The Universality of the Radon Transform: (Oxford Mathematical Monographs)

Written by a leading scholar in mathematics, this monograph discusses the Radon transform, a field that has wide ranging applications to X-ray technology, partial differential equations, nuclear magnetic resonance scanning, and tomography. In this book, Ehrenpreis focuses on recent research and highlights the strong relationship between high-level pure mathematics and applications of the Radon transform to areas such as medical imaging. The first part of the book discusses parametric and nonparametric Radon transforms, Harmonic Functions and Radon transform on Algebraic Varieties, nonlinear Radon and Fourier transforms, Radon transform on groups, and Radon transform as the interrelation of geometry and analysis. The later parts discuss the extension of solutions of differential equations, Periods of Eisenstein and Poincaré, and some problems of integral geometry arising in tomography. Examples and proofs are provided throughout the book to aid the reader's understanding. This is the latest title in the Oxford Mathematical Monographs, which includes texts and monographs covering many topics of current research interest in pure and applied mathematics. Other titles include: Carbone and Semmes: A graphic apology for symmetry and implicitness; Higson and Roe: Analytic K-Homology; Iwaniec and Martin: Geometric Function Theory and Nonlinear Analysis; Lyons and Qian: System Control and Rough Paths. Also new in paperback Johnson and Lapidus: The Feynman Integral and Feynman's Operational Calculus; Donaldson and Kronheimer: The geometry of four-manifolds.

Table of Contents:
PrefaceLeon Ehrenpreis: Chapters I-XLeon Ehrenpreis: I: Introduction I.1 Functions, Geometry and Spaces I.2 Parametric Radon transform I.3 Geometry of the nonparametric Radon transform I.4 Parametrization problems I.5 Differential equations I.6 Lie groups I.7 Fourier transform on varieties: The projection slice theorem and the Poisson summation Formula I.8 Tensor products and direct integrals II: The nonparametric Radon transform II.1 Radon transform and Fourier transform II.2 Tensor products and their topology II.3 Support conditions III: Harmonic functions in Rn III.1 Algebraic theory III.2 Analytic theory III.3 Fourier series expansions on spheres III.4 Fourier expansions on hyperbolas III.5 Deformation theory IV: Harmonic functions and Radon transform on algebraic varieties IV.1 Algebraic theory and finite Cauchy problem IV.2 The compact Watergate problem IV.3 The noncompact Watergate problem V: The nonlinear Radon and Fourier transforms V.1 Nonlinear Radon transform V.2 Nonconvex support and regularity V.3 Wave front set V.4 Microglobal analysis VI: The parametric Radon transform VI.1 The John and invariance equations VI.2 Characterization by John equations VI.3 Non-Fourier analysis approach VI.4 Some other parametric linear Radon transforms VII: Radon transform on groups VII.1 Affine and projection methods VII.2 The nilpotent (horocyclic) Radon transform on G/K VIII: Radon transform as the interrelation of geometry and analysis VIII.1 Integral geometry and differential equations VIII.2 The Poisson summation formula and exotic intertwining VIII.3 The Euler-MacLaurin summation formula IX: Extension of solutions of differential equations IX.1 Formulation of the problem IX.2 Hartogs-Lewy extension IX.3 Wave front sets and the Caucy problem X: Periods of Eisenstein and Poincaré series X.1 The Lorentz group, Minowski geometry and a nonlinear projection-slice theorem X.2 Spreads and cylindrical coordinates in Minowski geometry X.3 Eisenstein series and their periods X.4 Poincaréseries and their periods X.5 Hyperbolic Eisenstein and Poincaré series X.6 The four dimensional representation X.7 Higher dimensional groups Bibiliography of Chapters I-X XI: Peter Kuchment and Eric Todd Quinto: Some problems of integral geometry arising in tomography XI.1 Introduction XI.2 X-ray tomography XI.3 Attenuated and exponential Radon transforms XI.4 Hyperbolic integral geometry and electrical impedance tomography Index