {(, x)

(6.27)

= O. = -1. The integral transform

== (IXl + (2X2 + (aX3 + ~X4

where ( ranges over the hyperboloid of one sheet [(, () 'R' is given by the following formula:

'R'I«()

= 1(;2 ( I(x) 6( {(, x»

d:r2 d:r 3 d:r 4 , Xl

where 6(·) stands for the delta function on R; d. the definition for the hyperbolic plane in § 5. AB in the case of hyperbolic plane, the inversion formula for the integral tr&lllr form in question can be obtained as a consequence of the inversion formula for the Radon transform 'R.. on the projective space; see § 8 of Chapter 1. Namely, to a function I e C3 we assign the Coo function F on R4 \ 0 satisfying the homogeneity condition F(.\:r) = 1~1-3 F(x) and the conditions F Ilz,z)=1 = I and F = 0 for

[x, xl < o.

It follows from the definition of the operator 'R that 'R F«() = 'R'/«() for ((,(] = -1 and 'R..F(e) = 0 for le,e) > O. Therefore, the following assertion is an immediate coro1lary of the inversion formula (8.6) of Chapter 1 for the integral transform'R (see page 32). THEOREM 6.4. A smooth compactly supported junction I on C3 can be expressed in terms 01 the translorm 'R..' I rel4tetl to the completely geodesic sur/aces by the lollowing inversion formula:

(6.28)

I(x)

= _(211')-2

(

'R' J(e) 6"({e.

1[(.{)=-1

x» w(e),

where 6"(·) stands lor the second deritJative 01 the delta junction on R. For the meaning of this integral; see § 8 of Chapter 1. We emphasize that the inversion formula (6.28) is local, in contrast to the case of hyperbolic plane.

Appendix. HOl'08pherica1 tr8DSform for the hyperbolic space of an arbitrary dimension. The hyperbolic space en 01 an arbitrary dimension n is the upper sheet of the hyperboloid in Rn+1 given by

(6.29)

[x. x] == :rf - x~ - ... -

X!+l

= I,

Xl

> 0,

7. ANALOG OF THE FOURlER TRANSFORM

103

in which the motions are the linear transformations of the space Rn+! that preserve this hypersurface. By the horospheres we mean the sections of the hyperboloid by the hyperplanes

[{, x] := {I Xl

(6.30)

- ••• - ~n+!

Xn+l

= I,

where { is a point of the upper nappe K ({I> 0) of the cone [~,{] = O. The horospheriool translorm is given by 'R" I({)

(6.31)

= f I(x) 6([(, x] It!ft

1) dz,

(

e K,

where dz = xII dz2 ... dzn+l is an invariant measure on £n. The structure of the inversion formula depends on the parity of n. We present this formula without proof. If n = 2m + I, then

I(x) = 2\~!~;m

(6.32) where de then

(6.33)

= (1 1 de2'"

L

'Rh I«()

6(2m)([(, X] - 1) de,

den+l is an invariant measure on the cone K. If n

lex) = (-I)m(2m -1)1 f 'Rh I«()([e xl _1}-2m de (21r)2m

lK

'

= 2m,

.

Thus, as in the case of Radon transform, the inversion formula is local for odd n and nonlocal for even n.

7. Analog of the Fourler tramsform in the hyperboUc space, and its relation to the horospherical transform 7.1. DefInition of the Fourier transform. We define the Fourier integral for the hyperbolic space £3 in the same way as in the case of hyperbolic plane. Consider the Laplace-Beltrami operator 6 on £3,

2(fP fP fP) 0 OX2 + 8y2 + 1Jy2 -%8z'

A=z

It is known that the operator 6 on L2(£3} has continuous Lebesgue spectrum coinciding with the interval (-00, -I). Let Z be the unipotent subgroup of complex matrices of the form ( 1 By the Fourier integral on £3 we mean the expansion of a function / e L2(£ } in certain special functions related to the operator A, namely, in the eigenfunctions that are invariant with respect to the subgroup Z, and their shifts. Every eigenspace of the operator 6 corresponds to an eigenvalue of the form (8 e R) and contains two functions invariant with respect to the subgroup Z, namely, the functions z~. Take one of them, say, za;uj any translation sends it to a function of the form

J)'

4±l

(7.1)

•• (Wj() = (zlwe2 - (11- 2a;u )

= exp (2+iS --2- P(w;() ) ,

where p(w;~} is the distance &om the point W = X + yi + zj to the hor08phere h({). We c&ll the functions za;u the zonal horospherical /unctions, and their shifts (7.1) &re simply referred to &8 horospherical functions. If and are proportional, then the corresponding functions •• differ by factors only, and for 8 > 0, there are no other relations among the functions .,. If 8 ranges over the positive re&ls and ( over a set containing exactly one representative of every complex line in C2 \ 0

e

e'

3. INTEGRAL GEOMETRY IN

104

t:.2 AND t:.3

passing through the origin 0, then these functions form a complete set of functions in which every function in L2(£3) can be expanded.

I

DEFINITION. The Fourier translorm of a smooth compactly supported function on the hyperbolic space £3 is defined by the relation

:F 1(1.;8)

(7.2)

= Jt:.f 3/(W)+-.(WiF.) tb:dydz z3

(cf. the definition of :F I for £2 in (3.6».

+.

Since is homogeneous with respect to (, it follows that:F I is also a homogeneous function of

e,

:F I(~; 8) =

(7.3)

1.\li . -2 :F 1(1.; 8).

7.2. Inversion formula. The completeness of the family of horospherical functions +. means that an inversion formula for the Fourier transform :F must exist. We construct this formula by repeating the argument carried out in the case of hyperbolic plane. First, 88 in the case of hyperbolic plane, there is a simple relation between the Fourier transform :F I and the horospherical transform 'R,h I, namely,

:F I(e; 8) =

(7.4)

1-00+00

'R,h

l(e-),/2e)e¥), d.\.

Hence, by the inversion formula for the ontHlimensional Fourier transform we obtain (7.5)

'R,h 1(1.)

= (211")-1

1-00+00

:F 1(1.; 8) ds.

The inversion formula for the Fourier transform :F can be obtained from the inversion formula (6.21) for the horospherical transform 'R,h by substituting the expression for 'R,h I in terms of :F I (see (7.5» into (6.21). This gives (7.6)

I(w)

1 = -16,.-4

LC2 1+-00

00

L 2:F I(ei 8) 6(p(Wie» dsdv,

where

8

L = 1.1 8e1

8

-8

-8

+ e2 8F.2 + e1 8(1 + 1.2 81.2 + 2.

We simplify this expression. It follows from the homogeneity condition (7.3) for

:F I that L 2 :F I(e; 8) = -82 :F I(e; 8). Further, we pass in (7.6) from to the new coordinates F. = ~/, where .\ e C \ 0 and e' ranges over a surface r c C2 \ 0 intersecting once almost every complex line passing through the origin O. We obtain

e

I(w) =

1~ (~)2L:00 B2(ifcl.\I··:FI(eiB)6(2Iogl.\l+p(Wie» x d.\dl(e1 tIe2 - e2 tIe1)(e1 tIe2 - 1.2 tiel) ) ds.

Integrating with respect to .\, we finally obtain the following assertion.

8. REPRESENTATION THEORY FOR THE GROUP SL(2.C}

THEOREM

in tenns

7.1. A smooth compactly supported/u.nction / on (,3 am he ezpressed

0/ the Fouricr tmnsform F f

(7.7)

105

i loo i

1 . few) = lfur3

+oc

xcxp

= U:1T3 ~

by the following mversion formula:

s2 F f«(; 8)

2+iS) (~1~2-e2~1)(el~2-e2~1)d8 - - -( -~p(w;()

1: i 00

s2 F

f({;s)~"(W;{)({l ~2 -

{-

2~1)({1 ~ - (2~1)d8.

7.S. Symmetry relation and the Plancherel formula. Similarly to the case of hyperbolic plane. the shifts of the zonal horospberical funct.ions z:~ and z ¥ can be expressed in terms of each other,

zlT = c(8)~ where c(s)

=

*.

l
2 c

w-

{11-2)~~1 ~l'

This implies the following symmt'try relation for F

£ 1«(; 8)~.(W;()((1 F

de, -

I:

The integml

t., det}({l de2 - e,ded

is 4n even /tInction 0/8. By this relation, the nmction / on (,3 can be expressed in terms of t.he values of the function F s) for 8> 0 only,

(7.8) I(w)

t

fee;

= 8~3 ~ LOO a2 F I({i s)~.({; S)({l de2 -

(2del)((i tlf.2 - (.2 de0dJl.

REMARK. As in the case of hyperbolic plane, the symmetry relation for F can be obtained from the synametry relation (6.24) for 'Rh f.

The next assertion follows from tilt- definition of the Fburier transform F from the inversion formula (7.5). THEOREM 7.2. The following PlanchereJ fonnula holds pactly supported function I on c3 :

f

lea

(7.9)

for

I

I

and

tm1J smooth com-

l/(w)12dzdydz %3

=

~ ~ 100 82 (i 1F f(t.; 8)1 2(el ~2 - (2 ~l )(el ~2 - (2 ~l») ds

(cf. the Plancherel formula (3.14) Cor C2). Using the Planchercl formula, one can t.'Xtend the Fourier transform F f to all functlous f E L2(CJ) by the classical

scheme. 8. Relation to the representation theory for the group SL(2,C) By analogy with the case oC hyperbolic plane, we establish a relation between the results of § 7 and the represe.ntation theory for the group SL(2, C). We haw the HiJbert space L2(C3) and a unitary representation T of the group SL(2.C) ill this space.

Tgf(w)

= I(w 0 g),

9 e SL(2,C),

3. INTEGRAL GEOMETRY IN

106

e2 AND CS

where W ...... wog is the motion (of the hyperbolic space) corresponding to g. The problem of representation theory (harmonic analysis for the hyperbolic space) is to decompose L2(£,2) into the irreducible invariant subspaces. One must first describe the irreducible invariant subspaces into which the space L2(£,2) can be decomposed. Consider the eigenspace of the Laplace-Beltrami operator 4 on £,3 corresponding to an eigenvalue This subspace is spanned by the distributions •• (Wj() (see (4.1» and the elements of the subspace can be written 88

¥.

(8.1)

I.(w)

£

= ~ u.(~)•• (Zi e)(el fie2 -

(2 fiel)(el fie2 -

e2 fiel),

where u. satisfies the homogeneity condition u.(~)

(8.2)

= 1..\1,·-2u.(e),

..\eC\o,

and the integral is taken over an arbitrary surface r c C2 \ 0 that intersects once almost every complex line passing through the origin O. We equip the space of functions I. with the norm (8.3)

11/.112 =

i

lu.(e)1 2(el fie2 -

e2 fiel)(el fie2 - e2 fiel),

consider the space L2 with this norm, and denote the latter space by H.. The motions on £3 define a unitary representation T. of the group SL(2,C) in H •. In terms of functions u.(e), this representation is of the form

g=(~ ~). It is known that this representation is irreducible [8]. We introduce the operator p. : L 2 (£3) _ H. by the rule p.1 = I., where I. is given by formula (8.1) in which u.(~) = F I(e). This operator commutes with the action of the group SL(2, C) on the spaces L2(£3) and H •. Finally, using (7.8) and (7.9), we obtain the expansion of a function I e L2(£3) with respect to the irreducible suhspaces, 1 (00 I. = p./, (8.4) I 811'3 10 8 2 /. ds,

=

and the Plancherel formula (8.5) where

(

lea

lI(w)12dxd1ldz z3

= _13

(+00 s211/.11 2ds,

811' 10

11/.11 is given by (8.3).

9. Wave equation for the hyperbolic plane and hyperbollc space, and the Huygens principle 9.1. Two-dimensional cue. R.ecaIl that the Laplace-Beltrami operator on the hyperbolic plane £2 is 4 2 = y2 + and the spectrum of this operator in L2(£2) is the interval (-00, -1). Consider the following wave equation on £2:

(-£ I/;s),

(9.1)

lPu =. (lPu + lPu) + 41.1· 1 &2

.2 11

8%2

8y2

107

9. WAVE EQUATION

The term 1u is added for the spectrum of the operator on the right-hand side to coincide with the half-line (-00,0). We show how one can solve the following Cauchy problem for equation (9.1): u~(O,%)=/(%)

u(O,%) =0.

(9.2)

by using the horocycle transform; for simplicity we assume that / E V(£2). The argument is parallel to the process of solution of the Cauchy problem for the wave equation on Euclidean plane (see §3 of Chapter 1). Our solution uses the method of horocycle waves based on the following remarkable fact.

PRoposmON 9.1. 1/ !p(s) iI a smooth function

0/ one voriable,

then the func-

tion (9.3)

~(t, %)

= e:i:t/2!p(p(%; e:i:t/2(» = e:i:t/2!p(p(%; () ± t),

where p(%;() = log(1I- 11(2% - (112) iI the diltance from % to the horocycle h«(), a solution 0/ the wave equation (9.1) for each ( E R2.

u

The proof can be obtained by a direct verification. The functions (9.3) are called horocycle tlJtJve8. The method 0/ horocycle UHWe8 consists in representing an arbitrary solution of (9.1) as asuperposition ofborocycle

waves. We now proceed with the construction of a solution of the Cauchy problem (9.2). Let 'Rh / be the horocycle transform of a function / and let F(z;r) be the result of averaging the function 'Rh / over the borocycles equidistant from a point % E £2; see (2.11). Consider the function

(t ) u,z

(9.4)

= -~1+00 e~F(%,r+t)~_ 4

11' -00

It follows from (2.11) that

e~ F(%, r + t) = !e-~ 11'

J.

R2

. hr sln:l

UT.

'Rh f«() 6(P(%;() - r - t) d(l d(2,

and thus u( t, %) is a solution of (9.1) as a superposition of horocycle waves. We show that u is the desired solution of the Cauchy problem (9.2). By the symmetry relation (2.16), r ...... er / 2 F(z, r) is an even function of r; hence,

1 1+00 er / 2 F(%;r) u(O,z) = --4 sinb!: dr = O. 11' -00 2 Further, by the inversion formula (2.10) for the borocycle transform we have

.!..1+

00 (er / 2 F(z, r»~ dr = I( ) . hr Z. 11' -00 SID:I We want to retum in (9.4) to the original function /. To do this, we use the Asgeirsson relation (2.15) for the hyperbolic plane. From (2.15), for t > 0 we obtain '(0,z ) = _ 4

Ut

u(t,%) (9.5)

= _~

1+00

[00 F(z,s)sdsdr 211' -00 12181nh ~ 1 sinb !: ·/s2 - 4 sinb2 !:±! 2V'

1 [00 [

2

F(z,s)sdrds

= - 211' 10 12181nh i 1<8 sinh t=.! '/S2 2

V

-

4 sinb2 !: ' 2

3. INTEGRAL GEOMETRY IN ~I AND ~3

108

where F( %,8) is the result of awraging the function I OWl" the non-Euclidean circles in £2 centered at the point % (see (2.14»i the integral with respect to z must be understood in the principal value sense. We carry out the integration with respect to r in (9.5) by using the following relation: 5

J = -

1

= {O

6

tU: -6 sinh(z - a)Vsinh2 b - sinh2 z

for 0.< b, for a>b.

11'

Jelnha:-Blnha6

Here a, b > O. We obtain

u(t,%)

(9.6)

= -112alnb t 2 0

F{z, 8) ds

~sinh21- 82/4

.

Substituting here the expression for the function F(z, 8) in terms of I (see (2.14», we arrive at the following result. THEOREM 9.1. The solution 01 the Cauch1l problem (9.2) lor the watle equation (9.1) is 01 the lorm

u (t ,z) -

(9.7)

.!.1 411'

I(z')

cI(II,II')<&

~sinh2

i _sinh2 cI( lIt )

tU:'d1l' y'

2

'

where d(z, z') 8tands lor the non-Euclidean distance between the points z and z' (see (l.2». The formula thus obtained is similar to the Poisson formula for the wave equation on the Euclidean plane; see §3 of Chapter 1. AB in the Euclidean case, the solution of the equation at a point % at a moment t depends only on the initial data in the non-Euclidean disk of radius t, centered at the point % (due to the existence of the forward front of the wave). 9.2. Tllre.dlmenslonal case. We now pass to the hyperbolic space £3. Recall that the Laplace-Beltrami operator on £3 is

2(lP lP lP) 8 8z + 8y2 + 8z2 - 8;: ,

t::. = %

2

%

5ComputatloD of tbe Integral J. Obviously, J is an odd function of Cl. Hence, J

-! 1" - 2

_I>

(lIinh-l(z - Cl) -aiDh-l(Z+ a»dz -1" ainhClcoehzdz v'ainh2 6 - sinh2 Z -I> (ainh2 z - ainh2 a)v'ainh2 6 - ainh2 z .

Therefore, by elementary manipulatloDl we get J-

[

IDU

- elDU

(t2

-

ainhadt 11 sinhadt ainh2 a)v'ainh2 6 _ e2 - -1 (t 2 ainh2 6 - aiDh2 CI)v'f='i!

= aiDha 1"/2

tit

-71/2 ainh2 6ain3 t - aiDh2"

The Iaal integral can be obtained dlrectly: P. V. J

" > 6 (see formula 3.613.1 in (:l5».

=-1"/2

tit

-"/2 aiDh6ain t + ainh,,'

=0 for Cl < 6, IUKl J = - Jlialai a-slab " i for I>

9. WAVE EQUATION

and the spectrum of this operator in following wave equation on

.c3:

(9.8)

lPu lJt2

109

L2(.c3 ) is the interval (-00, -1). Consider the

lPu) = z2 (lPu ox2 + lPu 1Jy2 + OZ2

- Z

lJu

oz + u.

.c

AB in the case of 2 , the term u is added for the spectrum of the operator on the right-hand side to coincide with the half-line (-00,0). We solve the following Cauchy problem for (9.8) by the method of horospherical waves: (9.9)

u(O, w) = 0,

u:(O, w) = I(w)

(w = x

+ ui + zj).

The method uses the following property. PROPOSITION 9.2. 11 !p(s)

is a smooth function 01 one tJari4ble, then the func-

tion 4)~(t,w)

(9.10)

= e:i:c!p(p(w;{) ± t),

where p(w;{) = log(z- 1Iw{2 - {112) is the distance from a point w to a horosphere h({). is a solution 01 the wave equation (9.8) lor etJCh E C2.

e

The proof reduces to a direct verification. We refer to the solutions of the form (9.10) as horospherico.l waves. Let us proceed with solving the Cauchy problem (9.9). Let 'Rh 1 be the hornspherical transform of the function 1 and let F(w; r) be the result of averaging the function 'Rh lover the horospheres equidistant from the point w E .c3 ; see (6.20). Consider the function 1 d ,(9.11) u(t, w) = - 27f dt (e F(w, t». It follows from (6.20) that

e'F(w,t) = 7f- 2e-'

lea Rh 1({)cS(p(w;{»dv.

Since the function u( t, w) is a superposition of horospherical waves, it is a solution of the wave equation (9.8). We prove that u is the desired solution of the Cauchy problem (9.9). It follows from the symmetry relation (6.23) that e' F(w, t) is an even function of t; hence,

u(O, w)

= dtd (e.-F(w, t»I.=o = O.

FUrther, using the inversion formula (6.19), we see for the horospherical transform that

uao, w) =

-

2~ ~ (et F(w, t»I.=o = I(w).

We wish to return in (9.11) to the original function I. To do this, we use the ABgeirsson relation (6.22) for the hyperbolic space. From (6.22), for t > 0 we obtain

u(t,w) = sinhtF(w,2sinh ~),

.c3

where F( w, s) is the result of averaging the function 1 over the spheres in centered at the point W; see (6.23). Using (6.23), we obtain the following result.

3. INTEGRAL GEOMETRY IN

110

,2

AND "

THEOREM 9.2. The solution of tJ&e Cauch1l problem (9.9) for the wave equation (9.8) is of tJ&e form

u(t,w) =

8~ sinht

1,

2

sinh- 1 d(W w')

2

x 6 (4sinh2 d(W w') _ 4sinh2

I

= _1_ tanh-1 ! f (w')6 1611" 2 "

~) few') dx'~~:dzl

(sinh dew,2w') _ sinh!) dx'dy'dz' . 2 z,3

The formula thus obtained is similar to the Kirchhofl" fonnula in § 3 of Chapter 1 for the wave equation in Euclidean space. As in the Euclidean case, the value of the solution of the wave equation at a point w at a moment t depends only on the initial data at the points w' that are at the non-Euclidean distance t from the point w (the Huygens principle).

