Selected Topics in Integral Geometry

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1 Translations of MATHEMATICAL MONOGRAPHS Volume 220 Selected Topics in Integral Geometry I. M. Gelfand S. G. Gindikin M. I. Graev American Mathematical Society 'I Providence, Rhode Island

2 Contents Preface to the English Edition Preface Chapter 1. Radon Transform 1 1. Radon transform on the plane Radon transform on the Euclidean plane Inversion formula Remarks Radon transform on the affine plane Relation to the Fourier transform and another proof of the inversion formula 5 2. Radon transform in three-dimensional space Radon transform in Euclidean space Radon transform in the affine space Radon transform for a space of arbitrary dimension Wave equation and the Huygens principle Two-dimensional case Three-dimensional case Cavalieri's conditions and Paley-Wiener theorems for the Radon transform Cavalieri's conditions for rapidly decreasing functions Paley-Wiener theorem for the space 5(M 2 ) Paley-Wiener theorem for the space X>(M 2 ) of compactly supported infinitely differentiable functions Inversion of the Radon transform of a function / 2?(M 2 ) using the moments Reconstruction of unknown directions from known values of 72./ Poisson formula for the Radon transform, and the discrete Radon transform Poisson formula for the Radon transform on a plane Discrete Radon transform; relation to the Fourier series Problem of integral geometry on the torus Minkowski-Funk transform Radon transform of differential forms Radon transform of 1-forms on the plane Radon transform of 2-forms on the plane Radon transform of 2-forms in three-dimensional space Radon transform of 3-forms in three-dimensional space Radon transform for the projective plane and projective space 30 xi xiii

3 vi 8.1. Spaces P 3 and (P 3 )' Radon transform for P Relation to the affine Radon transform for M 3 and to the Minkowski-Funk transform for the three-dimensional sphere Inversion formula for the Radon transform on P On the inversion formulas for the affine Radon transform on R 3 and the Minkowski-Funk transform for S Description of the image of the Radon transform for P Radon transform for the projective plane P Radon transform for the projective space of an arbitrary dimension Radon transform on the complex affine space Definition of the Radon transform Relation to the Fourier transform Inversion formula for the Radon transform Case n = Relation to Paley-Wiener theorems for the affine Radon transform in M 2 and R 3 40 Chapter 2. John Transform John transform in the real affine space John transform in R John transform and the Gauss hypergeometric function Theorem on the image of the operator J Space S(H') Description of the image of (]R 3 ) in the space S(H') Proof of Theorem 1.1 on the image of the John transform Analogs of the operator K John transform of differential forms on R Definition of the John transform of differential forms John transform of 3-forms on R John transform of 2-forms on R John transform of 1-forms on R John transform in the three-dimensional real projective space Manifold of lines in P John transform in P Relation to the John transform in the affine space Description of the image of the John transform Another way to define the John transform Proof of the theorem on the image of the John transform John transform as an intertwining operator John transform in the complex affine space John transform in C Differential form Kip and the theorem on the image of the John transform Inversion formula Analogs of the operator K Problems of integral geometry for line complexes in C Problem of integral geometry for a complex of lines in C 3 intersecting a curve 71

4 vii 5.2. Definition of admissible line complexes in C Necessary and sufficient conditions for a complex K to be admissible Geometric structure of admissible complexes Description of admissible complexes 76 Chapter 3. Integral Geometry and Harmonic Analysis on the Hyperbolic Plane and in the Hyperbolic Space Elements of hyperbolic planimetry Models of the hyperbolic plane Horocycles Geodesies Horocycle transform Definition of the operator U h Inversion formula Asgeirsson relations Symmetry relation Inversion formula for the horocycle transform in another model of the hyperbolic plane Analog of the Fourier transform on the hyperbolic plane and the relation between this analog and the horocycle transform Fourier transform on R Fourier transform on the hyperbolic plane Relation to the horocycle transform and the inversion formula Symmetry relation Plancherel formula Relation to the representation theory of the group SX(2,R) Integral transform related to lines (geodesies) on the hyperbolic plane Definition and the inversion formula in the Poincare model Relation to the Radon transform on the projective plane Horospherical transform in the three-dimensional hyperbolic space Models of the hyperbolic space Horospheres Horospherical transform Inversion formula Symmetry relation Inversion formula for the horospherical transform in another model of the hyperbolic space Integral transform related to completely geodesic surfaces in Analog of the Fourier transform in the hyperbolic space, and its relation to the horospherical transform Definition of the Fourier transform Inversion formula Symmetry relation and the Plancherel formula Relation to the representation theory for the group SL(2,C) Wave equation for the hyperbolic plane and hyperbolic space, and the Huygens principle Two-dimensional case 106

5 viii 9.2. Three-dimensional case 108 Chapter 4. Integral Geometry and Harmonic Analysis on the Group G = SL(2, C) Geometry on the group G Group G as a homogeneous space Plane sections of the hyperboloid G Manifold of horospheres Embedding the manifold of horospheres H in the projective space Line complex in C 3 associated with the manifold of horospheres Manifold of paraboloids Integral geometry on the group G = SL(2, C) Integral transforms related to the space H of horospheres and the complex of lines K Symmetry relations for the horospherical transform Inversion formula for the integral transform TJ-o related to the line complex K in C Inversion formula for the horospherical transform Inversion formula for the horospherical transform on the hyperbolic space C? Integral transform related to paraboloids on G Harmonic analysis on the group G = SL(2, C) Laplace-Beltrami operator on the group G Horospherical functions on G Fourier transform on G Relation between the Fourier transform on G and the horospherical transform Symmetry relation for the Fourier transform Inversion formula for the Fourier transform Analog of the Plancherel formula Relation between the Fourier transform on G and the representations of the group GxG Relation to the representations of the group G Another version of the Fourier transform on G = SL(2, C) Functions $ x (g;,c) Fourier transform on G Relation between the above two versions of the Fourier transform Symmetry relation Inversion formula and Plancherel formula for the Fourier transform T Relations with representation theory 143 Chapter 5. Integral Geometry on Quadrics Integral transform related to the hyperplane sections of a hyperboloid of two sheets in M. n Definition Admissible submanifolds in the manifold of hyperplane sections of C n Operator K X Local and nonlocal operators K 151

6 ix 1.5. Inversion formula Examples Integral transform related to spheres in Euclidean space E n Definition Operator K X Inversion formula Examples 159 Bibliography 165 Index 167

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