RADON TRANSFORM

By CHARLOTTE GARCIA

CONTENTS CHAPTER I THE RADON TRANSFORM ON RN §1 INTRODUCTION §2 THE RADON TRANSFORM THE SUPPORT THEOREM §3 THE INVERSION FORMULA §4 THE PLANCHEREL FORMULA §5 RADON TRANSFORM OF DISTRIBUTIONS §6 INTEGRATION OVER D-PLANES X-RAY TRANSFORMS §7 APPLICATIONS A PARTIAL DI?ERENTIAL EQUATIONS B X-RAY RECONSTRUCTION BIBLIOGRAPHICAL NOTES CHAPTER II A DUALITY IN INTEGRAL GEOMETRY §1 HOMOGENEOUS SPACES IN DUALITY §2 THE RADON TRANSFORM FOR THE DOUBLE FIBRATION §3 ORBITAL INTEGRALS §4 EXAMPLES OF RADON TRANSFORMS FOR HOMOGENEOUS SPACES IN DUALITY A THE FUNK TRANSFORM B THE X-RAY TRANSFORM IN H2 C THE HOROCYCLES IN H2 D THE POISSON INTEGRAL AS A RADON TRANSFORM E THE D-PLANE TRANSFORM F GRASSMANN MANIFOLDS G HALF-LINES IN A HALF-PLANE H THETA SERIES AND CUSP FORMS CHAPTER III THE RADON TRANSFORM ON TWO-POINT HOMOGENEOUS SPACES §1 SPACES OF CONSTANT CURVATURE INVERSION AND SUPPORT THEOREMS A THE HYPERBOLIC SPACE B THE SPHERES AND THE ELLIPTIC SPACES C THE SPHERICAL SLICE TRANSFORM §2 COMPACT TWO-POINT HOMOGENEOUS SPACES APPLICATIONS §3 NONCOMPACT TWO-POINT HOMOGENEOUS SPACES §4 THE X-RAY TRANSFORM ON A SYMMETRIC SPACE CHAPTER IV ORBITAL INTEGRALS §1 ISOTROPIC SPACES A THE RIEMANNIAN CASE B THE GENERAL PSEUDO-RIEMANNIAN CASE C THE LORENTZIAN CASE §2 ORBITAL INTEGRALS §3 GENERALIZED RIESZ POTENTIALS §4 DETERMINATION OF A FUNCTION FROM ITS INTEGRAL OVER LORENTZIAN SPHERES §5 ORBITAL INTEGRALS AND HUYGENS’ PRINCIPLE CHAPTER V FOURIER TRANSFORMS AND DISTRIBUTIONS A RAPID COURSE §1 THE TOPOLOGY OF THE SPACES D(RN), E(RN) AND S(RN) §2 DISTRIBUTIONS §3 THE FOURIER TRANSFORM §4 DI?ERENTIAL OPERATORS WITH CONSTANT COE?CIENTS §5 RIESZ POTENTIALS