CHAPTER 4

Integral Geometry and Harmonic Analysis on the Group G = SL(2,C) In Chapter 3 we constructed integral geometry and harmonic aoaIysis on the hyperbolic spare equipped with the natural action of the group G = 5L(2, C), i.e., of the group of complex matrices 9 = (~ ~) with unit determinant, a6 -/h = 1. In this chapter \\"e present similar constructions OD the group G itself. The theory thus obtained naturally includes the theory constructed in Chapter 3 for the hyperbolic space. We regard the group G as a homogeneous space of the group G)( G, and from this point of view, we can simply treat the group G as the hyperboloid a6 -/h = 1 in et equipped with the action of the subgroup of linear transformations of C' that preserve this byperboloid.

1. Geometry

OD

the group

a

1.1. Group G as a homogeneous space. We regard G as a homogeneous space of the group G x G acting OD G by left and right translations,

9"'" g1"l g92 ,

(9t.!h)

e G)( G.

Since the isotropy subgroup of the identity element e e G is the diagonal subgroup, diag(G x G) ~ G, it follo\\'S that this homogeneous space is isomorphic to G x G/G, where Gc G x G is represented as the subgroup of elements (g,g),

geG. REMARK. The homogeneous space G is symmetric because the isotropy sul> group diag( a x a) is the subgroup of elements invariant under t,be involution 8(91,92)"'" (!h,g). The space G is not Riemaunian, but it is pseucio-Rimnannian, i.e., the metric OD this space is nondegenerate but not positive definite. This fact is of importance because integral geometry and harmonic analysis for symmetric spaces can be constructed in a more complete form than those for arbitrary h0mogeneous spaces. Nou> that the action of G x GOD G is not effective, and the inefFective kernel, i.e., the subgroup preserving all element!! of a, cousists of two eJements (E, E) and -(E, E). Therefore, the quotient group G x G/{±{E, E)} effectively acts on G. k, WDtI already said, in what follows, "ileD. considering G, we ignore the group structure and view G only as the hyperboloid in C4 with coordinates (o,P,-y,6) such that Q~-fJ'"Y=l.

The group of linear transformations of C3 that preserve the quadratic form o:~ - Ih. i.e., the group 50(4, C). act.s OD G effectively and transitively. The transformations corresponding to the elements of G )( G obviously preserve the III

112

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

En

form ocS - (J"Y. and we obtain an embedding of G x G/{±(E, in SO(4,C). It follows from the dimension considerations (the dimensions of these groups coincide) that we have in fact an isomorphism

SL(2,C) x SL(2,C)/{±(E.E)} ~ SO(4,C). Under the action of SO(4. C) on G, the isotropy subgroup coincides with SO(3. C), and another classical isomorphism arises,

SO(3,C)

~

SL(2,C)/{±E}.

1.2. Plane sectloDS of the hyperboloid G. We intend to relate the action of the group G x G on the hyperboloid G to the geometry of this hyperboloid. Namely, we study the sections of G by byperplanes in Col as orbits of some subgroups of GxG. It is convenient to present hyperplanes in C4 by equations of the form

(go,g)

(1.1)

= c,

90

= (oo,{Jo,"Yo,cSo),

where ( , ) is the polarization of the quadratic form ocS In the detailed notation, 1 2(600 - "Yot3 - Pu"Y + oocS) = c.

ih.

Thus, each hyperplane is given by a pair go, c. where 90 E C4 \ 0 and c E C, and this pair is defined up to a common factor. The group G x G naturally acts on the set of hyperplanes; namely, an element (91092) E G x G sends a hyperplane (go,g) c to the hyperplane (gllgo 92 ,g) c. We first describe the orbits of G x G in the set of hyperplanes. In general position «(go, go) :F 0) one can normalize go by the condition (go, go) = 1. Obviously, for any chosen number c, the hyperplanes (1.1) with (go,go) = 1 form an orbit of G )( G isomorphic to the original homogeneous space G. The remaining subset of hyperplanes (for which (go, go) = 0) is partitioned into two orbits of G x G. namely, the family of hyperplanes

=

(1.2)

=

(90,9)

= 0,

(go, go)

= 0,

tangent to the asymptotic cone K given by ocS - /3"Y = 0, and the family of hyperplanes (1.3)

(go,g) = 1,

(go,go) = 0,

parallel to the previous hyperplanes. Obviously, the second orbit is isomorphic (as a homogeneous space of the group G x G) to the asymptotic cone K with deleted vertex; the isotropy subgroup of this orbit consists of the pairs (6z 1 ,6- 1 z2 ), where %1 and %2 are matrices of the form ( Af ) and 6 is a diagonal matrix. The first orbit is isomorphic to the set of generatrices of the asymptotic cone K and has a stabilizer which is the subgroup of all pairs (6IZJ,~Z2) with ZI and Z2 as above and diagonal

cSl

and~.

Thus, the set of plane sections of the hyperboloid G is decomposed into the following orbits of the group G x G: the one-parameter family of orbits isomorphic to the homogeneous space G, the orbit isomorphic to the asymptotic cone K with deleted vertex, and the orbit isomorphic to the set of generatrices of the cone K. We now study the structure of plane sections of the hyperboloid G. It follows from the invariance considerations that it suffices to consider a single representative

1. GEOMETRY ON TKE GROUP G

113

on every orbit of G x G in the set of plane sections. For the repre5('.ntatives on the orbits in general position \\'e take the sections of G by hyperplanes of the form

(1.4)

0'+6 =c.

~ ±2. then the byperplane (1.4) intersects G along a t\\'O-dimensional hyperboloid. The hyperplane (1.4) is tangent to G in the special case c =< ±2, and the intenlection of this hyperplane with G is a cone with the vertex at the tangency

If c

point. The sections of G by the hyperplanes (1.4) have a simple group-theoretic sense.

Namely, since 0' + 6 = 1'1' 9 for 9 = (~~), the section of G by a hypcrplane + 6 = c is a conjugacy class; if c ~ ±2, then the conjugacy class is generic, and for c = ±2, the conjugacy class consist.s of elements of Jordan type (the vertex of the cone is assumed to be deleted). For representatiws of the two special orbits of G x G one can take the sections of G by the byperpJanes ..., = 0 and "'f = I, respectively. Obviously, the first section is a cylinder and the other is a paraboloid. Thus, there are four types of sections of G by hypp.rplanes in C', namely, hyperboloids, cones, paraboloids, and cylinders. Naturally, the scctioIUI of parabolic and cylindrical types are obtained by passing to the limit from scctioll.'f of hyperbolic and conic types. Every plane section of the hyperboloid G can be equipped with t.he structure of a homogeneous space. Let us first collsider the sections by plaue8 of the fonn 0' +6 = c. The subgroup diag( G x G) e! G transith'(>ly acts on each of these sections (in the case of conic sect-ion it. is assumed that the \"eI'tex of the cone is deleted); thus, 811 these sections, and hence all their shifts as well, are homogeneous spaces of G. Oil{' can readily 5CC that the group D of diagonal matrices is a stabilizer of any section of hYJ)(>.rbolic type, and the group ±Z, where Z is the subgroup of unipotent matrices (A l), is a stabilizer of every section of conic type. We now consider the sections of G by the hyperplanes 'l = 1 and 'Y = O. The subgroup Z x Z ~ freely and tral18itively on the 6rst section; thus, this section is equipped with the structure of the homogeneous space Z )( Zj{E}. The group Z x Z acts on tbe other section 88 well, but t.his action is no longer traDSitive. One can readily sce that the subgroup U C Z x Z of all pairs (6z 1,6- 1 Z2), where 6 E D and Z1,.4:2 E Z. 8oI."ts on the section t.raw.;t.il-ely. and the subgroup diag(±'Z.±Z) of pairs (±z, ±z), :z: E Z, which LOS isomorphic to ±Z, is a stabilizer ofthls scctl.on. Thus, the sectioILO"f of G of hyperbolk, conic, parabolic, and cylindrical type arc equipped \\;th the structures ofthe homogeneous spaces G/D, G/±Z, Z)( Z/{E} , and U I diag(±Z, ±Z), respecth-ely. Q

1.3. Manifold of horospheres. Consider the set of gener&trices of the bypcrboloid G. In our theory these lines plA\v the role similar to that of horospheres in tht! hyperbolic space, and we n!fer to these objects 88 horospberes by analogy with the hyperbolic space. Note that the horospheres are the generatrices of all plant> sections of the hyperboloid G. The group G x G naturally acts on the set of Iwrospheres H. auc:I this action is transitive (to prove this assertion, it suffices to consider the set of horospheres passing through an arbitrarily chosen point 9 E G and to show that the stabilizer of this point transitively acts on this set of horospheres). The horospheres on G have a simple group-theoretic interpretation. Namely. the subgroup Z c G of matrices of the fono (A f ) is one of the horoolpberes. Since

4. INTBOIlAL GEOMETRY ON THE LORENTZ GROUP

114

all horospheres are obtained from this horosphere by translations, it follows that the set of horospheres H coincides with the set of double cosets 91 1ZI/2. Therefore, HE!! (G x G)/V,

where V is the subgroup preserving the horosphere Z. One can readily see that the elements of V are all possible pairs (6z1,6z2), where Zl,%2 E Z, 6 E D. Since we obviously have dim V = 3, it follows that dim H = 3, i.e., the dimension of the space of horospheres is equal to that of G. We introduce a coordinate system on H. To this end, we first consider the set Ho of the generatrices of the asymptotic cone K given by et6 - P"1 = O. The cone K admits two one-parameter families of tWCHlimensional planes, namely, the family elet + e2"1 0, (1.5)

=

elP+e26 =0

and the family

"1 6 - 'l2"1 = 0,

(1.6)

-"lP + 'l2et = O.

In the matrix notation, this gives the family of planes eg = 0 and the family of planes g"J. = 0, where = Every generatrix of the cone K can be obtained as the intersection of a pair of planes (1.5), (1.6) and is thus given by a pair of vectors E C 2 \ 0, each defined up to a factor; we denote this generatrix by l(e, ,,). The manifold of horospheres H forms a natural bundle over Ho, with the fiber over l(e,,,) being the set of all horospheres parallel to l(e, ,,). We claim that every horosphere parallel to l(e,,,) is given in C4 by the system of equations

r L":J

e,,,

e~ et +

eh = ,,~,

eiP + e~6 = rh, ,,~6 - rh"1 = e~.

(1.7)

-fliP + rh et = e~,

e',,,'

e =

where E C2 \ 0 are vectors proportional to and ", respectively. Indeed, consider the first two and the last two equations of (1.7). On G we can represent them in the form e'g fI', ",g-1 e'; hence, these two systems of equations are equivalent on G. Hence, system (1.7) defines a line in C', and this line belongs to G. Further, it is clear that this line is parallel to l(e,,,).

=

REMARK. The horospheres parallel to the line l(e,,,) are generatrices of the cylinder obtained 88 the section of G by the hyperplane which is tangent to the asymptotic cone K along the line lee. ,,). The equation of this hyperplane is (1.8)

'l2(elet + e2"1) - '11 (elP + (26) = 0,

or e9'lJ. = 0 in the matrix form. Thus, every horosphere on G can be given by a system of equations i.e., elet + e2"1

= 'It.

F.IP + F.26 = 'l2,

eg = ",

1. GF..oMETRY ON THE GROUP G

where

115

e.." E (:2 \ 0, or by the cquh-aleut system 719- 1 =~, i.e., fl 16 - '12, = {I.

-fl18

+ rn o = {2.

Denote this horosphere by h(~,,,) and regard~. '1 as paraDlCters on the manifold of horospherM H. Since \\"e obviou.'!ly have h(~ •.\tl) = h({.fl) for each .\ #; 0, it is natural to refer to fI 311 homogeneous coordifuJtes on H. The action of an element (91.92) E G x G on H is given in the homogeneous coordinates 8.'1 follows:

e.

e,,,

h({. fI) ...... 1&(e9lt TJ92).

Note that the borosphere Z has the homogeneous coordinates ~ = '10 = (0.1); hence, every horosphere g1 1 Z92 has the homogeneous coordinates = eog}, fI = {092· It ilf of prime impurtanc(! in what follows that. along with the action of the group G x G, the manifold of horospheres H is equipped with an action of ex; namely, corresponding to each number ~ E ex is the transformation

e

h(e, ,,) ..... h({, A'I)

which takes every horosphelt' to a parallel horosphere. These transformations are called left translatimu on H. Obviously. they commute with the action of the group

GxG. We now describe the families of generatrices (horospheros) on the plalle sections r of the hyperboloid G in the homogeneou8 coordinates By virtue of the aforesaid, ea<'.h cylindri<'.aI section r is the iot.ersection of G with a hyperplanc of the form (1.8), 8Jld the horospheres on r are h(~, .\,Il, where .\ ranges over ex. Consider now the plane sections r of tbe other types. For each or these sections, the equation of the secant hyperplane can be written as

e, ".

2{go,g}

(1.9)

= 1 + (go,go).

Hence, since (g,g) = 1 on G, it follows that the section the eq1aation

r

itself is defined

011

G by

i.e.,

(1.10)

(o - 00)(6 - 60) - (a -

Bo)h -

'"YO)

= o.

=

This section is a cone. a paraboloid, or a byperboloid if detllo 1. det 90 = 0, or det!Jo ~ 0, 1, respectively. The equations of generatrices (horospheres) OD r immediately follow from (1.10): the horospheres of onc family arc gi\o"ell by the equations (0)

+ (2h -

(1(.8 - ~)

+ ~2(c5 -

(I (0: -

i.e., (g

=:

= 0, 60) = 0,

"'/1)

e90, and tbe borospberes of the other family by the equations 111 (6 -

60) - '12h -1'0) = 0,

-1h (,fj - ~)

i.e., TJg- 1 =

'190,

where jl

+ '12(0' -

= (_~ -::).

0'0)

= 0,

Thus, the following assertion is proved.

116

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

PROPOSITION 1.1. The families of horospheres belonging to the section of the hype"boloid G by a hyperplane (1.10) are the family of horosphere.s h({.{90) and

the family of horospheres h('Iio, 'I), where io = set C2 \ O.

(!~o ~r:) and~. r] m1lge over the

COROLLARY 1. For any chosen 90 E G and belonging to the section of G by the hyperplane

~

::f O. the families of horospheres

= 1 + ~2

2~(go.g)

are formed by the horospheres h(~,~90) and h(~.~-1{90)' where { mnges over C2 \O. COROLLARY 2. The horospheres passing through an arbitmrily chosen point go E G are the horospheres h({,{90), where ~ mnges over C2 \ O.

1.4. Embedding the manifold or horospheres H in the projective space. Assigning to each horosphere h(~. 'I) the point of the projective space ep3 with the homogeneous coordinates ({J,{2. '11, 'l2), we obtain an embedding

H

'-+

cpl.

The image of H under this E'mbedding is the entire space Cp3 from which a pair of disjoint lines is deleted, namely. the line 0 and the line 'I O. Thus. Cp3 is a natural compactification of the manifold of horospheres H. The points at infinity on H have the following simple mE'aning. The set of points at infinity on G is a two-dimensional hyperboloid having two families of generatrices. Corresponding to these generatrices are the points of the lines = 0 and " = 0 in cp3, i.e., the points at infinity on H.

e=

=

e

REMARK. We explain how this construction must be modified when passing from the group G = SL(2, C) to its quotient group by the center PSL(2, C) = G / {±E}. i.e.. when antipodal points of the hyperboloid 06 - 111 = 1 are identified. The manifold G = G/{±E} can be embedded in the projective space era by assigning to each matrix ( ~ ~) the point of ep3 with the homogeneous coordinates

(0.,13.1'.6). The image of G under this embedding is the entire projective space era except for the quadric r given by the equation 06 - ,131' = 0; the points of this quadric can be treated as the points at infinity on G. One can readily see that the horospheres on Gare taken by this embedding to all possible lines in Cp3 tangent to the quadric r. The equation of each of these lines can be written in the following

form: (1.11)

where { ::f 0 and 'I ::f 0 (the rank of this system is equal to 2). A compactification of this manifold of lines can be obtained by adding all generatrices of the quadric r; these generatrices are defined by (1.11) with either ~ = 0 or 'I = 0 (two families of generatrices). The resulting variety is isomorphic to the projective space CP3: namely, corresponding to each line (1.11) is a point of cp3 with the homogeneous coordinates ({1,{2.'Il.'l2).

I. GEOMETRY ON THE GROUP G

117

1.5. Line complex in C3 associated with the manifold of horospheres. It L'i convenient to pass from the manifold of IiI1t!$ (borospberes) 011 tbn hyperboloid G to the manifold of liuCH ill C 3. To do this, we use the projection of tbt> hyperboloid a6 - 8-t = 1 to the (',oordinate 8ubsp8CC C 3 with the coordinates (a, (J. 6). The image of the bypcrboloid u. the entire space C3 except for the pomt-s (a, 0, 6) such that ct6 f:. 1. \Vhen projecting the hyperboloid G to Cl, the generfttrices of G that are not parallel to the hyperplam.> {J = 0 are taktm to the lines int.erscct.ing the hyperbola (:J = O. 0'6 = 1 and not belonging to the plane fJ = O. Thus, with the manifold of hOlUlllbP.res on the hyperboloid G we 8HSociatoo a. tJtroo.dbnensional manifold K of lines in C3. In difW.n>..ntial geometry, three-dimemriollal manifolds of lines in C' arc usualJy caUed line complexes. Only one point oCtbc hypcrboloid lies over any point (0,,8,6) e C 3 On the other hand, IM.'.r any point. such that P = 0 alld 0'6 = 1, lies the entire line. Thus, the hyperboloid is obtained by pasting lines, by means of a blow-up process, in place of points of the hyperbola given by the equations f1 = O. 06 = 1. The lines of our complex are just the lines t.bat can be lift.ed to the hyperboloid as 8 result of this blow-up process. REMARK.

with {j

of: O.

1.6. Manifold of paraboloids. Now wt' describe another \"enUOIl of borospheres ill G. Consider paraboloid." in G, i.e., t~lSionaJ surfaces Cllt out in G by il:;utropic hY]:lCrpl.anel in C' not pa8lring through 0, (!Io. g)

= p,

geG,

where 90 is a nonzero point of the asymptotic cone

(go. go)

E

det(go)

= 0,

and p of: 0 is an arbitrary nOD7.CJ'O number. 1 We introduce coordinates in the space IP' of all paraboloids; these coordinates an- similar to the homogeneous coordinates in the spare of borospberes H. Set (..l = for each 2-vcctor ( = ('1'~)' Thus, ~<.1. = _«(..l = ~1'2 - ~I is an antisYIDJOetric bilincar form on C'J. Obviously, this form is invariant ,,'itb respect to the group G, I.e.,

(5U

({g)("g).1.

= erl

fur each 9

e G.

<

lb each pair of vectors {. E C'l \ 0 and every number p of: 0 we assign all isotropic hyperplanc in the four-dimeo."ionaJ space of matri<',e[I g. This hyperplanc is given by the equation (1.12)

~g(J. = p.

P:F O.

or. in the detailed notatiou. by the equatioll (1.13) wherf.> { = ({t.{2), <; = (1, (2)· One can readily see that the t.'C)nations of isotrop~ hyperplaDel not passing through the origin 0 and only these equatious can be rcpl'Cfir..nU>c:1 in tbe form (1.12). lSometimes borospheres in our _ horospberes.

are mlled horocydcs, and paraboloids are called

4. INTEGRAL GEOMETRY ON THE LORE!'lTZ GROVP

lJ8

Denote by h«(. <.p) the paraboloid in G given by (1.12). Obviously. (1.14)

h(~. p<. '\pp)

= h«(.<,p)

for any numbers '\,P:F O.

REMARK. In (1.13) one might assume that p = 1 and write h({. () instead of h«(. (.1). Then h(~. ,\-1<) = h«(. <) for each ,\ :F o. However. it is more convenient for us not to do this.

In the coordinates (.(.P. an element (91.92) E G x G acts on HP as follows: h«(.

<. p) ...... h({gl. (92. p). cl ).

For instance. the paraboloid ZsZ. where s = (~ has the coordinates (0 = <0 = (0.1). Po = 1. Hence. the paraboloid g,1 Z s Z 92 has the coordinates ({Ogl. <092.1). It is of prime importance that the space of paraboloids HP (as well as the space ofhorospheres H) is equipped with an action of Cl( that commutes with the action of G x G: namely. corresponding to any ,\ E Cl( is the transform

h«(, (. p) ...... h«(. (, '\p). which sends every plane parabolic section to a parallel section. Every paraboloid h({.(.p) admits two one-parameter fanlilies of horospheres. namely. the horospheres parallel to the planes eg = 0 and g(.!. = O. respectively. which belong to the asymptotic cone in We describe these families of planes in the homogeneous coordinates on the manifold of horospheres H. The first family consists of the horospheres h({.,,) where" satisfies the relation ",1. = p. The other one consists of the horospheres h(".() where" satisfies the relation {".!. = p. Indeed. a horosphere h«(.,,) is the set of matrices 9 E G satisfying the relation (g = ". and the condition that the horosphere is contained in a paraboloid h«(. <.p) is equivalent to the relation f1(J. = p. A horosphere h(". () is thf.> set of matrices 9 E G satisfying the relation "g = or. equivalently. the relation = g<.l.. Therefore. the condition that this hor08phere is contained in a paraboloid h«(.(.p) is equivalent to the relation ("J. = p. Note that exactly one horosphere of each family passes through any given point of the paraboloid. Conversely. two one-parameter families of paraboloids pass through any given hor08phere h«(.,,). namely, the family of paraboloids h«(.<.I) where 7I(J. = 1. and the family of paraboloids h( 1) where «1. = 1. We now discuss the analogy between paraboloids and hor08pheres. Similarly to the case of horospheres. every paraboloid is an orbit of the subgroup Z x Z of G x G or of a subgroup conjugate to Z x Z. Conversely. every orbit of the subgroup Z x Z or its conjugate subgroup in G is either a horosphere or a paraboloid. Indeed. let LeG be an arbitrary orbit of Z x Z. Le.. the set of matrices of the form %1 gO %2, where gO = (~: ~) is an arbitrarily chosen element of G, and %1 and %2 range over the subgroup Z. Obviously. if 10 = O. then L is a horosphere. If 10 :F O. then L is the section of G by the hyperplane 1 = 10. i.e .• a paraboloid. Thus, the manifold of orbits in G x G of the subgroup Z x Z and of the subgroups conjugate to Z x Z is the union of two homogeneous spaces of the group G x G. namely. the space of horospheres H and the space of paraboloids HP.

ca.

<.

<. ",

".l.

2. INTEGRAL GEOMETRY ON THE GROUP G

= SL(2.C}

liS

Let US compare the manifold of horospheres H with the manifold of paraboloids HP. Both manifolds are equipped with the structures of a bundle, with the fiber ex over the manifold HC of cylindrical sections. The mapping H --. He takes every horosphere to the cylindrical surface passing through this horosphere, and every fibel' of the bundle consists of borospheres parallel to one another. The mapping HP --. HC takes every parabolic section to the parallel cylindrical section, and every fiber of the bundle consist.s of parabolic sections parallel t() one another. In the coordinate form, to every horospbere h((,,,) and every paraboloid h({,,,, p) we assign the cylinder given by the equation f.grf = O. The group G x G permutes the fibers of each of these bundles. and the group ex acts on the 6bers. Note that H and HP are not isomorphic when regarded as homogeneous spaces of the group G x G. Indeed, their stabilizers in G x G are the group V of pairs (6%1,6%2) and the group U of pairs (6- 1 %1.«5%2). respectively. where 6 E D and %1, %2 E Z. These subgroups are not conjugate in G x G; they are not even isomorphic. REMARK. The space HC of cylindrical sections is naturally isomorphic to the product ClPl x epl of two complex projective lines. and the bundles H and HP over HC wit,h the fiber C \ 0 coincide \\ith the line bundles 0(-1,+1) and 0(-1, -1). well known in algebraic geometry.

2. Integral geometry on the group G = SL(2, C) 2.1. Integral transforms related to the space H of horospheres and the complex of Unes K. To the manifold of horospheres H we usociate the integral transform sending a compactly supported COO function on G to the integrals

9.

of this function cn-er the horospheres. 1 Z!J2, where Z is the subgroup Each horosphere h(~.,,) is a two-sided coset of matrices (A I) and 91, 92 E G are arbitrary matrices satisfying the condition

(2.1)

{o9t

DEFINmoN. Let us h(~,,,) by the formula

(2.2)

= {.

{o92

= ".

where (o

= (0,1).

define the integral of a function

'RI(~.,,) = ~

f

1(91 1Z92)dtdi.

%=

I

(~

over the horospbere

!),

where 91.92 E G are arbitrary matrices satisfying (2.1). The function 'R./(e.,,) is called the horosphencal troflsloml of I.

Formula (2.2) can be written 88 (2.3)

'R./{{.,,)

= 1a'(9)6«(9-'1)dp(g),

where!S(·) is the delta function on C2 and dp(g) is an invariant measure OD G; in the coordinates 0', P. 6 on G, this measure is

dp(g)

= (~r 1131-2 ,lI.uliidndPtMd6.

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

120

The equality of integrals (2.2) and (2.3) is verified directly. It follows from the definition that the function 'R. I(~. '1) satisfies the homogeneity condition (2.4)

f(e.

Thus. the function 'R. '1) depends not only on the horosphere h({. '1) but also on the homogeneous coordinates '1 in which this horosphere is given: therefore. we can regard 'R. I as a section of a one-dimensional vector bundle over the manifold of horospheres H. It also follows from (2.2) that the operator 'R. commutes with the action of the group G x G. i.e., if 11 (g) = 1(91 1g92). then 'R.11({. '1) = 'R./(~gl. '192). Introducing affine coordinates on H. one can define the horospherical transform as a function on H with numerical values rather than as a section of a vector bundle over H. It is convenient to introduce the following affine coordinates on H:

e.

e1

'12

'11

= -, X2 = -. ~ = -. e2 '12 {2 In these coordinates every horosphere h({, 'I). where e2 :F 0 and '12 the equations {, = -x 1 i3 +~. (2.5)

Xl

:F o. is given by

REMARK. The condition e2 :F o. '12 :F 0 is equivalent to the condition that the horosphere h(e, '1) is not parallel to the hyperplane /3 = 0 in C 4 • Taking

Q.

/3. {, as

the local coordinates on G. one can define the horospherical = I(Q. 8. 6) by the following relation:

translorm of any function I(g)

'R.o I(xt. X2.~) =

(2.6)

~

J

f(x2/J + ~ -1 ./J. -XIJ3 + ~) dJ3 dlJ.

Geometrically. the passage from 'R. to "Ro means that instead of the integral transform related to the manifold of horospheres H. we consider the integral transform related to the line complex K in C 3 associated with H. i.c.. the transform taking a function on C3 to the integrals of this function over the lines in C3 intersecting the hyperbola /3 = O. Q6 = l. Let us establish the relation between the functions 'R. I and "Ro I. In (2.2) one can set gl

9 = (~

= ((f (~) and 92 = ('1,f ~); and z = (A n. implies that

n

Q

then the relation 9 = 91 1z92. where

e2 = 'I1e2 t + -,

'12 Finally, the expression for 'R. I in (2.2) becomes (2.7)

'R. I({. '1) =

1{2'121-2~ /

f(x2/J + ~-1.8. -x18 + ~)d/Jd73.

e.'1

where X1.X2.~ are connected with by (2.5). Comparing this expression with that given by (2.6) for "Ro I. we obtain the desired relations between 'R. I and "Ro I: (2.8) (2.9)

= 1{2'121- 2 "RoI (~:,:. ~), 'R.o I(xt. X2'~) = 1>'1 2 'R. I(Xlo 1. ~2' >.).

'R./(e, '1)

2. INTEGRAL GEOMETRY 0:-1 THE GROUP G"" S£(2. C)

121

2.2. Symmetry relations for the horospherical transform.. Every func"p = 'R. 1 satisfies some additional symmetry relations. Let \18 describe the structure of tlu.>J;c relations. Assume that r is an arbitrary section of the byper. boloid G by a h..VJlel"planc in C4. Since r is a quadratic surface, it followB that tion

r.

in general position there are two families of generatrices (borospberes) on A symmetry relation is that the integrals of the function V' = 'R. 1 over t·hese families of horospberes (with respect. to an appropriate measure) coincide because both integrals arc equal to the illt~ of the original fUllction / ~ the surface r. Wc VtTite these relatioll8 in an explicit fonn. Choose an element 90 E G and 8 number A e C \ 0 arbit.rarily and consider two families of horospheres, namely, h(e, ~!Jo) and h(~q901, q). It was already shown in 1.3 tbat these horospheres are gencratrices of the surface r c G obtained as the section of G by the byperpIane

2>'{go,g} The surface

r

= 1 + A'J.

±l and a colle for A = ±1. The !unction r.p = 'R. I soh8jie8 the symmetry relation

is a hyperboloid for A:F

PROPOSITION 2.1.

1r;({,~go)w({)w({) =1

(2.10)

"le lor any 90 e G and .\

r.p().f19Ql,q)w(r1)w(q)

"t"

e C \ o.

wee)

stands for {I d{2 - {2 d{l, and 7E and 7" are arbitrary surfaces ill 0 that interBect once almost every line passing through the origin 0; it. follows from the homogeneity condition (2.4) for 'P that the integrals do not depend on the Here

(:2 \

choiCE' of ')'E and ')'". For the proof, it suffices to substitute the expression for 'R. I ill terms of I (see (2.3» into each integral (2.lO). Simple manipulations transform each of these integrals to the Kame integral

L

/(g) 6( -2), l.!Al , g)

+ 1 + ).2)dp(g) =

L

1(g)6«g -

Note that relation (2.lO) is meaningful only for)' section r is not. a roUt>. REMARK.

form:

:F

±1, in which case the

The symmetry relation (2.10) can also be represented in the following

1'P'(~,~go)w(~)w(~) 1 =

"re

where go =

).9O.9 - ).90» dlJ(g)·

",("go, '1)w(q)w(q).

-r"

(~t) is an arbitrary nonsingular matrix Ilnd 90 = (_\. -;!). In

this form the relation remains valid for singular matrices !ID ". 0 as weD. Note that, for any singular matrix 90, the horospheres h({.~) and h("iO.q) arc gcoeratriccs of tbe paraboloid obtained as the section of G by the hyperpIaoo 2{90. g} = 1.

no

2.3. Inversion formula for the integral transform related to the line complex K in C3 • Consider the integral t.ransform 'R.o given by (2.6); this transform takes every compactly supported function 1(0.. $, 6) OD ca to the integrals of this function over t.he lines in C 3 intersecting the hyperbola {j = 0, 0:6 = 1. We intend to reconstruct f fronl the function r.p = 'R.o I. Recall that iD Chapt-er 2

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

122

we solved a more general problem of reconstructing a function on C3 from the integrals of this function over the lines intersecting an arbitrarily chosen algebraic curve in It. C C 3 • Therefore, it is possible to apply the results of Chapter 2. However, we give here another solution, which is independent of Chapter 2. We first present the solution of the problem under the additional assumption that 1 = 0 if (J is sufficiently small. Under this assumption, the forthcoming manipulations are correct. In what follows we get rid of this unnecessary assumption. We first pass from the function !p(xl, X2, ~) to its Fourier transform with respect to Xl and X2,

f

~({l'{2'~) = (211")-2 (~r !p(XJ,X2,~)eiRe«(lzl+(aza)dzldXtdz2dXi. Let us find an expression for (2.11)

~

~({a.{2'~) = (211")-2 (~r

in terms of the original function

f

I(X2 t +

I. We have

~-l, t. -xlt +~)

x eiRe«(lzIHaza) dt dldz 1 dXt dz 2 dXi

= (211")-2 (~r

f

1(0, {J,cS)eiRe (¥(I+=F(a) IfJI- 4 doood{JdPd6d6.

We now pass from coordinates cS PI = -{i'

0,

{J. cS on C3 to the new coordinates a 1 P2

= {i'

1'3

= (i

and set 1(0, fJ, cS) = IP3I 4 !l(Pl'P2.P3). In this case, formula (2.11) becomes

~({J,{2'~) = (211")-2 (~r f !l(PllP2,P3) x e iRe (PI(I+P2(a+II3().(I-).-1(2» dPl djii dP2 dpidP3 d;J3. We thus obtain a simple relation between the function form F of!l,

F({J,{2,{3)

= (211")-3 (~) 3

f

~

and the Fourier trans-

1(Pa.P2,P3) x e iRe (P'(I +P2(a+II3(:s) dPl djii dP2 clfi2 dP3 d;J3,

namely, (2.12) The desired inversion formula readily follows from (2.12). For that, one must apply the inversion formula for the Fourier transform of h,

IfJI 4 /(o,fJ,cS)

= h(Pl,P2,P3) = (211")-3

(~) 3

f

F({1,{2.{3) x e- iRe «(,p,+(aP2+(:SP:s) d.{1 d.{l d.{2 d.{2 d.{3 d.{3.

123

2. INTEGRAl. GEOMETRY ON THE GROUP G .. SL(2.C)

Making here the change of variables ~3 = ~I

/(o,P,6)

see by (2.12) that

f ~(elo!2,A)I~1 + 1 21fJ1 -4/ - (A-6 O-A-I) P

= ~(211")-4 (~y IPr"

A- 2!21 2

x e-iRe(-~1!+(3it+(~1-~-1~3)l) ~l del d(2d(2d>.dX

(2.13)

= 2(211')

where L

- A-l~2' ll'e

=k

_2i

LLcp -{3-'

+)..-2~ and L=

,A tUtU,

-k +r2~.

It remains to simplify the expression obtained. Since P-IL + ,(pIT. + -/r.> '1'( A;6 , ~,A) with respect to >. (to X), one

is the full deri~'8.ti"'e of thl" function 1

it

--1-

can replace P- Land [J obtain r.

/(a.{J, 6)

L in (2.13) with -lU: and

121 i -2 = 811'2 .81

f

/I

cp~.>'

it -ax' respectlveiy.

We fioa1ly

(AT' -6 8A-I) ,A dAd.\.. 0 -

The desired inversion formula follows by changing the variable, x formulate the final result.

= ¥.

We now

THEOREM 2.1. Let /(o,tJ,,,() be a compactly supported Coo function on CS that vanishes Jor the sufficiently smalllJfJlUe8 of fj, and let cp = R.o / be the integral fronsform of f given by (2.6). Then the foUoUling inversion formu14 holds:

(2.14)

f

/(a,f~.6) = ~2 ~ .p~.X (x, ~::~ ,px + 6) dz~.

Note that the differentiation with respect to A in this formula is the differentiation with J'e!ipect to the set of parallelUnes intersecting the h.l-perbola fj = 0, 06 = 1, and thl" integral is taken over thE- set of lines intc.-rsecting this hyperbola and passing through the point (0, p, 6). As in the C8lie of the hyperbolic space, the inversion formula thus obtained is local, namely, to reconstruct the ~'8lue of a function / at a point. (a, p,~) one must know only the integrals of this function over the lines intersecting the hyperbola fJ = O. 06 = 1 that are infuUtely close to this point. REMARK. The line complex K is admissible in the sense of Chapter 2, §5. Therefore. the inversion formula (2.13) can he obtained using the operator It.

2.4. Inversion formula for the horospherical trausform. Assume that leg), 9 = (~~). is a compactly supported CtXl function on G. The problem is to express f in terms ohlle horospberical transform 'R. f(~,,1) defined by (2.3). We temporarily assume that / = 0 for all sufficiently small values of p. In tu case, the function /, viev.'ed as a function of 0, /1. 6, can be extended to a compactly support.ed COQ flmction on the entire space C 3 , and this ext.ension vanishes for all sufficiently small values of p. Hence, the extended function can be expressed in terms of cp = 'Ro / by the inversion formula (2.14). Hence, to obtain the desired result, it suffices to use the expression for 'Ro / in terms of 'R /; see {2.9}. We thus obtain (2.15)

J(9)

= 8r.12 2i /

LL'R/(x, l.crx + "(,{Jx + 6)dxdX,

4. INTEGRAL GEOMETRY ON THE LORENTZ GROllP

124

where L = (ox + '})~ + (/3x + 6)~ + 1. Using the homogeneity condition (2.4) for 'RI, one can write the inversion formula in a more symmetric form,

I(g)

(2.16)

= 8~ ~

1

L"L" 'R I(~.,,)

I,,=~g w(~)w(~).

"11rn

where L" = + 7121.;; + 1, w(~) = ~1 ~2 - ~2 ~J, and the integral is taken over an arbitrary surface '} C C 2 \ 0 which intersects once alma&1: every line passing through the origin O. Since the horospherical transform 'R is invariant with respect to the action of G x G, it follows that the initial assumption on f is unessential: namely, if the inversion formula (2.16) holds for the functions vanishing for all sufficiently small values of /3, then this formula remains valid for any compactly supported Coo function on G. This proves the following assertion. THEOREM 2.2. Let I be an arbitrary compactly supported ex> function on G. If 'R I is the horospherical transfoma of I defined by (2.3), then the inversion /omaula (2.16), which expresses I in temas o/'RI, holds.

In formula (2.16), the integral is taken over the set of horospheres h(~. ~g), i.e., the horospheres passing through a point g E G. The operator L" = 7111.,. +712/.;; + 1 occurring in the formula can be given as follows:

L"cp(e.71) = :>. (>. cp(~. >.,,»

1.\=1 .

Recall that h(~,,,) 1-+ h(eo, >.,,) is the left translation taking every horosphere to a paraUel horosphere; see 1.3. Thus, L" is the operator of the infinitesimal left translation. REMARK.

It follows from the homogeneity condition (2.4) for 'Rf that

+ L,,)'RI = (L~ + L,,)'RI = O. = el ili + ~2 Jl; + 1. Therefore, the operators L" and L" in the inversion (L~

where L~

formula (2.16) can be replaced with L L, respectively.

= ~ (e1 ~ + e2 Jl; - '1l-k - 712 ~)

and

2.5. Inversion formula for the horosphericaI transform on the hyperbolic space £3. The inversion formula for the horospherical transform 'R on G implies an inversion formula for the horospherical transform 'Rh on the hyperbolic space £3 (for the definition of 'Rh ,see Chapter 3). Namely, since C3 = U\G. where U = SU(2), it follows that each function f on £3 can be lifted to G, i.e., I can be viewed as a function 1on G satisfying the condition l(ug)

= l(g) for each u E U.

Since U is compact, it follows that the image j of a compactly supported function f on C3 is a compactly supported function on G: therefore, the space of compactly supported functions on £3 is embedded in the space of compactly supported functions on G. FUrther, one can readily see that, under the mapping G - U \ G = £3, the horospheres on G are taken to horospheres on £3 and the integrals 'R j of a function lover the horospheres on G are equal (under an appropriate agreement between the measures on the horospheres) to the integrals

:2 I"TF.GRAL GEOMETRY ON THE GROUP

a-

SL(2,C)

126

'Rh I of the function

I over the images of these horospheres. Thus, to obtain an inversion formula for 'Rh. one must only find an explicit expression for 'RI in terms of 'Rh .f and substitute this expression into tbe inversion formula (2.16) for'R.. Let us state several assertions readily following from the definitions of the borospbcres and the horospberical transforms OD G and r.3. 1. Uocler the mapping G ~ U \ G = £3. t!''ery horosphere h«(, '1) on G goos to the borosphere h(lI) on £3. 2. The transform 'R as a function of depends on]y on the variable

l.el = (I~J p~ + 1~213)1!2.

J(e, ,,),

e.

3. The functiom; 'R j and Rh I are related as follows: 'R/(-l,O;I'h.'l2)

= 'R 10 1(1I1,'l2). terms of 'Rh I follows

Tbt' desired expression for 'Ri in from tht> homogeneity condition for 'RI,

'Rj(~,7J) = 1{1-

(2.17)

4

'R/a

from 2) and 3) and

C~I)·

Substituting this expression into the inversion formula (2.16) for the borospherica1 tramd"onrl 'R on G for 9 = e, after elementary manipulations we obtain the followiog ioversion formula for the horospherical transform 'Rh 00 £3:

11

I ('IL'I) = 87r2

(2.18)

-h "Y~ L"I,,'I 'R. f(,,) w(,,) w(,,),

c.

where U'c,J e 3 is the point corresponding to the identity coset in K \ G. The expression for I at an arbitrary point of w e £3 is obtained from (2.18) by the group translation. The resulting inversion formula coincides with the in\wsion formula (6.17) of Chapter 3.

2.6. Integral transform related to paraboloids OD G. Recall (sce 1.6) that each paraboloid on G can be obtained as the section of G by a hyperplane in :R4 of the following fonn: (~lQ + ~"")'2 - (~1{:1 + ~26)'1

= p.

P'l= Ot

or, in the matrix fonn.

{g(1. =P.

(2.19)

p;i:O.

where ~ = (el. &) and , = «1. {2) are noozero vectors in C 2• and <=1. stands for the column vector Thus. each paraboloid on G is given by a pair of vect~ e C2 \ 0 and a number p :F 0 and is defined as the set of matrices 9 E G satisfying (2.19). We denote thit! paraboloid by h{{.(.p). Obviously, h(.\1('.\~'.\1;\2P) = h(e. c, p) for any -'10 -'2 :F O.

e.'

(5,J

DEFINITION. We define the integral of a COO smooth function paraboloid h{{,(,V) by the formula

(2.20)

'RP I(~,{.p)

=

lar 1(g)6(eg{1. -p)dl'(g),

where 6(·) is the delta function on C.

lonG over the

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

126

Obviously, the function 'RP I satis6es the following homogeneity condition: (2.21) Thus, 'RP 1(~,C,p) depends not only on the paraboloid h(~. (,p) itself but also on the parameters p defining this paraboloid. There is a simple relation between the integral transform 'RP and the horospherical transform 'R. Namely, the passage from the function I(g) to 'RP I(e."p) can be carried out in two steps. We first integrate I over the horospheres h(e, rJ) for fixed e; we thus obtain the function 'R I(e • .,,). Then we integrate 'R I(e, .,,) over the set of horospheres into which the paraboloid h(e.c.p) is 6bered, i.e.• over the set of horospheres h(~, .,,). where is fixed and ." satisfies the relation '1(.l = p. We express the result by an explicit formula.

e, ,.

e

2.3. The tronslonn 'RP I spherical tronslonn 'R I as lollows: THEOREM

'RP l(e.C,p) =

(2.22)

can be ezpressed in terms 01 the horo-

(~r k2 'R/(e,rJ)6(."e -

p)d."dij.

It follows from (2.22) that. for every ~, the function 'RP I(~ •.",p) is the twodimensional Radon transform of the function 'R/(e •.,,) viewed as a function of.". Thus, the operator 'RP is the composition of the operator 'R of horospherical transform and of the two-dimensional Radon transform on C2. We clarify the relation between the transform 'RP and the Radon transform. This Consider the family H~ of the horospheres h(e,.,,) with a chosen value of family consists of all horospheres parallel to some two-dimensional space of the form (1.5) on the surface of the asymptotic cone K. The mapping h(e •.,,) >-+ ." defines an isomorphism H~ - C2 \ O. On every paraboloid h(e".p) one of the families of generatrices is contained in H~, and the isomorphism H~ - C2 \ 0 takes this family to a line on C2. Thus, due to this isomorphism, the integration of the function 'RI(~,.,,) over the set of generatrices of the paraboloid h(~.,.p) can be interpreted as the integration of a function on C2 over a line in C2. Using (2.22) and the inversion formula for the two-dimensional Radon transform in C2 (see § 9 of Chapter 1), we obtain the following expression for 'R I in terms of 'RP I:

e.

'R/(e • .,,) = (2.23)

= _,..2 where 6(1.1)(s)

le ~ (~) 1J

_,..-2~

2

'RP I(e.c,p)

Ip=",J. w«)w«)

'RP I(e. ,. p) 6(1·1)(.,,<.l. - p) dpdpw«) w«),

= 6~:n6(8).

REMARK. In (2.22) the function 'R I is averaged over one of the families of horospheres on the paraboloid h(e, "p). Averaging the function over the other family of horospheres, we obtain a relation similar to (2.22):

(2.24)

'RP l(e.C,p)

= (~) 2

12

'R/(.",C)6(e."J. - p)d."dij.

2. INTEGRAL GEOMETRY ON THE GROUP G = SL(2.C)

127

By the inversion formula for the Radon transform in C2 , this implies that

'RI(~,,.,) = -7I"-2_2i

1~

'RP I«,,.,,p)

-rc "1"'1'

I

.I.

w«)w({).

P={'

We now find an inversion formula for the integral transform 'RP. Since 'RP is the composition of the horospherical transform 'R and the two-dimensional Radon transform, it follows that the inversion formula for 'RP can be obtained 88 the composition of the inversion formulas for these two transforms. Substituting (2.23) into the inversion formula (2.16) for 'R, we obtain I(g)

= - 8!5

(~) 3

ic ie fc x

'RP f(e,(.p)

~L"L,,6(,.,(.l. -

p)

I

I1:z{g

"'1"'1'

dpdpw«)w«)w(~)w(~).

After elementary manipulations we obtain the following assertion. THEOREM 2.4. A function I on G can be e:rpressed in tenns 01 the integral translorm 'RP I by the lollowing inversion lormula:

(2.25)

I(g)

(i)211 1Jp8p(lpl IJpIJp

1 = -871"5 2

'lC'lc

{fl

2

{fl

'RP I(~,(.p»)

1p={g(.1.

x w(e) w(e) w({)w({). REMARK. The integral transform 'RP of functions on G produces an integral transform of functions on the hyperbolic space C3 • Namely, when projecting G to the hyperbolic space C3 = 5U(2)\G, every paraboloid h(e, (,1) (which is a manifold of real dimension four) is mapped onto the domain C, C C3 which is the exterior of the horosphere (in the hyperbolic space) given by the equation It!I(3;'11 3 = 1. The integral transform 'RP on G becomes the integral transform of functions on C3 that is given by the following relation:

fll«1.(2) =

)-1 I(w)dzd:dz,

f (IW(2 - (11 2

1£.(

z

z

-la

where w = x+yi+zj (see Chapter 3). The inversion formula for'R readily follows from the inversion formula for 'RP.

Appendix. Remark on the horospherica1 transform on the group

Ro introduced in this section can be extended to the real case without any modifications. However, in contrast to the complex case, in the real case the mappings I ..... 'R I and I ..... Ro I have nonzero kernel, and therefore a function I cannot be reconstructed from its image. It suffices to show this fact in the case of the transform Ro. In the real case, the operator Ro takes a function 1(0, /3, 6) on R3 to the integrals of lover the lines in R3 intersecting the hyperbola /3 = 0,06 = 1, i.e., 5L(2, R). The integral transforms 'R. and

Ro/(xltX2'~) = [:oc I(X2 t + ~-1 ,t, -xlt + ~)dt. The arguments used in 2.3 to obtain the inversion formula can be repeated for the real case. We pass to new coordinates PI ~, P2 1'3 = on R3

=-

="

i

128

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

and set /(0, (J.6) = 11'31 2h(P.,P2.P3). Denote by F(~I'~2.~3) the Fourier transform of h and by ,p(~"~2 • ..\) the Fourier transform of the function ",(XI.X2."\) = 'R.o /(XI. X2 • ..\) with respect to the variables X2. The following relation can be established as in the complex case:

x.,

(2.26)

0;;«(., (2 • ..\)

= (211') 1/2 F«(I. (2 • ..\(1 -

..\ -I (2)'

We face the question of whether one call reconstruct the function F from the function,p. It is clear that this is impossible in general. The condition ~3 = ..\(1 - ..\-1(2. where ..\ is real, implies the relation ~l + 4(1(2 > 0, and hence the function F is defined by (2.26) in the domain (l + 4(1~2 > 0 only. Note that if f is a compactly supported function vanishing for the small values of 8, then it can be reconstructed from the function 'R.o f. Indeed, in this case the above function h(PI,P2.P3) is also compactly supported. and hence its Fourier transform F is an entire analytic function. Therefore. the function F. which is defined on the domain (l + 4(1(2 > 0 on R3. can be thus defined on the entire space R3. It is also possible to construct a complex version of the parabolic transform on SL(2, R). This transform has no kernel and admits an explicit inversion formula (see 117]).

3. Harmonic analysis on the group G

= SL(2.C)

In this section, following the same scheme as the one used in the case of hyperbolic space (see Chapter 3), we construct an analog of the Fourier transform on the group G. The construction uses the eigenfunctiollB (of the Laplace operators on G) that are constant on the orbits of the subgroup Z x Z c G x G. where Z is the subgroup of matrices (A I). There are two types of such orbits (see § 1), namely, the one-dimensional orbits (horospheres) and the two-dimensional orbits (paraboloids). Correspondingly, one may consider eigenfunctions (of the Laplace operators) constant on the orbits of the first or second type. This leads to two versions of the Fourier transform on G. In this section we present in detail the first version of the Fourier transform. i.e., the one related to horospheres. The main result is the expansion of a function on G in an analog of the Fourier integral; it is obtained below as a corollary of the inversion formula for the horospherical transform; see § 2. The other version of the Fourier transform on G is considered briefly in the next section. 3.1. Laplace-Beltrami operator on the group G. We begin with presenting two second-order differential operators 4 and 4 on G that commute with each other and with the action of the group G x G. Operators of this type in the space of all matrices 9 = (~~), i.e.. in C4. indeed exist. namely,

(3.1)

lP

lP

4 = 8086 - 881h'

lP

lP

A = 8ii.86 - 88frY'

To define such operators on the hyperboloid G given by 06 - /3-y = 1. we introduce generalized spherical coordinates x = pg on C 4 • where p E C and y E G. and consider our operators only on the functions that do not depend on p. The operators 4 and 4 thus defined are called the Laplace-Beltrami operators on G.

3. HARMONIC ANALYSIS ON THE GROUP G = S£(2, C)

129

We present the expressions (omitting the calculations) for the operators A and A in the coordinates 8, t, .\ on G that are given by the Gauss decomposition

(~ ~) =

~) (~ .\~l) (~ ~). It follows from this decomposition that .\ = a, = ;, and t = !. We obtain

(!

S

(3.2)

A

1(8

1

= --4 .\+ 1)2 -.\28s8t - tP + 0.\ 4'

and a similar expression for A. 3.2. Horospherica1 functkms OD G. Following the scheme presented in Chapter 3 for the hyperboUc space, we must first introduce an aoalog of the subgroup Z c G of matrices of the form (~ ~ ) for the group G x G. For such a subgroup it is natural to take the group Z )C Z. Let us find on G the eigenfunctions of the operators A and ~ that are inwriant with respect to the subgroup Z x Z c G x G. Th do this, note that the orbits of the group Z x Z in G are the two-dimensional subsets of matrices (

~ ~)

with a cbcsen

:-1)

'Y :# 0 (paraboloids) and the one-dimensional subsets of matrices (: with a eho8en 0' (horofipheres). Hence, the distributions. on G invariant with respect to the subgroup Z x Z are

.(g)

= uh) + ~(-y)v(a),

g=(~ ~),

wbere 'U and v are arbitrary functions and cS(·) is the delta function. Wc must find out what functions u(;) and 6(-y)I1(a) are eigenfunctions of the LapJace-BeJtrami operators. After simple manipulations we obtain the following assertion. PROPOSITION 3.1. The /ollowing di.ttributions on G are Z x Z -imIoricmt eigmfunctions of the Laplace- Beltrami operators A and~:

(3.3)

.,,(9) = 6h) X(Q),

(3.4)

'1I,,(g)

= x-1h) br 2 ,

9

= (~ ~),

u.t/aere cS(·) is the delta /undion on C and X is an arbitnlTfl multiplicative clumu:ter on C. i.e., a Junction of tlu! form (3.5)

X(t) =

tAIr-

= Itl'B2 t"'-"2,

nl -

ft2

e Z.

Moreover, thejunction8 tt x' .,,-,. ~x' and .x-1 belong to (the 8Gme) ei.gen.space. 0/ A and A corresponding to the eigentJtJlUe8 -l(n~ -1) and -1(nl-1), rcspeditJely.

Thus, v.'C have tv.oo types of functions, namely, the functions .x concentrated on a ~mensional submanifolds of G and the functions .x' Correspondingly, there are two versions of harmonic analysis on G; the first version uses the functions .x alld the second the functions IJI x' Here we pre&ent the first vet'8ion in detail. The second version is considered in the next liIl'Ction. DEFINITION. The distributions tt,,(g), where X is an arbitrary charaA:ter of the form (3.5), are called the zorwllwrospMriool functio'f18 on G and their shifts .,,(91992 1), 91,!/2 E G, are simply referred to 88 hor08pheric4l:functions on G.

130

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

As we shall see later, the horospherical functions .,,(91992"1) where X ranges over the unitary characters (X(t) X(t) == 1), form a complete family in L2(G), i.e., each function in L2(G) can be expanded in these functions. By (3.3), the zonal horospherical functions are concentrated on the cylindrical surface 'Y = 0 in G formed by the horospheres of the form OZ, 0 e D, and the functions are constant on each of these horospheres. Hence, every horospherical function 9 ...... • ,,(91992"1) is concentrated on one of the cylindrical surfaces in G and is constant on the generatrices of this surface. DEFINITION. We define the horospheric4l functions ." (g,~, 'I) on the manifold of triples formed by 9 E G and ~, 'I e C2 \ 0 by the foUowing relation:

(3.6) where gl,!/2

.,,(9,{,'1)

= .,,(91992"1),

e G are arbitrary matrices satisfying the conditions

e= eoglt

(3.7)

'1 = {o!/2,

where (o

= (0,1).

The right-hand side does not depend on the choice of gl and 92. In particular,

.,,(9, ~o, eo) = .,,(g).

e,

For any chosen vectors {and 'I, the distribution (g, 'I) t-+ .,,(g, (, 'I) is concentrated on a unique cylindrical surface in G passing through the horosphere h«(, 'I), and it is constant on all horospheres h«(, ~'1), ~ e C \ 0, forming this surface. For a chosen 9 e G and for 1{11 2 + le21 2= 1, this distribution reduces to a function on the manifold of horospheres H, as can be seen from the foUowing formula (3.9). PROPOSITION

e,

3.2. The horospherical functions .,,(g, 'I) satisfy the relGtions

(3.8) for any 91,!/2 (3.9)

.,,(91g92"I,e, '1)

eG

= .,,(9j {gl, '1!/2)

and .,,(gj ~l~' ~2'1)

= X(~11 ~2) 1~1~21-2 .,,(9,{, '1)

for any ~1'~2 E C\O. We give an explicit expression for the horospherical functions.

(3.10)

where 0(.) is the delta function on C. Indeed, one can set 91 =

((f :,) and!12 = ( ",ftl ~) in (3.6) and (3.7). This

gives 91992"1 = (~;~:), where a' = e2"l( -'lIP + '12a) and '11 (edJ + (20). Hence, using (3.6) and (3.3), we obtain

.,,(9,

i = '12(ela + e2'Y) -

e, '1) = ob') x(a'),

which gives (3.lO). REMARK. On the cylindrical

we have

surface 'Y' = 'I2({la + e2'Y) - '11 ({IP + (20) =

e1 a + {2'Y = {tP + e2 0 _ '1t

'12

el _ {2 - '11 0 - '12'Y - -'lIP + '12 a .

0

131

3. HARMONIC ANALYSIS ON THE GROUP G = SL(2.C)

Therefore. the ratio IJua'l2Q (the argument of the character X-I in (3.10» can be replaced by each of these rati08. PROPOSITION 3.4.

The distribution

~K(g,~.'1)

can be eqressed by the integral

Ic

~:\:(g.~,'1) = ~ 6(~g - ~'1) X-1(~)d~dX,

(3.11)

where 6(.) is the delta /unction on C2. Indeed. it suffices to prove this relation for a zonal horoepherical function, i.e.,

for to

e= '1 = (0,1). However, for these values of eand fI, the right-hand side is equal ~fc6h,6-~)X-l(~)d~dX =

s.s.

Fourler trausform

OD

6h)x-1(6)

=

6h)x(0)

G.

Let us define the Fourier transform F on G 88 follows:

DEFINITION.

Coo function

I

(3.12)

F f(e,'1,X) =

= ~,,(g,~,'1).

(~,,(-,e, '1), f) =

la ~,,(g,~,

I

of a compactly supported

'1) I(g) dp(g),

where ~:\:(g,e,fI) is the horoepherical function on G. Using the explicit expression (3.10) and (3.11) for the hor08pherical functions, 88 follows: (3.13)

we can represent F f

F I(e, '1, X) = where 9

(3.14)

la 6('12(~lo + ~2'Y)

»

e:

- '11 (edJ + (2 6 X ( -'11P

'120) I(g) dp(g) ,

= (~~), or F I(e, '1. X) =

~

laic

6(eg -

~'1) X-I(~) l(g)d~dXdp(g).

We emphasize that the integral (3.12) is in fact two-dimensional; namely. it is taken over the cylindrical surface in G on which the function ~:\:( . ; '1) is concentrated. Some facts immediately follow from the definitions. 1. The function F I satisfies the homogeneity condition

e.

F I(~I~' ~2'1, X) = X-l(~l) 1~11-2 X{~2) 1~21-2 F f(e, '1; X). 2. The operator F commutes with the action of the group G x G on G and on the space of pairs (e, '1); namely, (3.15)

= 1(91 1g92),

F It(e,.,,;x) = F f(~9lt'192;X)· We now establish another property of the operator F. Chooee an arbitrary unitary character X (thus. XX == 1) and consider the space of functions tp(e.'1;X) (3.16)

if It(g)

then

satisfying the homogeneity condition (3.15). Equip this space with the structure of a C·-algebra by setting

132

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

where Wee) = (1 cl(2 - (2cl(1 and the integral is taken over an arbitrary surface "Ye in C2 \ 0 that intersects once almost every line passing through the origin O. PROPOSITION 3.5. The mapping I ..... F I( . ; X) is a C· -algebm homomorphism, i.e., F(h • h) = F h • F 12 and (F f)0 = F 1", where

(h • h)(g) and I" (g)

=

f

h (9) h(9- 19) dl'(9)

= 1(g-1).

PROOF. The relation (F f)" = F I" immediately follows from the definition of the Fourier transform. We now prove the first formula. By (3.16) we have (3.17)

F(fl •

h)(~, ,,; X) = fa h (9) F h(Cg. ,,; X) dl'(9)

= (~) 2faL h(9)F 12«, ";X)cS(Cg = (~) 2fa

1, fc

()cl(d(dl'(9)

h(9) F 12«. ,,; x)cS(Cg - A()x- 1 (A) dAd:\w«)w«)dl'(9),

where cl( = cl(1 cl(2; the last formula follows by passing to the "polar" coordinates with respect to (. namely, ( = A('. where A E C and (,' E "Ye' Since

~ fa

fc h(g)cS(~g - A()

it follows from (3.17) that

:F(h • h)(e. ,,; X)

COROLLARY.

(3.18)

X-I (A) d.\dJ.dl'(g)

= F h(e. (; X),

1

= ~ :F h (e, (,;x) :F 12«, '1: x) wee) w«) '" h)(e,,,;x)· =(F h.F

We have

F(f. I")(e. ,,; X)

=~

1

F

f(~,(;x)F 1('1,(, x)w«)w«).

'"

3.4. Relation between the Fourier transform on G and the horospherica1 transform. THEOREM 3.1. The Fourier tmnJJlorm F I 01 a function I is related to the horosphericol tmnslom& 'R I of I as lollows:

(3.19)

F

I(e, 1I;X) = ~

fc 'R I(e,

A,,>x- 1 (A) dA dJ..

Indeed. using the explicit expression for 'RI given by (2.3) (see page 119), we can represent (3.14) in the form (3.19). Thus. to pass from a function I on G to the Fourier transform F I, we first integrate I over the horospheres on G. i.e., we pass from I to the horospherical transform 'RI. and then integrate the function 'R/(e. A,,) X- 1(A) with respect to A. REMARK.

Using the homogeneity condition 'R/(~. A,,)

= IAI- 4 'R/(e. 11),

3. HARMONIC ANALYSIS ON THE GROUP G = SL(2.C)

133

where ~ is an arbitrary nonzero number, one can represent (3.19) as follows: (3.20)

F

I(~.,,; X) = ~

k I(~.,,) X(~) 'R.

d>..d>..

We see that F I is obtained from R I by using the complex Mellin transform. Since x{>.) ei(pa+,,..,,), where 8 log I~I and IP = argA. this transform is reduced to the composition of the Fourier transforms on the real line and on the circle, and therefore, conversely, applying the classical inversion formula for the Fourier transform OD IR and on the circle S. wt' can express RI in terms of F I. Namely, it follows from (3.19) that the following assertion holds.

=

=

The horosphericol translorm R I of a junct.ion I can be ezpressetl 01 the Fourier trnns/orm F I by t,he fomavJa

COROLLARY.

in

temlS

(3.21)

R

I(e. 'I) = (211")-2

Ix

F I(e,,,;~:) dx,

where the integml is taJren over the set X 01 unitary characters X(t) = t!.I!Pl¥ , p E K, m E Zj the int~tion over X is understood QJI the usual integration over p and summation over m.

3.5. Symmetry relation for the Fourler traDSfonn. In §2 (see (2.10»

proved the symmetry relation for the horospherical transform 'R I. Substituting the expression for 'R I in terms of F I (see (3.21» into this relation, v.-e obtain the following formula for the function F I: ~

1

1

f F I({,~~g:x)w(t.)w({)dx;:: f F 1(~flfJ-1 ,,,;x) w(,,)w('1) dx Jx '" Jx "" for any 9 E G and A E C \ o. Let. us write this relation in a simpler form. Replacing '1 by (g and X by X- 1 on the right-hand side and applying the homogeneity condition (3.15) for F I, we see that

where (3.22)

Z(g,x)

=

1

F I({, (g;x)w(e) w(e).

"c

Since t.his relation holds for each ~ ". 0, the following assertion is true. PROPOSITION 3.6.

The function Z(g, X) gtrle"

~

(3.22) SGl:ilJjies the frmcaDnol

equation (3.23)

lor any 9 E G and X E X.

3.6. Inversion formula for the Fourler traD8fonn. The completeness of the family of borospherical functions ~)(g;~,,,), where X ranges OWl' the unitary characters, means that there must be an inversion formula for the Fourier transform. Such a formula exists indeed, and it immediately follows from the invenJioo formtlla

134

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

for the horospherica1 transform 'RI; see (2.16). Namely, substituting the expression for 'RI in terms of F I into this formula, we obtain

(3.24)

where c(X) = til + m 2 for X(t) = t~l·ei''': the last relation foUows from the homogeneity condition (3.15) for F I because condition (3.15) implies that L"L,,:F I(e.,,; X) = -l(til + m 2 ) :F I(e.,,; X), By the symmetry relation (3.23) for:F I. the integral over X can be replaced by the double integral over an arbitrarily chosen subset Xo c X containing exactly one representative of almost every pair of characters X and X-I. This gives the foUowing assertion. THEOREM

in tenns

3.2. Each compactly supported Coo function lonG can be expressed by the lollowing inversion lonnula:

01 the Fourier tronslonn F I

(3.25)

= -2-67r-1I~ f

I(g)

1

Jxo 'Ye

c(x) F I(e.eg;x) w(e) w(e)dx.

where c(X) til + m2 lor X(t) t~l·ei'" and Xo C X is an arbitrarily chosen subset that contains exactly one representative 01 almost every pair 01 characters X and X-I.

=

=

Recall that (as was assumed everywhere in §§ 2 and 3) the surface 'Y( in C \ 0 intersects once almost every line passing through the point 0; by the homogeneity condition for F I. the integral does not depend on the choice of this surface. One can write the inversion formula (3.25) similarly to formula (3.12) expressing F I in terms of I. Namely,

I(g)

6

= -2-

7r- 1I

(~)2

f JXo

11 C(X).~-I(g;e.,,) 'Y..

'Ye

x F I(e,,,; x) w(e)w({)w(,,)w(,,) dx (3.26)

= -2-67r-1I (~)2 f JXo

XX (el where 9

= (~ ~).

~ (3.27)

11

Q

'l'..

:.

6('12(el Q + {2'Y) - '11 (elP + e26

"rc

»

e2'Y) F I(e.,,; x) wee) w(e)w(,,)w('1) dx.

To prove (3.26). it suffices to show that

1

F I(e, eg; x) wee) wee)

"re

= (~)211 .,,-1 (g;e,,,) :F l(e,'1;x)w(e)w(e)w(,,)w(,,). 'Y..

'Yc

3. HARMONIC ANALYSIS ON THE GROUP

a = SL(2.C)

135

Webave

~

1

F I(F., F.g; x) w(F.) w(F.)

"'E

= (~r

(~) 3

=

le fC2

F I(F.. ,,;x) 6(F.g - ,,) d"dijw(F,,)w(F,,)

1e 1'1 fc

F I(F",,,: x) 6(F"g - A,,) X(A) dA dXw(,,)w(,,)w(F.)w(F,,),

where the last equality is obtained by passing to the "polar" coordinates " = where A e C and rt e Integrating over A, we obtain (3.27) by using (3.11).

"Y".

88

Art,

3.7. Analog of the Plancberel formula. The following assertion is obtained a corollary to the inversion formula for the Fourier transform. THEOREM

3.3. The lollowing relation holds lor each compoctly supported

Coo function lonG: (3.28)

la

I/(g)1 2 dlJ(g)

= _2-671"-5 (~)2

(

11

lxo ..,,, "'E

2

c(x) IF I(F",,,; x)1 w(F,,)w(F,,)w(,,)w(,,) dX

(the notation is the some os in 3.6). For the proof, consider the integral fa/(g)/(g)dp.(g) and replace the factor I(g) in the integrand by the expression for I(g) in terms of F I given by the inversion formula (3.26). Since ~X-I(g;F",,,) = ~x(g;F",,,), we obtain

( 1/(g)1 2 d1J(g) la

= _2-671"-5 (~)2 = _2-

{(

11 c(x)/(g)~x(g;F",,,)

lalxo "'r" "'rE X F I(F",,,; x)w(F,,)w(F,,)w(,,)w(,,) dxdlJ(g)

671"-5 (~)2 ( 11 c(x) F I(F",,,; x)F I(F",,,; x) lxo "'r"

"'rE

X

w(F,,)w(F,,)w(,,)w(,,) dx.

Denote by Hxo the HUbert space of functions tp(F",,,; X), where F",,, e C2 \ 0 and X e Xo, satisfying the same homogeneity conditions (3.15) 88 those for F I, where H Xo is equipped with the nonn

IItpll2 = _2- 671"-5 (~)2

(

lxo

11 "Y"

c(x) Itp(F", '1; x)1 2w(F,,)w(F,,)w(,,)w('1) dx.

"YE

By the Plancherel formula, the operator F, which was originally defined only on the compactly supported Coo functions, extends to the entire space L2(G) and defines an isometric mapping

F: L2(G) ..... Hxo' One can prove that this mapping is in fact a HUbert space isomorphism.

136

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

REMARK. One can define the space HXo = H without choosing Xo C X, namely, as the function space on the entire set X of unitary characters with the norm

1I'P1I2 = _2- 1 11"-5 (~r

Ll.,lE c(x)I'P(~,'I;~:)12w(e)w(e)w('1)w('1)dX,

assuming that the functions 'P satisfy the additional symmetry condition (3.23).

3.S. Relation between the Fourler transform on G and the representations of the group G x G. Consider the Hilbert space L2(G). It is equipped with the natural unitary representation of the group G x G by the translation operators:

(T(glo92)f)(9) = l(g1 1g92). The problem of harmonic analysis on G is to decompose this representation into irreducible ones. We first construct the irreducible representations of the group G x G. Take an eigenspace of the Laplace-Beltrami operators a and a generated by the horospherical functions ~~-l(' ;e,'1), where X(t) = t~l·eim. The elements of this space can be represented as (3.29) where

I~(g) = (~r u~(~, '1)

l.,lc u~(~,'1)~~-l(g;~''1)w(~)w(e)w('1)w('1),

satisfies the homogeneity condition U~(~l~' ~2'1) = x-1(~d 1~11-2X(~2) 1~21-2u~(e, '1)

for any nonzero ~l and ~2' The integration is carried out over arbitrary surfaces 'Y., and 'Y( in C \ 0 that intersect once almost every line passing through the point 0;

the integral does not depend on the choice of these surfaces. In the space formed by the functions 1~ we introduce the norm (3.30)

1I/~1I2 = (~r

i., iE lu~(~,'1)12w(~)W(~)w('1)w('1)'

consider the L2 space with this norm, and denote the latter space by 'H.~. The translations by elements of G define a unitary representation T~ of G x G in 'H.l(' which acts on the functions ~(~, 'I) as follows:

(Tl( (910 92)~)(~, 'I)

= u~ (~g1 , '192)·

Obviously, Tl( is the tensor product of the unitary representations of the factors of G x G. Namely, 'H.~

= H~-l ® H~,

Tl(glo 92) = T~-l (gd ® Tl(92).

where Hl( is the Hilbert space of functions 'P(e) on C 2 \O satisfying the homogeneity condition with the norm

3. HARMONIC ANALYSIS ON THE GROUP G = SL(2.C)

137

and T;I(-l and T;I( are the natural representations of G in the spaces H;I(-l and H;I(' respectively,

(3.31)

=

It is known from the representation theory of the group G 8L(2, C) (see [8]) that the representations T;I( are irreducible, and two representations T;I(l and T "3' XI ", X2, are equivalent if and only if X2 = Xli. Hence, the representations T;I( of G x G are irreducible, and the representations TXl and T"3' XI ", X2. are equivalent if and only if X2 = Xli. We want to decompose a function le L2(G) in the functions 1;1( e 'H;I(' To this end, we define an operator P;I(: L2(G) - 'H;I( by the rule

where

P"I = I". 1;1( is given by formula (3.29) in which u;I((' '1) = F I(e, '1; X)·

Obviously, this operator commutes with the action of G x G on the spaces L2(G) and 'Hx. Moreover, one can readily see that PlC ", 0 for any X. The next assertion immediately follows from the inversion formula (3.26) and the Plancherel formula (3.28) for the Fourier transform F. THEOREM 3.4.

I" = P"I; namely.

The /unctions I e L2(G) can be decomposed in the /unctions the lollowtng relations hold:

I = _2- 6 71'-5 f

(3.32)

11/(g)1

(3.33)

Jxo

2 dp(g)

G

c(X) I" dX,

= _2- 6 71'-5

f c(x>ll/xIl 2 dX,

Jxo

where c(X) (il + m 2 lor X(t) t~t"im ; the integration is carried out over an arbitrarily cholen character subset Xo c X containing exactly one reprelentatille 01 almost every pair 01 characters X. X-I.

=

=

COROLLARY. The representation T 0/ G x G in the space L2(G) hu simple spectrum, i. e.. it is decomposed into pairwise nonequillalent irredudble representations.

Note that the simplicity of the spectrum of T (i.e., the fact that the integration in (3.32) and (3.33) is only over Xo rather than over the entire character set X) follows from the symmetry relation for the Fourier transform F. REMARK. Simplicity of the spectrum is a usual fact observed when one decomposes representations related to symmetric spaces (see Chapter 3, where the representations related to the hyperbolic plane and hyperbolic space were consid-

ered). 3.9. Relation to the representations of the group G. We now discuss the previous results from the viewpoint of the representation theory for the group G. Consider the unitary representation of G in the space L2(G) by the operators of right translation, (T(9o)",,)(g) = ",,(g9o)

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

138

(the fKH:8Iled right regular representation). The projection TX of this representation 1tx = HX-I ® Hx is the tensor product

to the space

TX = I®Tx. where I is the identity representation in H x - I , and T x is the irreducible unitary representation in Hx given by (3.31). Hence, Tl( is a multiple of the irreducible unitary representation Tl( of G. Thus, relations (3.32) and (3.33) define the decomposition of the (right) regular representation of the group G into pairwise nonequivalent representations which are multiples of irreducible representations. In terms of representations of the group G the Fourier transform F I hu the following meaning. For any X E X, consider the irreducible representation Tl( of G in the space H x' To each compactly supported Coo function I on G we assign the following operator on the space H x: (3.34)

Tx(f)

=

fa

l(g)Tl(g) dlJ(g)·

This operator acts on the functions I{J E Hl( by the rule

(Tl((f)IP)(~) =

(3.35)

la

I(g) 1{J(~g) dlJ(g)·

PROPOSITION 3.7. The integral kernel 01 the operator TX (I) (in the homogeneous coordinates ~, 'I) is the function F I (~, 'la -I), i. e.,

(3.36)

(Tx(f)I{J)(~) = ~

1

F

I(~, 'la-I) 1{J(1]) W{1]) W{1]).

"r"

Indeed, we integrate over G in (3.35) in two steps, namely, we first integrate over the horospheres {g = '1, which gives the horospherica1 transform 'R I{£". '1) "'('1), and then over 1]. We obtain

= (~) 2fC2 'R I{f.. 1]) "'(1]) d1]dij. dll = dllld1l2. Here we pass to the "polar" coordinates 1] = >.,.,', where>. E C and 1l' E ,.". Since 'f'(>'''') = x(>')I>'I- 2'f'(1]) and d'1 = >. d>.w(,.,), it follows that we obtain (3.36) after (Tx(I)",)(f.)

integrating with respect to >.. COROLLARY. TIu! operator TX(f) has a trace. and this

(3.37)

Tr Tx(f)

= ~1

F

trace is equal to

I{~,~;x-l)w(f.)w(~).

"re

Using this result, we can evaluate the trace of a unitary operator Tx(g), 9 E G. Certainly, this operator hu no trace in the usual sense. However, it is natural to define Tr TX(g) U a distribution on the space of test functions on G, (Tr

T X {')'

f)

= Tr Tx(f)·

PROPOSITION 3.8.

(3.38) where >.~l) and >.~2) are the eigentJalues 01 the matriz g.

4. ANOTHER VERSION OF THE FOURJER TRANSFOR.'f ON G = S£{2.C)

PROOF. According to the definition of F

~

1 ~

F I({,e. X-l)W({)W(~)

=~

11 ~

Hence, if for ")'( Vt-e take the line {l =

1'1' T1«g)

f ~f

=~

6(to

=

6«s -

(3.39)

f, we have

6({2({1 0 +(2"') -

G

x

139

~1«(IP +e26»

X(ell3 ~ e~) l(g)dp{9)W(e)W(e).

t. e2 = I, then

+.., - t(tP + 6» X(fP + 6) dtdl ~~I»(s - ~~2»)X(s)dsdi

(the last equality is obtained by changing the variable immediately follows from (3.39).

tp + cS = s).

Fbrmula (3.38)

Since F I(e,,,; X-I) is the integral kernel of the operator T,,(J), we can modify the Plancherel formula. Namely, consider the operator 1',,(1)1';.(1) = fX(1. where r;,(f) is the operator conjugate to T')(/). By (3.18), the integral kernel of this operator is

r).

'flee,,,: x) = ~

1

F

1«(, (;x) F 1(". (. x)w«) wC,)·

"re

Hence,

Thus. the Plancherel formula (3.28) can be written (3.40)

f

II(g)1 2d#l(g) = 2- 6 71'-5

ko

88

c(x)Tr (1'1«(f)T;(/)) dX·

REMARK. The expansion (3.25) IIl&b8 sense not only for the test functions but also for the distributions on G. Note that it is natural. to view the delta function 6(g) on G 88 the character of the regular representation. Therefore, the inversion formula (3.25) (or 6(9) can be ?Titten as

c5(g)

(3.41)

= 2- 6 71'-5 f c(X) 1'1' TX(g) dX. Jxo

4. Another version of the Fourier transfOrm on G = SL(2, C) In § 3 we have const.ruct.ed the projections of functions on G to the eigenspaoes of the Laplace-Beltrami operators ~ and~. The projection was carried out by using the zonal. horoepherica1 functions ~'!«g) 6(')') x(o). However, one can do it in a different way, by using tht' zonal. functions (I:t given by

=

(4.1)

(I')((g)

= x- 1(1.) 1..,1- 2 ,

where 9

= (~ :)

(see 3.2). This leads to another version of the Fourier transform on G, which we now briefly describe.

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

140

4.1. Functions 'ir7(g;e, (). Define analogsofthe functions .,,(g;e, ,,) (see §3) by the formula (4.2)

where

e, ( E C2 \ 0 are given by the relations

e= eogl,

(4.3)

(= (092,

where

eo = (0 = (0,1).

One can readily see that the left-hand side of (4.2) does not depend OD the choice ofthe matrices 91192 E G satisfying (4.3). In particular, 'ir,,{g)

= 'ir,,(g; eo, (0).

Formulas (4.1) and (4.2) lead to the foUowing explicit expression for these functions: (4.4)

We note the foUowing two properties of 'ir 7( (which are immediate consequences of the definitions): (4.5) for any g1l!12 E G; 'ir,,{g; ~le, ~2() = X-1{~1~2) 1~1~21-2'ir,,(g;

(4.6)

e, ()

for any ~t. ~2 E C \ O. THEOREM 4.1. The functions 'ir" are related to the horospheriaJl functions ." (see § 3) as follows:

'ir,,(g;

(4.7)

e, () = ~

1

.,,{g;

e. ,,) x- {,,(.!.) 1J1(.!.1- w(,,) we,,); l

2

'l"

(4.8)

.,,{g;

e, ,,) = (2'11")-4 c{x) ~ 1'ir,,(g; e, () X{,,(.!. ) 1J1(J.1- w«) w{(), 2

'le

where c{x)

= jl + m 2 for x(t) = t.l'ei

m

• The integration is carried out over arbitrary two-dimensional real surfaces 'Y" and 'Ye in C2 \ 0 intersecting once almost every complex line passing through the origin O.

Both integrals must be understood as integrals of distributions. We do not present rigorous definitions of these integrals and restrict ourselves to a formal proof of (4.7) and (4.8). We first Dote that it suffices to prove these relations for a chosen element g and for a chosen = By setting 9 = e and = (1,0), we obtain

e eo.

."{e,eo,,,) = 6{'12) X{"l); If in (4.7) with g

eo

'ir,,{e,eo,()

= X- l «2) 1(2r2.

= e and e= eo we take for 'Y" the line "1 = 1, then the integral

becomes

~ fc 6{'12) X- l {(2 -

'12(d 1(2 - '12(11- 2 d'12 dTi2'

Obviously, this integral is equal to X- l «2) 1(21-2, i.e., to 'ir,,(e,eo,().

... ANOTHER VERSION OF THE FOURIER TRANSFORM ON G ... SL(2.C)

Obviously, the distribution

"'('12) = ~

L

X(I- '12(1) 11

satisfies the homogeneity condition 'P{>''12)

141

- '12(11- 2 (/(1 del

= 1>'1-2 '1'('12). This implies that 'P('12) =

c6('12). We leave the evaluation of the factor c to the reader as a useful exercise integration of distributions.

G. DEFINITlON. We define the Founer tronsloTTn ex' function lonG by the rule 4.2. Fourier transform.

(4.9)

OD

OD

FI

of a compactly supported

FI«(.(;xJ = ("'~(·.e.(), I) = L/(gh- 1 «(g(i)legt;J.1- 2 dp(g).

There is a simple relation between the Fourier transform FI and the integral transform 'R." I in § 2 related to paraboloids on G. Namely, comparing the definition of FI with the de6nition of 'Rl'l (see (2.20)). we obtain the following assertion. THEOREM 4.2.

The operators F and'R,P are related as lollows:

F/(e,(:x) = ~

(4.10)

f

'R,P l(e,(;p)x-l(p)lpl-2dpdp.

Thus, FI is obtained from 'R,1' I by using the Me1lin transform with respect to p; cr. the relation between F I and 'R, I in 3.4. Hence, applying the i.nvenion formula for the Mellin transform, we obtain the following assertion. COROLLARY. The

junction 'R,P I can be upressed in tenns 01 Ff as follows:

'R,P I({,(,p)

(4.11)

= (211')-2

Ix

FI«(,(,X)X(P)dX,

where the integral is taken over the set X of unitary chamcters.

4.3. Relation between the above two versions of the Fourier transform. It follows from the relations between the borospberical functions tt ~ and .x (see 4.1) that the functions Ff and F I satisfy the same relations, where F is the Fourier operator introduced in § 3. THEOREM 4.3.

The Fourier operators F and F ~ related as /ollouzs:

F/({, (; X) = ~

(4.12)

1

F I({, 71a)x.-l(flC.l.) l«il-2 W(71) w('I);

'Y.,

(4.13)

F

I(~, 'I; X) = (211')-4c(X)~

1FI(e,(; 'Y(

where c(x) =,; + m 2 for x(t)

= t~l"i'" .

x) x(<<.1) 1«.11-2 w«)w«),

142

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

REMARK. These relations can also be established immediately by using the known relations between the integral transforms "R and F. between "R and "RP. and between "RP and F.

4.4. Symmetry relation. The next assertion follows from (4.12) and the symmetry relation for F THEOREM

I.

4.4. The function F

Z(g,X)

(4.14)

I satisfies the symmetry relation

= Z(g,X- 1 )

lor any 9 E G.

X E X,

where

Z(g.x)

= (~r

L, Le

FI«(.(a) x«(g(.1.) l(g(.1.1- 2 w«()w«().

4.5. Inversion formula and Plancherel formula for the Fourier tr8D8form F. 4.5. EtJery compactly supported Coo function lonG can be ezpressed in tenns 01 the Fourier translorm FI by the following inversion lormula: THEOREM

(4.15)

l(g)

= _2- 811"-7 (~)2 f

JXo

11

c2 (x)FI«(.(;x)\)x- a(g;(.()

"Yc "Ye

x w«() w«() w«) w«) dx, where Xo is an arbitrary subset 01 the set X 01 the unitary characters x( t) = t~z'ei'" , pER, me Z, such that Xo contains precisely one representative 01 almost every pair 01 characters X, X-I. Indeed, it immediately follows from the inversion formula for the integral transform "RP related to paraboloids on G (see (2.25» and from the above relation between "RP I and FI that

I(g)

= _2-'11"-7 (~r

Ix 1, le

c2(x)Ff((.(; x)x«(g(.1.) l(g(.1.1- 2

x w«() w«()w«) w«) dx

(4.16)

= _2-'11"-7

(~) 2 Ix

1, le

c2(X)FI«(.(; X)\));-I (g;(,()

x wee) wee) w«) w«) dx. By the symmetry relation (4.14) for FI, the integral over X in (4.16) can be replaced by the doubled integral over the subset Xo C X. Note that there is an analogy between the inversion formula (4.15) and the original formula (4.9) expressing 1:1 in terms of I. As in the case ofthe Fourier transform F, the inversion formula (4.15) implies the following assertion.

4. ANOTHER VERSION OF TIlE FOURJER TRANSFORM ON

COROLLARY.

(4.17)

la

a". SL(2,C)

Th.e following Plancherel formula holds for functioJUJ

OR

143

G:

If(g)1 2 dp(g)

= _2- 8 11'-7 (~) 2 Ix.,

l( le ~(x) IFf({.<,x)12w(~)w(~)w«)w(C)dX.

According to the corollary. the Fourier transform F. which was originally defined for compactly supported coo functions on G only, can be extended to all functions in L2(G).

4.6. Relations with representation theory. The arguments iD § 3 about the n>.lation between tbe Fourier t.rans£onn F and representation theory can be applied to the Fourier transronn F "ithout subst.antial modificatioos. Namely, to decompose the representation T of G x G in the space L2(G) into irreducible representations, we consider the eigenspace (of the Laplace-Beltrami operators ~ and 3) generated by t.he horospberical functions .x-1(' ;{,(). where X(t) The elements of t.his subspace can be written as (4.18) where fiX

h:(g) =

= t~t¥.

1,

(~) 21c u,,(~.') f.lX-l(g;~,')w(~)w(~)w(')w«),

(e. C) satisfies the homogeneity condition UX(~le. ~2()

= X-l(~1~2)1~1~2r26x(e, ()

for any ~t. ~2 e C \ O. Equip the space of functions

111,,11 2 = (~)

2

ie le

A with the norm

16x«(.{)12w(e)w«()w«)w«),

consider the L2 space with this norm, and denote this space by fi". The translations on G define an irreducible unitary representation Tx of G x G in fix. and this representation acts on the functions 6x({,() as follows: (T~(gl,92)6x)(~,') = 6x«(91o(92)·

ix.

\Ve further construct the operator P,,: L2(G) -+ fix by the rule P"f = where is given by relation (4.18) in which U't«(,() = Ff(~,<:x). The desired decomposition of the representation T is given by the formulas

A

(4.19) (4.20)

f

= -2-8 11'-7

1

Xo

c2(x)A dx,

h

= Px/'

f If(g)1 2 dlJ(g) = 2-1$11'-7 f c2(x) IIhll 2 dx. la lxo

These relations immediately follow &om (4.15) and from the P1anchere1 formula (4.17) for the Fourier transform F. Since the decomposit-ion of the representation T of G x G into irreducible represent-ations is unique (because the spectrum is simple), it l'ollows that the dec0mposition thus obtained coincides with the one fmmd in 3.8. Namely,

h = c-1(x)f",

IIAIl = c- 1/'2(x)lIfx ll·

4. INTEGRAL GEOMETRY ON THE LORENTZ GROUP

Note that the relation /'" between F / and j:/.

= c- 1 (x)f1(

is equivalent to the above relations

CHAPTER 5

Integral Geometry on Quadrics This chapter is devoted to the problem of reconstructing a function f OD a quadric X C lRn + 1 &om the integrals ! 0Vl".r the byperp1ane sections of the quadric. This is an example of an O\--erdetermined problem of iDtegrai geometry; namely, we reconstruct a function of n variables (n is the dimension of X) from a function of (n + 1) variables (n + 1 is the dimension of the space of sections). Another example of an overdetermined problem (the John transfonn in R3 and CS) was already treated in Chapter 2. As in any O\rerdetermined problem, tbe main tMk here is to find n-dimensional submanifolds r of plane sections (admissible submanifolds) such that the function ! can bt> reoonstmcted from the rest.riction of the function to The reconstruction of f from llr is carried out by the general method of the operator 1'1':. which was first applied in [1]. This method was explained in Chapter 2 using, as an example, the John transform in In the case considered in chapter. for any point x EX, we assign to a function a closed differential form Kz! of degree n on the manifold of plane sections. The integrals of this form over various cycles are equal to the value of the original function! at the point x. This is a rare case where the method of the operator" is developed for a nonlocal problem of real integral geometry. Many fomlUlas of this chapter can be written for arbitrary quadrks; however, the choice of appropriate cycles giving the inversion formulas is diJlicult and often impossible. For this reason ...-e restrict oun!elves to the case of a byperboloid 01 two sheets and a sphere (which are projectively equivalent). The case of a byperboloid of two sheets gives another proof of the inversion formulas in problems of integral geometry related to the hyperbolic space (see § 3). The case of a sphere is equivalent to the problem of integral geometry related to spheres in Rft. The operator K for quadrics was COWItrUcted by Gindikin (15, 16] and Goncharov [24). The problem of sections of the quadric can be interpreted as 8 conformally invariant extension of the Radon transform. In this interpretation the Radon transform (Chapter 1) and the horospherical hyperbolic transform are conformally equivalent.

lof

1 r.

cs.

1

tu

1. Integral transform related to the byperpIane sectloDs of a byperboloid of two sheets In RII+1 1.1. Deftnition. Let en be the upper sheet of the hyperboloid of two sheets in Rn+l given by

= ~ - ~ -'" - x~ = 1, Xo > O. Consider the sections of en by the h}1)erplanes [~. xl = pin RM·l, where (~.x] = ~oXo - ~lXl - ••• - enxn· (1.1)

[x.x]

148

146

5. INTEGRAL GEOMETRY ON QUADRJCS

We recall that en is one of the models of the n-dimensional hyperbolic space. and corresponding to the hyperplane sections of en are various spheres in the hyperbolic space and their limit cases, namely, the geodesic hypersurfaces and the horospheres. Generic spheres occur as sections of en by the hyperplanes

[e,X]

= p,

where p ::/= 0 and le,e] ::/= O. The sections related to the limit cases p = 0 and le, el = 0 are the geodesic hypersurfaces and the hor08pheres, respectively. Thus, the manifold of spheres is (n + 1)-dimensional, and the manifolds of horospheres and of the geodesic hypersurfaces are n-dimensional. Note that the horospheres and the geodesic hypersurfaces are the only ndimensional families (in the manifold of all hyperplane sections) that are invariant with respect to the motion group 50(1, n) of the hyperbolic space. We introduce a differential n-form #leX) on Rn+! satisfying the formula

d([x, x]) " #leX)

= dx,

where dx

= dxo""'" dxn

(i.e., #leX) is an interior product ofthe differential forms dx and d([x, xJ)). Such a form on Rn+l is not unique; for instance, it can be given by the relation 1

#leX)

= 2 le, (I

n+!

. L(-I)'ei

,=0

However, the restriction of this form to on en, this restriction is

(\dei'

);i,

en is uniquely defined.

In the coordinates

Xl, •.• ,Xn

#leX)

... " = dx. " 2.xo

dx n

•

We introduce the integral transform 'R taking a function I on en to the integrals of lover the sections of en by the hyperplanes [e,x] = pin Rn+l, (1.2)

'RI«(,p) =

1 t:. ..

I(x) 6(1(, xl - p) #leX).

Assume that the functions I are compactly supported and infinitely differentiable on en. Obviously, 'RI(e,p) = 0 if the hyperplane le.x) = p is disjoint

from en.

REMARK. The integral transform 'R can be regarded as the Radon transform ofthe distribution I(x) 6([x, xl -1) on Rn+!.

It immediately follows from (1.2) that satisfying the homogeneity condition (1.3) and the differential equation

(1.4) Note that (1.4) is the wave equation.

I{J

= 'RI is a function

,\ ER, 0,

011

Rn+2 , 0

1. INTEGRAL TRANSFORM RELATED TO HVPEItPLANE SECTIONS

147

1.2. Admissible submanlfolds in the manifold of hyperplane sections of en. An n-dimensional 8ubmanifold r of the manifold of hyperplane sections of en is said to be admissible if any function I on en can be reconstructed from the restriction of the function tp = RI to r. In Chapter 3 we obtained inversion formulas (for n = 2 and n = 3) reconstructing a function I on the hyperbolic space e" from the integrals of this function over the horospheres or over the geodesics. In particular, we have established that the manifold of horospheres and the manifold of geodesics are admissible submanifolds of the manifold of all hyperplane sections. We now present, for an arbitrary n, three examples of admissible submanifolds in the space of hyperplane sections of e". In each of these examples, the proof of the inversion formula is reduced to the inversion formula of Chapter 1 for the Radon transform in the space Rn. EXAMPLE

1.

r

is the manifold of sections of e" by the hyperplanes

[e,x]

= P.

where

eo = O.

Denote by ,,(e1,'" ,(",p) the restriction of the function

f{J

= RI to r, i.e.,

"«(1, .. ',(n,P) = Jr." f 12(X) 6«(lX1 + ... + {nX" + p) dx 1 /\ '" Xo

/\ dx".

IJ!;!

We see that" is the Radon transform of the function viewed as a function of Xl, ... , Xn on Rn. Hence, by using the inversion formula for the Radon transform (see 2.3 of Chapter 1). we obtain the desired inversion formula:

(1.5) 2xo

11

00

.,

-00

"«(1, ... ,(n,p) «(IX1 + .. ·+enXn +p-iO)-n dp/\a«() = en I(x),

where en = iJr:l~)" a«() = E:=t (-I)i-1(i I\#i de:l, and 'Y C R" is an arbitrary surface intersecting once every ray issuing from the origin O. In particular, we have proved that the manifold r is admissible. EXAMPLE

2.

r

is the manifold of sections of

[(.x] Denote by

= P.

where (0

en by the hyperplanes = (1.

"(e1 .... ,(n,P) the restriction of the function

"«(1 •...• (n,P)

= f{J«(lt(I •... ,(n,p) = f I(x) 6«(1 (XI Jr."

xo) +

(2X2

f{J

= RI to r, i.e.•

+ ... + (nXn + p) p(x).

We find the inversion formula reconstructing I from the function ". To this end, we introduce a new system of coordinates on

en.

111

= Xl -

Xo.

Y2

= X2 • •••• 1/n = X n •

and denote by 11 (fI) the original function I in these coordinates. It follows from the assumptions imposed on I that the function 11 is defined on the domain 111 < 0 and is compactly supported in this domain. Assume that 11 is extended to the entire space Rn by setting l1(y) = 0 for Y1 ~ O. In the coordinates fI. the differential form I' becomes

p(y) =

148

S. INTEGRAL GEOMETRY ON QUADRICS

and hence (1.6)

1/1(~1o ••. '~n'P) = - f

la..

121 (y) J/I

6(~IYI + ... + (nXn + p) dYI A. ••• A. dJ/no

Thus, 1/1 is the Radon transform of the function F(y) = _/~~!). Therefore, it foUows from the inversion formula for the Radon transform that

-2YI

11

00

..,

-00

tJ1(~I' ... '~n'P) «(iYI + ... + ~nYn + P -

io)-n dp A. w(~)

= en I(x) •

where en, w(~), and "'1 are defined in Example 1. Returning from Y to the original variables x, we obtain the desired inversion formula (1.7)

2 (xo - XI)

11°C tJ1(~I' ..,

... ,(n,P)

-00

x (ei(XI - xo) + ~2X2 + ... + ~nxn + P - io)-n dp A. w(~)

= en I(x).

REMARK. The manifolds r in Examples 1 and 2 have a simple geometric description; namely, they consist of the sections of en by the hyperplanes passing through the points at infinity with the homogeneous coordinates (1,0, ... ,0) and (1,1,0, ... ,0), respectively. The first point does not belong to the projectivization of the hyperboloid, whereas the other point belongs to the projectivization. EXAMPLE 3. r is the manifold of geodesic hypersurfaces in en, i.e., the submanifold of sections by the hyperplanes passing through the origin, I~, x] = O. Denote by tJ1(eo, ... '~n) the restriction of the function r.p = 'RI to r, i.e.,

1

t/J«(o, ... , en) = -2

1

fez) cS([e. zl>

"..

dz l A. ••• A. dz n

.

Xo

One can view I(x) as the restriction to en of a homogeneous function on Rn+l \ 0 defined by the relations I(>.x) = I.\I-n I(x) for any .\ :/: 0 and x e en and I(x) = 0 for Ix.x] o. We also note that the differential form (xo)-ldz 1 A. ••• A. dz n in the integrand is the restriction to en ofthe differential form Wn +1 (x) = E:'::o(-l)iXiAj",idzJ on Rn+l. Hence (see 8.8 in Chapter 1). the function 2t/J«(o, ... ,en) is the projective Radon transform ofthe homogeneous function I(x). Thus, in view of the inversion formula for the projective Radon transform (see 8.8 in Chapter 1), we obtain

:s

(1.8) where

2 it/J(~o •... '~n)(le,X]-io)-nwn+l(~)=en/(X),

en = ijr:~~)!·

1.3. Operator ,,~. Let us proceed with constructing a universal inversion formula for the integral transform r.p = 'RI. For any point x e en, we construct an operator ,,~ from a function space on the manifold of hyperplane sections to the space of differential forms of degree n on this manifold. If r.p = 'RI, then the dif£erential form "~r.p is closed. We also claim the following inversion formula for various cycles r : if r.p = 'RI, then

i "~r.p =

c/(x).

1. INTEGRAL TRANSFORM RELATED TO HYPERPLANE SECTIONS

149

We first introduce the following notation which is useful in what follows. We denote by 100, ... , Bn+ll the determinant of order n + 2 with columns ai some of which consist of 1-forms. We expand the determinant from left to right and assume that multiplication of I-forms is the exterior multiplication. Such a determinant is symmetric with respect to the columns consisting of I-forms and may contain identical columns of 1-forms without being equal to zero. For instance, under this notation, the differential form ,,+1

w(e)

= L(-I)~ei A~j ;.Fi

i=O

becomes

1

wee) = (n + 1),I{,~, ... , ~I, or, briefly,

wee) = {n+1 1)1' le ~("+1)1 , where ~("+1) stands for the sequence of (n + 1) equal columns~. For convenience, in what follows we write e,,+l instead of p and 'P{e) instead of 'P(e,p), where e = (eo, .. . ,e,,+1) E R,,+2 \ O. Let 'P{e) be an arbitrary infinitely differentiable function on R,,+2 \ 0 satisfying the homogeneity condition 'P(.\e) = I~I-l 'P{e) for ~ ~ D. To the function 'P and to an arbitrary point % E e" we assign the follOWing distributions on Rn+2:

%i 'P(e)

(1.9)

.,(e, %) = ([e, %] - e"+1 - iD)"

+ _1_ n- 1

Ei'P(, (e) . , aO),,-l i = 0,1, ... , n + I,

{le, %] - e"+1 -

=

where EO = I, Ei = -1 for i > 0, and %,,+1 1. Recall that the distributions ([{,%] - e"+1 - iO)-n (see [10]) were introduced in Chapter 1 and used there in the inversion formulas for the Radon transform in the affine and projective ndimensional spaces; see 2.3 and 8.8 in Chapter 1. Note that %) can be given as

.,(e,

(1.10)

.,«(,%) = Ei n~ 1 {([e,%1 -e"+1- iO)-,,+l,'P(e) }i'

where the braces stand for the Poisson brackets

{'PI, 'P2h

8'P2

= 'PI 8ei

/Jtpl - 'P2 8ei .

en

To every point % E we assign the following operator I£z from the space of functions 'P(e) to the space of differential n-forms whose coefficients are distributions:

(1.11) where .((,%) is the column with the elements .i({,%) defined by (1.9). Expanding the determinant (1.11) in the elements of the 6rst column, we obtain the representation of I£z'P in the form ,,+1

(1.l2)

I£z'P =

L .i(e,%)Wi({), i=O

11. INTEGRAL GEOMETRY ON QUADRlCS

150

where n+l

WO(e)

= E(-l)'-I~i

"

d{;.

j~O.i

i=1

and the subsequent forms Wi are obtained from WO by a cyclic permutation of the coordinates ~n+l. and the sign depends on the parity ofthe corresponding permutation. It follows from the definition that the differential form "ztp is homogeneous of degree zero; moreover. this form is orthogonal to the fibers of the bundle Rn+2\0 5"+l. where 5"+l is the (n + I)-dimensional sphere. Hence. this form is defined on the manifold 5"+l of rays in Rn+2 issuing from the origin.

eo •..••

THEOREM 1.1. The differential/orm /Cztp is closed on Rn+2 for any point x E en and any function tp(e) on Rn+2 satisfying the homogeneity condition tp(~) = 1-'1- 1 tp(e) Uor -' :/: 0) and the differential equation

{fltp

Dtp

= a~~ -

{fltp

a~l

(fltp

- ... - ae~+l = o.

The proof of the theorem results from the following general lemma on determinants. LEMMA. [laCe) is a column o/length n+2 whose elements ao(~), ... , an+l(e) are homogeneous functions 0/ degree -n - 1, then

dla(e),e,~(n+1)] = __1_ n+ 1

(E aeJ

8aj (e»)

;=0

[e,~(n+1)].

This lemma can be applied to the determinant (1.11) because the elements of the column +(e.x) are homogeneous of degree -n - 1. Hence, by the lemma we have dtcztp

= __1_

(E

n+ 1 ;=0

M;(e, X») [e,d{(n+1»). a~

On the other hand, it follows from the definition of the functions +j(e. x) in the form of Poisson brackets that

E

Mj(e.x)

;=0

8e;

= ([e. x]- e"+1 Hence. d("ztp) = O. COROLLARY.

io}-n+1 Otp(e) - tp(e) o([e. x]- en+l - iO)-n+l

= O.

1/ a function tp belongs to the image 0/ the integral transform 'R.,

then the differential/arm /Cztp is closed.

In conclusion we present the proof of the lemma. It suffices to prove it for the case in which only one element of the column aCe) is nonzero. say. the element ao(e). Then

u ==

[aCe). e. d{(n)1 = n! ao(e) Wo(e).

I. INTEGRAL TRANSFORM ftELAnV TO HYPERPLANE SECTIONS

1st

where the form wo({) was defined above. Hcncc.

1 n+1 800({) -. du :;: ~ d{. /\ wo(~)

L

n.

.=0

v ...

+ ao({) /\ dwo({).

By the Euler theorem on homogeneous functions, ao«() Thus, since dwo«()

= __1_

n+ 1

E

8ao({) (1'

0

.=

8(;

= (n + 1)d{1/\ "'/\d{n+l, it follows that

ao(~)/\dwo(e):;: -

n+18a o«() L ~ei~I/\···/\den+l' ,=0 .,...

Substituting this expression into the formula for duo we obtain

..!.. du = OaO«() (d(o /\ '"-'G({) - eo d{1 /\ ... /\ d{n+d n!

8(0

+

n+l OaO(~)

t; ~(d{i

/\ wo«() -{.d(l/\"· 1\11(11+1)'

It remains to note that the coeffident of ~() is -rn:hJy[{,d«"+l)] and the coefficients of the other deri\'&tivcs vanish. REMARK. One can extend both the definition of the operator K-z and the proof that the differential foml Kzl{) is closed to any nondegenerate quadric in R"+2.

1.4. Local and nonlocal operators K.. The distl'ibution (t-io)-n occurring in the definition of the operator K-z cau be represented 88 a tinear combination of the distributions t- n and 6(n-l)(t), (t - iO)-n

(1.13)

= t- n + ur. ( -

l)n-l

6(n-l)(t)

(n. - I)! (see, for instance, 110». Moreover, depending on the parity of n, one of the terms is an ('"\"en distribution and the other is odd. Accordingly, the operator K-z can also be represented 88 the sum of loca.l and uonlocaJ part.s. Namt>Jy, dt-..6nt> the operators K~ and K; by (1.14)

K~'; :;:

n+l

L +:«(, X)Wi«(), ,=0

n+l

K~I{J

=L

+:'({,x)Wt(e),

i=O

where

, i",(-I),,-1 ( n-I .i(~' x) = (n _ I)! Xj !p(~) 6< )([e, xl - (,,+d

(US)

- Ei'PE< ({) 6("-2)([~.xl - (,.+I»); .:'(~,J:)

= x, r;({)([{,x]- {n+1)-n + Ej
Then it follows from (1.13) that (1.16)

U the function

I{)

K-z'P = ,,~lP + ~I{J. satisfies the equation Dl,C = O. then botb differential forms

K.~I{J and K;'P are closed.

5. INTEGRAL GEOMETRY ON QUADRlCS

1&2

When integratiDg the differential form ICz'P over various cycles, we shall restrict ourselves to the cycles 8 ft +1 symmetric with respect to the transforms ( - -(. For these cycles. the integral of the odd part of the form ".'P vanishes. Note that the form "~'P is even and the form IC~'P is odd if n is odd. The picture is opposite if n is even. Thus,

re

t

ICz'P =

1r IC~'P

for odd n;

1.6. Inversion formula. Recall that the differential form "z'P is defined on Rft+2 \ 0, and it can be pushed down (under the natural mapping R,,+2\ 0 _ 8"+1) from R,,+2 \ 0 to the manifold 8"+1 of rays in Rft+2 issuing from the origin.

THEOREM 1.2. If'P = 'Rf, then the following inversion formula holds for each n-dimensional cycle r c 8"+1:

1r "z'P =c(r) f(x).

(1.17)

c(r)

r.

where depends only on the homology class of the cycle mUBt be understood as the integral ot/er an arbitrary section where is the preimage 0/ r in RftH \ O.

r

The integral over r

0/ the bundle r - r,

All cycles r treated below belong to the same homology class, and we present the proof of the theorem for one of these cycles, namely, for the cycle that we are going to introduce. Since the form "z'P is closed, the theorem will thus also be proved for each cycle c 8"+1 which can be obtained from by a continuous deformation. Let = {( E R,,+2 \ 0 I (0 = (I} and let r = be the image of in 8"+1. Denote by 1/I(~h"" (,,+1) the restriction of 'P 'R/ to in other words, 1/I«(1 •...• ~,,+1) 'P«(I,(I"",(,,+1)' The inversion formula expressing f in terms of 1/1 was already obtained in 1.2 (see (1.7); recall that this formula for I immediately follows from the inversion formula for the Radon transform in R"). Thus, to prove the inversion formula (1.17) for = it suffices to show that the integral "z'P coincides with the left-hand side of (1.7). Consider the expression for "z'Plr in (1.11). Since (0 (1 and d(o d(1 on r, we see, after subtractiDg the second row of the determinant (1.11) from the first, that

ro

r

ro

r

ro

=

r;

=

r

Ir

r ro,

=

ICz'Plr = ( Since

(xo -

Xl)

'P«(~

{[(,x] - (,,+1 - 10)"

=

+ _1_ 8'P«()/8(0 + 8'P{~)/8(1 ) Wo«()lr. n -1 ([(,x] - (,,+1 - aO),,-1

(:to + Gfa)lr = 1£, we have

(1.18) "z'Plr =

(

(xo - xJ) 1/1«() ([(,x] _ (,,+1 - iO)"

r-

1

+ n-

8t/J«()/8(1 ) 1 {[(,x1- (,,+1 _ iO),,-1 Wo{()lr·

For a section of the bundle r we take a cylinder of the form {e E R,,+1 I «(h ..• ,(,,) E 'Y}, where 'Y is an arbitrary surface in R" intersectiDg once every ray issuing from the origin. The restriction of Wo«() to this cylinder is w«() A d(,,+1, where w{() = E~=l(-l)i-l("\j~id(j.

1. INTEGRAL TRANSFORM RELATED TO HYPERPLANE SECTIONS

153

Integrating (1.18) over this section and applying the integration by parts in the second term, we obtain

f "ztp =

11'

2 (zo - ZI)

1foo 'l

",(eh ... , (nH)

-00

X ({i(Zl - Zo) + (2Z2

+ ... + (nZn + (nH -

Thus, it follows from the inversion formula (1.7) that

en

= 1"2~21r)")1 n-1 •

io)-n d{n+l "w({).

II' "zIP = en J(z), where

The inversion formul88 obtained by using the general formula (1.17) for various cycles r take into account not only the value on r of the function IP = 'RJ itself but, 88 a rule, also the values of some derivatives of tp which cannot be expressed in temis of the restriction tplr. The terms with derivatives can be expressed in terms of tplr for some special cycles r only. Just these manifolds r are admissible. Several examples of admissible manifolds can be found in the next subsection. REMARK. The above definitions and formul88 for the functions on c. n remain valid for functions J defined on the entire hyperboloid of two sheets Hn.

1.8. Examples. Consider the submanifold r = r tJ of sections by the hyperplanes passing through a point a with homogeneous coordinates (00, ai, ...• OnH)i if OnH = 0, then a is a point at infinity. We first obtain the inversion formula for an arbitrary point a and then specify the formula for various points a that belong or do not belong to the hyperboloid. Let = Cl be the preimage of rain n +2 \ 0, i.e.,

r r

a

n+l

r = ra = {{ E R ,,+2 \ 0 I L Giei = O}. i=O

r

The manifolds a corresponding to distinct points a can be obtained from one another by a continuous deformation; in particular. each of them can be continuously deformed to the submanifold considered in 1.5. Indeed, let 9h 0 :S t :S 1, be a continuous curve in GL+(n + 2,R) such that 90 = e (the identity element), and let 91 send a to b. Then tJ is transformed to by the continuous deformation tJ 9i 0 :S t :S 1, where 9i stands for the transform conjugate to 9t. Hence, the inversion formula (1.17) holds for each of these manifolds. We find the explicit form of the inversion formula. If a is not a point at infinity, i.e., if OnH ". 0, then one can set OnH = 1. We modify the expression for "zIP in (1.11) by adding the ith row (of the determinant) multiplied by 4i to the last row for all i, i ". n + 1. We obtain

ra,

r

"ztp I-

r

r"

( 1 + E~=o aixi) tp(e)

I' = ([{, z1 - (n+l -

iO)"

_1_ 00 IP~o ({) - E::l GiIP~1 (e) - IP~n+1 ({») (e)l+ n -1 ([{,x1- (n+l - io)n-l WnH 1"

where n

WnH({)

= (_I)n+l Eei i=O

A j;U.n+l

d{j.

154

5. INTEGRAL GEOMETRY ON QUADRlCS

(1.19)

=

If a is a point at infinity, i.e., anH = O. and if ao ".0, then one can set ao 1. Adding the ith row (ofthe detenninant (1.11» multiplied by aj to the first row for all i, i ". 0, we obtain

rpl_

K

z

r

=(

Thus, the inversion formula for (1.20)

+

(E:'-Oaix,)rp(e) ([e,x)- en+1 - io)n

ra

_1_ rp~o(e) - E~I ai"'t(e») Wo(e)ln - 1 ([e.x)- en+1 - io)n-I r'

becomes

f ( (E~=oOiXi)rp(~) +_1_ ~o(e) -

ir..

«e,x)- en+1 - lo)n

n-

E:'..I

a~~.(e»)lUo(e)=en/(x).

1 ([e,x)- en+1 - 10)n-1

Consider some examples. EXAMPLE 1. Let a = (0•...• O. l), i.e., r is the manifold of sections by the hyperplanes passing through the zero point, le, xl = O. Recall that r is the manifold of geodesic hypersurfaces on C.". In this case, the inversion formula (1.19) becomes

f (
1

orp(e)/oenH)

ir (le,x)-iO)" - n-I ([e.x)-io)n-I where IUnH (e) = (-1 )nH E:=o( -1)' ed\hh dej· (1.21)

IUn+l(e)=cn/(x).

Along with the function rplr. the formula contains the derivative 8:"~1 Ir transversal to r. To obtain an inversion formula depending on ."Ir only. we replace the integral transform of functions on n by an integral transform on the full hyperboloid Hn; this transform takes a function on Hn to its integrals over the hyperplane sections of Hn. Both the definition of the differential fonn Kz and the inversion formula remain unchanged in this case. Let I be a function on n • let h be the even extension of I to Hn. and let rpl = 'Rh. Since h is even. it foUaws that the function rpl is an even function with respect to en+ I. and hence ..!!:£L.lr = O. Therefore, the second term in the inversion 8(,,+1 formula (1.21) for the integral transform = 'Rh vanishes. Moreover, note that 'Pllr = 2rplr. Therefore, the inversion formula for the transform 'PI = 'Rh on H" implies the following inversion formula for the original integral transform rp = 'RI on n :

c.

c.

"'I

2

i

c.


= en I(x).

Note that this inversion formula coincides with (1.8). EXAMPLE 2. Let a = (1,0, ... ,0. 1); thus, r is the manifold of sections by the hyperplanes passing through the vertex of the upper sheet of the hypeorboloid. The inversion formula is (1.22)

f (x o + 1)~(e) + _1_ 8'P(e)fof.o -

ir

(Ie,xj-IO)"

n-l

8.",(f.)/8en+1 ) IUn+de) (le.xj-aO)n-1

= Cn I(x).

I.

INTEGRAL TRANSFORM RELATED TO HYPERPLANE SECTIONS

155

Hence. the function I can be reconstructed from the restriction'" of the function 'P to r only. tP(eo •... ,en) = 'P(eo •...• en, -eo). Indeed. since ('P~o - ip~"+1 )Ir = tPEo' it follows that the inversion formula (1.22) can be written as f + l)tP({) 1 lJtP({)/tJeO) lr (le.xl- io)n + n -1
(xo

Using the integration by parts in the second term. we obtain 2 (xo

+ 1)

t

'P(e)([{. xl - iO)-n wn+! ({)

= c,. I(x).

EXAMPLE 3. Suppose a = (1.0..... 0). i.e., r is the manifold of sections by the hyperplanes passing through a point at infinity that does not belong to the projectivization of the hyperboloid. The inversion formula becomes

f(

Xo 'P(e)

(1.23) lr ([e.x)- {n+1 - io)n

+n

1 IJv;(F,.)/tJ{o ) -1 ([e.xl- {n+1 _ io)n-I wo(e)

= c,. I(x).

Along with the function 'Plr, the formula contains the derivative of 'P transversal to

r.

To obtain an inversion formula expressing I in terms of 'Plr only, we pass to the integral transform of functions on the full hyperboloid Hn. as in Example 1. Extend the function I on t:. n to a function which is even with respect to Xo on the full hyperboloid Hn, and let 'PI = 'RI be the integral transform on HR of the function I thus extended. Since I is even with res~ to xo, it follows that 'PI is an even function with respect to eo, and hence ielr = O. Therefore, the term with the derivative ~ vanishes in the inversion formula for the integral transform 'PI = 'Rf. Since ~Ir = 2 Ifolr. it follows from the inversion formula for the integral transform 'PI = 'RI on Hn that the following inversion formula holds for the original integral transform

2xo

t

'P(e)(le.xl- en+1 - iO)-nWo(e) = c,. I(x).

Note that this formula was already obtained earlier; see (1.5). EXAMPLE 4. Suppose a = (1. 1,0, ... ,0). and r is the manifold of sections by the hyperplancs passing through a point at infinity on the projectivization of the hyperboloid. The inversion formula becomes (1.24)

1(

(xo

+ XI )'P(e~

r ([( • .1')- (n+1 - aO)n

+ _1_ tJ'P(~)/tJ~o - tJ'P(~)/tJel )Wo({) =c,./(x).

n - 1 ([(. xl - (n+l - aO)n-1 Similarly to Example 2. this inversion formula can be written as

2 (xo

+ XI)

t


= c,. I(x).

EXAMPLE 5. Let r be the manifold of horospheres. i.e., let the submanifold C Rn+2 ,Obe given by the equation [e, F.I = O. Note that the manifold r can be deformed to the manifold in Example 1 by the continuous mapping eo - 0 (1 - t) 0 ~ t ~ 1. Hence, the inversion formula (1.17) holds for the manifold r. Let us find an explicit form of this formula.

f

eo.

~.

156

INTEGRAL GEOMETRY ON QUADRlCS

To do this. we modify the determinant expressions for K. z '" I r. Subtract the = I, ... , n multiplied by (i/(o, respectively. from the first row of the determinant. Since le, e] = 0 and ((. de] = 0 on r. we obtain

rows with indices i

K. z

cpl r = ((ol[e,x] cp(e) ([e,xl- enH - io)-n +

n

~I Gl(Eei:~)(le.xl-en+l-io)-nH)1 WoU)· i=O'" r

Note that the homogeneity condition yields

Eei OIP = -enH --.!!:L - r,p = _0({n+1CP). i=O

Oei

Oen+l

Oen+l

Passing to the integral, we can integrate by parts with respect to second term. This gives

(n+l in the

Thus, the inversion formula related to the manifold of horospheres is

t

eo 1r,p(e) «(e, xl- enH

- iO)-nH wo«() = c..f(x).

In particular, we have proved that the manifold of horosphcres is admissible. The reader can verify that this formula is equivalent to the inversion formula in Chapter 3.

2. Integral tr8DSform related to spheres in Euclldean space ER We consider here another overdetermined integral transform 'R. that takes a function on n-dimensional Euclidean space ER to the integrals of this function over all possible (n - 1)-dimensional spheres in ER. This transform can be regarded as an integral transform taking a function on the n-dimensional sphere C ER+l to the integrals of this function over the hyperplane sections of the sphere 8 n • Namely. let x and x' be two antipodal points of the sphere S". Then. when projecting the sphere S" from the point x to the hyperplane ER C ER+l tangent to S" at the point x', the hyperplane sections of the sphere go to spheres on En. There is a simple relation between this integral transform and the transform treated in § 1. Namely, under the projective transformation

sn

(XO.x., ...• x n) ...... (l/xo,xIixo •... ,xn/XO) the hyperboloid of two sheets xg - xi - ... - x~ = 1 becomes the sphere xg + xi + 1 and the hyperplane sections of the hyperboloid become the hyperplane sections of the sphere. Due to this connection, one can obtain the main formulas for the integral transform on ER related to spheres in ER, by using the results of § I. Here we give an independent proof of these formulas.

... + ~ =

2. INTEGRAL TRA!I/SFORM RELATED TO SPHERES

157

2.1. Definition. Let H be the manifold of ('1 - 1)-dimensional spheres in Euclidean space en. It is assumed that H contains the hyperplanes, which must be viewed as spheres passing through a point at infinit)·. We present the spheres by equations of the form (2.1)

u(e,x)

~

"

j=1

j=1

== eo LXj2 + EeiXi + (RH = O.

e;* o.

Thus. the spheres are identified with points of the projective space pn+l that are given by the homogeneous coordinates ,(,,+1). Note that equation (2.1) defines an element of the manifold H if and onJy if

«o,e.....

..

E{i 2 ~ 4(0{n+l' i-I

Every homogeneous equation with respect to (i defines an n-dimensiooal submaoifold of the manifold of sphert'S. The simplest examples are

(1) ~ = O. the submanifold of all hyperplanes: (2) +l = 0, the submanifold of spheres passing through the point 0; (3) {, = 0 (i = 1, ... ,11), the submanifold of spheres centered at theooordinate hyperplane Xi = 0; (4) E~:ll = 4(oeR+lo the submanifold of spheres tangent to the hyperplane X tl = 0; (5) ,.2 E:=I~} = (,.2{O + {tI+l)2, the 8ubmanifold of spheres tangent to the spheres :q + ... + x! = (6) L~I (1 = 4(o(tI+l +ra~, the submanifold of spheres of a chosen radi\l8 r. Introduce the integral transform 'R. taking any compactly supported ~ funotion I on BY' to its integrals over the spheres in En by the formula

e..

er

r;

(2.2)

'RICe) = {

lEn

eo

J(x)cS(u(e,X»dxl A .. • A dx".

For ;* 0, this integral coincides with the integral with respect to Euclidean measure OD the sphere up to the factor (21~1 ,.)-1, where r is the radius of the sphere. The function 'P = 'RI thus defined is nonzero only in the domain ~"=l ~,2 ~ 4~en+l'

It immediately follows from the definition that the function t.he homogeneity condition ~;*

(2.3)

!p

= 'R. I

satisfies

0,

and the differential equation

/Pip

2.2. Operator

~ /Pip

Oip == 8(08( - L- 8f~ ft+1 0=1 ...,

(2.4) Kc.

= O.

We construct (repeating the arguments of § 1) the unhrer-

sal inversion formula that recovers a function I from the values of the function 'P 'R. I OD various 71-dimensional submanifolds of H. Let ;p«() be an arbitrary infinitely differentiable function on R"+2 \ 0 satisfying the homogeneity condition 'P(~) = I~I-J cp«() for ~;* O. To the function 'P and to

=

5, INTEGRAL GEOMETRY ON QUADRlCS

158

an arbitrary point x

e En we assign the following distributions on an+2 : ",(e)

1

+o({.x) = (2.5)

+,({ x) ••

2 (u(e.,r) -

=-

+ 2(n -

x.",({) __1_ 8.,,(£.)/o{. (u({.x)-io)n n-l (u(e.x)_io)n-l

1

+n+1({.x)

8'P({)/8en+l 1) (u({.x) - io)n-l;

1

io)n

=2

(E:=1 X;2) ep(e) (u(E.,,r) - io)n

1

(1 :::; i :::; n);

8.,,(e)/8eo io)n-l'

+ 2(n - 1) (u(E..x) -

Note that one can represent +i(e.X) in the form

(2.6)

+0 (E.. x)

= 2(n ~ 1) {(u({.x) -

+;(e.,r)

= -~1 {(u({.x) n-

+n+1(£',x)

;o)-n+1. "'({)}n+I' io)-n+l. 'P(£.)}.

= 2(n ~ 1) {(u({,x) -

(1:::;

i:::; n),

io)-n+1, 'P(O }o'

where the braces are the Poisson brackets, i.e.,

{'PI. '/J2};

O'P2

= 'PI 8ei

Oepl - 'P2 O{i .

en

To each point ,r e we assign the following operator Kz from the space of functions 'P(e) into the space of differential n-forms whose coefficients are distributions:

(2.7) here -t(~, x) is the column with the elements -ti (£., x) defined by (2.5). Expanding this determinant in the first column. we obtain the following representation of Kz'" in the form of a sum: n+l (2.8) Kz'P = -ti({, x) Wi(£').

L

where Wi({) are differential n-forms; for their explicit expressions, see § 1. As in § 1. the form Kz'P is defined on the manifold 5 n + 1 of rays in a n +2 issuing from the point O. THEOREM 2.1. The differential fann Kz'P is closed on Rn+2 for any point x e and any junction 'P({) on Rn+2 satisfying the homogeneity condition 'P(.\£.) = 1,\1- 1 ",(£,), ,\ f:. 0, and the differential equation (2.4).

en

PROOF. Since ou({,x)

= 0, it follows from the definition of the functions -t;

as Poisson brackets that

E~i =

(u(e,x) - io)-n+1o." - ",0 (u(£..x) - iO)-n+l

= O.

• =0 " ...

Therefore, the assertion of the theorem follows from the lemma in 1.3. COROLLARY. If a junction 'P belongs to the image then the differential form Kz'P is closed.

0/ the integral transform 'R,

169

2. I!,(TEORAL TRANSFORM RELATED TO SPHERES

2.3. Inversion formula. By analogy ",;th § 1. the following inversion formula holds: THEOREM

2.2.

If I{) = 'RI, then

i

(2.9)

"z'P

= c(r) f(x)

for any point x E ER and any n-dimensional cycle r cH, where c(f) depends only on the homology cltJ88 of the cycle r.

=

=

We prove (2.9) for the submanifold f fo ghren by the equation 6J 0, i.e., for the manifold of hyperpianes in En. Since the form K"ZI{) is closed, this will imply that the iuw.rsion formula (2.9) holds for any n-dimensional submanifold f C H obtained by a continuous deformation of ro. In particular, using a continuous deformation, one can transform fo to a manifold r(-j of spheres that is defined by an arbitrary linear relation E"~ Qi(i O. Hence, corresponding to each of these manifolds is an inversion formula (2.9), where c(r) = e(ro). Let 1LI({lt .•.• (n+.) be the restriction of the functicm 'P = 'RI to r,

=

"'({It ....

e..+1) = lEf f(x) 6«(IXI + ... + {AX" + e.. +l) tUl'\ ... A un.

Since t/J is the Radon transform of I, the following inversion formula holds: (2.10)

1[ ,.

~({l""

,en+d ({IXI

+ ... + enxn + {.. +l - io)-n ~n+l

fI. u(e)

= t'.... I(x),

-00

where t'", = in(~:~~)i and u{{) = E:=I(-l)l-lei/\"i~j. It remains to show that tbe integral fr KzIP coincides \\itb the IeIt-band side of (2.10). It follows from the expression for #\.;cC{) in the detenninant form that

#\.;cl{)lr = 4'o(e, x) Lo.'o({)lr =

~ ( tP(~) ({IXI + .'. + ~nXR + ~"+1 -

iO)-"

+ --.!.-} ~:(e) (~lXl + ... + ~nxn + {nH - 20)-.. +1) Wo«() n-

I.

r Passing to the integral over rand tLoring the integration by parts in the second integral, we see that

t t 11"'" K.z;p =

(2.11)

=

tJ;({)«(1.:r1

"'«(1." ..

."

u .... +1

+ ... +~..xra + (n+1 -

iO)-"Wo«()

{n+l)(~lXl + ... + enXn + {n+1 -

io)-n ~"+l A u«() .

-<:IQ

Comparing (2.10) and (2.11). we obtain (2.9).

2.4. Examples. Let us find tbt> explicit form of the inversion formulas for the simplest cycles r cH. EXAMPLE 1. Let r be the submanifold of spheres passing through the point o. Wc write the equation of r in the form en+! = O.

&. INTEGRAL GEOMETRY ON QUADRlCS

160

It follows from the expression for K. z4{J in the determinant form that

= +n+l(~,x)wn+l(~)lr = ~ (tx~)1P(~)(U(~.X) -

K.z4{J Ir

iO)-"

;=1

n

+ ~ 1 ~;) (u«(,x) -

io)-n+1) Wn+l(e)

Ir'

where 1P is the restriction of the function 4{J = 'RI to f. Passing to the integral over and using the integration by parts in the second term. we obtain

r

( K.z'{J=

lr

(tx~) i=l

('{J«()(u«(,x)-io)-n Wn+1«().

lr

Thus, the inversion formula related to f becomes

(~xn

(2.12)

£

'{J(e)(u(e,x) - io)-n wn +1(e)

In particular, this shows that the submanifold

r

= en I(x).

is admissible.

EXAMPLE 2. Let r be the submanifold of spheres centered at the hyperp1ane Xl = O. We write the equation of r in the form (I = O. It follows from the expression for K.z'{J in the determinant form that K. z

4{Jlr

= +l(e,X)Wl«()lr = (Xl '{J«()(u«(.x) -

io)-n

+n~1

a;i!)

I

(u(e,x) - io)-n+l) WI«() r .

Thus, the inversion formula related to f becomes

Xl

l '{J«()(u(~,x)

(2.13)

- iO)-n W1 (e)

+n~1

l a;g) (u(~,

x) - io)-n+1 wd~) =

en I(x).

According to this formula, to reconstruct a function I, one must know not only 4{Jlr but also ~Ir, i.e., the derivative in a direction transversal to f. The inversion formula can be simplified for functions I(x) satisfying some additional conditions. We consider two cases of this kind. Case 1: I(xt. ... ,Xn ) is an even function with respect to Xl. Then the function 4{J 'RI is even with respect to (t. and hence Gtlr = O. Thus, in this case the inversion formula becomes

=

Xl

£4{J(~)(u«(,x)

-

iO)-nWl(~) = en I(x).

=

=

Note that 4{J 'RI is an odd function of el, and hence :pI r 0 if I(Xl, . .. ,xn ) is an odd function of Xl. Thus, in this case the inversion formula becomes

l a;g)

(u«(, x) - io)-n+1 Wl «()

= (n -1)en/(x).

Case 2: the function I(xt. ... ,xn ) vanishes in the domain Xl < o. Let It(x) be the extension of the function I to the domain XI < 0, which is even with respect

181

2. INTEGRAL TRANSFORM RELATED TO SPHERES

to Xl. and let 'PI = R/t. Then 'Pllr = 2'Plr and lfalr = inversion formula to the function 'PI. we obtain 2Xl

£'P(~) (u(~.x)

-

o. Therefore, applying the

iO)-"Wl(~) = en/(x).

3. Let r be the submanifold of spheres tangent to the hyperplane we take L.'::ll ~~ = 4~O(n+l. We represent r 88 the union of the oriented submanifolds of spheres. r + and r _. formed by the spheres tangent to the hyperplane X" = 0 and belonging to the subspaces Xn > 0 and X" < O. respectively. Let us show that each of the submanifolds r + and r _ admits a continuous deformation (taking account of the orientation) to the manifold ro of all hyperplanes in E". This will yield EXAMPLE

X"

= O. For the equation of r

(2.14)

{ "z'P

Jr

=2

(

Jr.

"z'P

= 2c" I(x).

where the coefficient c" coincides with that in the inversion formula for ro. We prove this assertion, say, for r +. Using the stereographic projection, we can interpret the spheres and hyperplanes in E" 88 hyperplane sections of the n-dimensional sphere S" c E"+l. In this interpretation, corresponding to the hyperplane Xn = 0 is the (n - 1)-dimensional sphere (1 C S" passing through the pole p of the sphere S", and to the manifolds r :t, the manifolds of (n - l)-dimensional spheres in S" tangent to the sphere (1 and belonging to the hemispheres S;, respectively. into which the sphere (1 partitions the sphere S". We contract the sphere (1 to the point p in such a way that the hemisphere SO!. contracts to p and the hemisphere ~ becomes the entire sphere S". Then the manifold r + is deformed to the manifold of (n - 1)-dimensional spheres on S" passing through the point p, i.e., in the original interpretation. to the manifold ro of hyperplanes on E". We modify the expression for "z'Plr in the determinant form. Take the first row (i = 0), subtract from it the next (n - 1) rows (with indices i = 1, ... , n - 1), each multiplied by the respective factor ~d2{n+ 10 and add the last row multiplied by (O/~n+l·

Since E'::11 ~~ - 4(o~n+l

r. we see that

= 0 and E':::11 (, d(i -

Substituting here the expression for the functions Kz

cp

2{o d(n+l - 2{n+l d(o

~i«('X)

(see (2.5», we obtain

Ir = _1_ (u(~,x) -("x,,)cp«() + _1_ EiPn(i'P(.«(») 2{,,+l

(u«(,x)-iO)"

= 0 on

n-l(u«(,x)-iO),,-1

(C) WO ..

Ir·

It follows from the homogeneity condition for 'P that

L (, 8cp = -If' _ (n 1Jrp = _8«(n'P) . ,pn

8(.

Hence, "z Ir 'I'

= _1_ (u«(, x) 2{n+l

/)(n

8(n

I

(nxn)'P«() _ _1_ 8«(n'P)/8(n ) Wo«() (u«(,x) - io)n n - 1 (u«(,x) - io)n-l r·

162

S. INTEGRAL GEOMETRY ON QUADRlCS

When evaluating the integral with respect to this differential form over r, one can use the integration by parts in the second term. We finally obtain the following inversion formula:

( {;;.!1 (u({,x) -

ir

2{n xn)Y'«()

(u«(,x) _ io)n

(C) Wo..

=4

en

I() x.

In particular, we thus proved that the manifold of spheres tangent to the hyperplane Xn = 0 is admissible. EXAMPLE

4. Let

r

be the submanifold of spheres tangent to the sphere x~ +

... + x~ = 1. For the equation of r we take E:':.1 e1 = «(0 + (,,+1)2.

As in Example 3, one can readily see that the inversion formula (2.14) holds for the cycle r. We modify the expression for ItzY'lr in the determinant form. Take the first row of the determinant, subtract from it each of the next n rows multiplied by the corresponding factor of the form (eo + {,,+tl- 1 {i, i = 1, ... , n, and add the last row. Since «(0 + (,,+1)2 - E:':.1 = 0 on r, we see that

(1

n

ItzY'lr = (4-0«(, x) - «(0 + (n+d- 1

E ~1

(i

4-i «(, x) + 4-n+1 (e, x») wo(e)

Ir .

Substituting here the expressions for the functions 4- i «(,x) (see (2.5», we obtain

1 ( P({, x) Y'(e) "zY'lr= 2«(0 + (n+1) (u«(,x)-iO)"

1

+ n-l

I

Q(e) ) (u({,x)-iO),,-1 wo«() r'

where n

P«(,

n

x) = (eo + (n+1) (Ex~ + 1) + 2 EeiXi, i=1

i=1

Q(e) = «(0 + (n+1) (Y'(o + Y'("+I) + 2

"

E (i Y'(i' ~1

We transform the expression for Q«(). Passing from eo and e"+1 to the new variables '10 = {o + e,,+1 and '11 = eo - {n+1 and using the homogeneity property of the function Y', we obtain

Q(e) = 2

('1oY'~ +

t{i

Y'(.) = 2 (-Y' -

'IIY'~l) = -2 8<;:;;).

i=1

When evaluating the integral over r with respect to the differential form "zY', we can use the integration by parts in the second term. We finally obtain the following inversion formula:

( (eo + {,,+1)-1 ({o + E:-l {iXi + {,,+1 E~-1 xn Y'«() (C) = 4 I() ir (u«(,x) _ io)n Wo.. en x. In particular, we have proved that the manifold of spheres is also admissible. REMARK. In Examples 2 and 4, r is not a cycle, but the inversion formula still holds in these cases. One way to see it is to pass to sections of the sphere S", in which case r becomes a cycle.

2. INTEGRAL TRANSFORM RELATED TO SPHERES

163

REMARK. Using motions of the space El", one can transform the inversion f0rmulas in Examples 1-4 into inversion formulas for the manifolds of spheres passing through a given point, or centered at a given hyperplane, or t-angeot to a given hyperplane or sphere, respectively.

Bibliography [11 J. 8emstein and S. Gindikin, Integrnl pometly on rrtIIm/Dlb &/ ~ Lie RJOUPI!I and symmetric spaces, Preprint. 2003. [21 I. M. Gelraud &Dd S. G. Gindikin (<
98. (11( _ _ • Intqral geometry on SL(2;R). l\rlath. Rea. Lett. ,. (2000),411-432. (18) S. Gindikm (eel.). Applied pru6lenu 0/ Radon trafUl/orm. AlDer. Math. Soc. naasJ.. Series 2, .."01. 162. Amer. Math. Soc., Provid8nal. RI, 1994. (19) S. G. Gindlkin and G. M. Henldn, The Penro.e trrJnsform and complu iftkgral ~ Currt!DI. problema in mathematics, vol. 11, VINm. Moscow. 1980, pp. 57-112; Eaglish tnuuJl. in J. Matb. Sci. :U (1983), no. 4.

1611

166

BIBLIOGRAPHY

(20) S. G. GindikiD and N. D. Vvedeoslcaya, Ducme RtuIon trGna/cnm and image reanutruction, Mathematical Problems of Tomography, 1\-aos1. Math. Monographs, vol. 81, Amer. Math. Soc., Providence, RI, 1990, pp. 141-188. (21) A. B. Goocharov, Three-dimensional reconstruction 0/ orintnlrily afTUfl9ed identical particlu gillen their pro;edioM, Mathematical Problems of Tomography, Transl. Math. Monographs, vol. 81, Amer. Math. Soc., Providence, RI, 1990, pp. 67-95. [22) _ _ , InUgral geometry and mmai/olds 0/ minimal degree in cpn, Funktaional. Anal. i Prilozhen. 24 (1990), no. 1,5-20; Eoglisb trausl., Funct. Anal. Appl. 24 (1990), no. 1,4-17. (23) _ _ , Adrrriuible double bundlu, 75 Years of Radon Transform, S. Gindikin and P. Michor (eds.), International Press (Boston), 1994, pp. 129-152. (24) _ _ , InUgral geometry and'D-modulu, Matb. Res. Letters 2 (1995), 415-435. (25) I. S. Gradshtein and I. M. Ryzhik, Tablu 01 integraU, ""nil, 6enu, and prociuct6, Fizmatgiz, Mosc:ow, 1963; English transl., Academic Press, San Diego, CA, 2000. (26) S. Helguon, The RtuIon trGna/orm, Birkhiuser, Boston, MA, 1980. (27) - - , Geometric analyN on qmmetric 6J1GCU, Amer. Math. Soc., Providence, RI, 1994. (28) F. John, The ultn&hyper60lic diJlef'efltiIIJ equation with lour independent vanablu, Duke Math. J. 4 (1938), 300-322. (29) A. A. Kirillov, On a prohlem 011. M. Gelland, Dold. Akad. Nauk SSSR 131 (1961), DO. 2, 276-277; English trans1., Soviet Math. Dokl. 2 (1961), 268-269. (30) F. Natterer and F. WuebbeUng, MathemotiaJl methods in image reanutruchon, SIAM, Philadelphia, 2001. [31) J. Radon, Oiler die ButimmuR9 lIOn I'Unktionen durch Ihre Integralwerte 1Gng6 getAllNef" Mannigfoltigl:eiten, Ber. Verb. Konig1. Siichs. Geael1sh. Wisseoach. Leipzig, Math. Naturwi88. KI. 89 (1917), 262-277.

Index

Abet transform. 2 admissible tiDe complex, 73 d.cription, 76 geometric structure. 75 admillJlbJe submanilolds of hyperpIane aections in C..•• 147 Asgeirsaon relatiOllll for the hyperboW; plane, 85 for the hyperboIfc space. 101 for three-dimensional space. 9 on the EuclideaD plane, 3

homogeneity condition. 14 horoc:yde. 79 horocycle lunctioDs, 88

n.

_ai,88 horocycJe transrorm, 83 poop-theoretic meaning, 92 iJMnioD formula, 85, 86 horocycle _ - . 107 ~.97,

108,128,129

c:enter.98

oriented distaac;e., 98 parallel, 98 horospherical function. 103 on SL(2,C), 129

back from. 13 bMlctor,59 dual,64

ZODBl.I08 honxJpheric:aI transform, 98, 103 IIMII'8ion formula, 126

CavaIicri', c:ouditio1ls, 14

011

electron microsc:opy, 18

1:", 108

on 5L(2,R), 128 on (.3

forward front. of the waw, 13 Fourier integral on (.2. 88 on (.3,103 Fourier transform in the hyperbolic: apace, UN ilM!lSion I'onnula. 104 PlaDcherel formula, 105 relation t;o the horospheric:al trabsform,

inwrsicm I'onnula, 100, 101 horoapherical_-. 109 HuygeDI PriDcipIe. 1.3 ~ plane, 79,80 area element. 80, 81 geodesicII, 82 bOlOcyde, 79, 81 center,81 oriented dia&aDce, 82 pamllel, 81 iDVariant diataDce, 80, 81 IiDeS,82 motions, 79, 80

104 lepI_At_ion theory, 106 symmetry relation, 105 011 R2, 86 011 5(H), 49 011 the hyperbolic plane, 88 imer'8iOD fonnula, 89 Plaoc:berel formula, 91 relation to representMioII theory,92 relation to the horocycle t.raosi'onn, 89, 92 symmetry relation, 90 on the Schwanz apace. 49

Poinari model, 79 Riemannian metric. 80 hyperbolic space, 96 of an arbitrary climeDalon,

integral geometry for line complexes in 011 t.he torus, 21 167

un

ca. 71

168

INDEX

integral transform for spheres in E", 157 inversion formula, 159 operawr ", 157 of difl'erential forms, 53 related w completely geodesic surfaces in 1;3, 102 completely podeaic surfaces in 1;3: Inwrsion formula, 102 byperp1aDe aections of 1;", 146 lines on the hyperbolic plane, 93 related w geodesics on the hyperbolic plane, 93 on the hyperbolic plane: inversion formula, 96 on the hyperbolic p1aDe: relation w the projective Radon transform, 96 inwrsion formula for a complex of lines intersecting a curve in (:3, n for the Fourier transform in the hyperbolic space, 104 for the horocycle transform, §. 86 for the horospberical transform on 1;3, 100, 101 on 1;", 103 for the Radon transform ewn-dimensional case, 11 for the complex aftine space, jQ for the projectiw space, 32 for the projective space of arbitrary dimension, 38 in three-dimensional space, 8 locaI,11 DOnlocal, 11 odd-dimensional case, 11 of arbitrary dimension, 11 on the Euclidean plane, :I on the afBne plane, lJl locaI,38 DOnlocaI, 38 on C', 152 isotropic cone, 75

a.

Jolua transform, A 46. 68. 88 an intertwining operawr, 68 group-theoretic meaning, §§. ti6 image, ~ in C3, 6Z image, fiR inversion formula, 68 in p3, fiD. 63 the af&ne John transform, 61 image, fi3 in the afBne space, 4:i image, jfi inWl'llion formula, IHl

John transform of I-forms on R3, 58 image, 58 kernel,58 of 2-forms on R3, 56 image,52 of 3-forms on R3, M image,5Ii of differential forms, 53 operawr ", &Z complex case, 68 the differential form "'P. 4Z relation w GaUII hypergeometric function, 4:i Kircbholf formula, 13 Laplare-Beltrami operator on SL(2.C). 128 on 1;2, 88 on 1;3, 103 line complex, ZL 117 admissibility conditions, 73 in C3, 76 critical plane, m critical point, 76 Lorentz group (5 L(2. C», III manifold homogeneous coordinates, 115 left translation, 115 of horospheres of lines inp3,59 in R3, 4Z in the space, .a on the plane, 1 of one-ciimensionalsubspaces, 47. method of horocycle waves, 107 of borospheres, 93 of plane waves, 12 Minlwwski-FUnk transform for the three-dimensional sphere, M OD two-dimensional sphere, 22 inwrsion formula, :u opera&or re for the John transform, 47. analogs, 52 analogs in complex case, 69 complex case, fi8 in E", 157 on 1;", 149 inwnion formulas, 152 local and DOnlocal, 151 paraboloid, 129 Pliidcer coordinates, 59

169

INDEX

P1ancberel formula for the Fourier transform. 91 plane waves, 12 Poisson formula, 13 pullback, 53

pushdown.53 Radon transform "complex". 11 discrete, 19. 21 for the complex afIlne space, 38 of arbitrary dimension: Fourier transform. 39 of arbitrary dimension: inversion formula, 40 for the projective plane, ;ID. 36 image. 37 inversion formula, 36 for the project.ive space. ~ 32 iIIlqe. as in,'eI'IIion formula, 32 of an arbitrary dimension, 38 of an arbitrary dimension: Im-ersion formula, 38 the aIIine Radon transform, 32 the Minkowski-Funk transform, 32 in thraHiimensional affine space, 9 in tlm!oe-dimenslonal space, 7. Fourier tranaform, 9 inversion formula, 8 inversion formula local, ll..lj nonlocal,9, l j using the moments, lZ of I-forms on the plane, 25 on the plane: inversion, 2li of2-forms in tJtree.djmensional space, 28 in thretMlimeo9lonal space: inwnion formula, 28 on the plane, 2B. on the plane: iIIlqe, 2Z of3-forms in three-dimensional space, 29 in three-dimensional space: im-ersion, 30 of arbitrary dimension, 10

combined in\'el'llion formula, 11 inversion formula, 11 on the afline plane, fA inversion formu1a, 10 on the Euc1idean plane, 1 imago, IS. lfi inversion formula, a Paley-Wlener theorem, 15. lJi Poisson formula, 19

Radon tTaDSform relation to Fourier series, 21 the Fourier transform, li tomography problems. , WIn'!! equation on the plaDe, 12. 13 rapidly decreasing func:ticm. on a vector bundIe, 47 reconstruction of unknown directions, 1B right regular representation, 138 SL(2,C),111 as a bomogeneoua space, 111 as a hyperboloid. Ul Fourier transform, 131, 141 inversion formula, 134. 142 Planc:herel formula, 135, 142 representation theory, 137, 138, 143 symmetry relation, 133, 142 transform relatA!d to paraboloids, 141 borosphere. 128, 129 borospberlcal function, 129 expression, 130 on the manifold of triples, 130

zonal,l29 borospherical transform, 119, 120 inversion formula. 124 symmetry relation, 121 integral transform line complex usoc:iated with borospberes, 120 related to paraboloids, 126 related to paraboloids, and an integral transform on the hyperbolic space, 121 related to paraboloids: Inwrsion

formuia,l21 related to the line complex 888OC:i ated ~'ith borospheres: ilm!rBion fonnula, 123 Laplace-Behraml operator. 128 mani£old of horoepberes, 113 associated Hne complex, 117 embedding in projective space, 116 manifold of paraboloida, 118 orbits and sectioos, 112 paraboloid, 129 zonal borospberical function, 129 symmetry relation for the Fourier transform on SL(2,C). 133, 142 on the hyperbolic plane, 90 for the hyperbolic plane, 86 for the hyperbolic space, 101

twlMiimensional Fourier series,

:n

ultrahyperbollc dUferential equation, ""

INDEX

170

_ve equation for the hyperbolic: plaDe. 106 for the hyperbolic spaa!. 109 three-dimensional. 1.3 ~imeDllional.

12

X -ray transform. 43

zonal horocycle functions. 88 zonal boroepherical function. 103 on SL<2,C), 129

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