1 Distributions and Fourier Transform.- 1.1 Introduction.- 1.2 Distributions and generalized functions.- 1.3 Tempered distributions.- 1.4 Homogeneous distributions.- 1.5 Manifolds and differential forms.- 1.6 Push down and pull back.- 1.7 More on the Fourier transform.- 1.8 Bandlimited functions and interpolation.- 2 Radon Transform.- 2.1 Properties.- 2.2 Inversion formulae.- 2.3 Alternative formulae.- 2.4 Range conditions.- 2.5 Frequency analysis.- 2.6 Radon transform of differential forms.- 3 The Funk Transform.- 3.1 Factorable mappings.- 3.2 Spaces of constant curvature.- 3.3 Inversion of the Funk transform.- 3.4 Radon's inversion via Funk's inversion.- 3.5 Unified form.- 3.6 Funk-Radon transform and wave fronts.- 3.7 Integral transform of boundary discontinuities.- 3.8 Nonlinear artifacts.- 3.9 Pizetti formula for arbitrary signature.- 4 Reconstruction from Line Integrals.- 4.1 Pencils of lines and John's equation.- 4.2 Sources at infinity.- 4.3 Reduction to the Radon transform.- 4.4 Rays tangent to a surface.- 4.5 Sources on a proper curve.- 4.6 Reconstruction from plane integrals of sources.- 4.7 Line integrals of differential forms.- 4.8 Exponential ray transform.- 4.9 Attenuated ray transform.- 4.10 Inversion formulae.- 4.11 Range conditions.- 5 Flat Integral Transform.- 5.1 Reconstruction problem.- 5.2 Odd-dimensional subspaces.- 5.3 Even dimension.- 5.4 Range of the flat transform.- 5.5 Duality for the Funk transform.- 5.6 Duality in Euclidean space.- 6 Incomplete Data Problems.- 6.1 Completeness condition.- 6.2 Radon transform of Gabor functions.- 6.3 Reconstruction from limited angle data.- 6.4 Exterior problem.- 6.5 The parametrix method.- 7 Spherical Transform and Inversion.- 7.1 Problems.- 7.2 Arc integrals in the plane.- 7.3 Hemispherical integrals in space.- 7.4 Incomplete data.- 7.5 Spheres centred on a sphere.- 7.6 Spheres tangent to a manifold.- 7.7 Characteristic Cauchy problem.- 7.8 Fundamental solution for the adjoint operator.- 8 Algebraic Integral Transform.- 8.1 Problems.- 8.2 Special cases.- 8.3 Multiplicative differential equations.- 8.4 Funk transform of Leray forms.- 8.5 Differential equations for hypersurface integrals.- 8.6 Howard's equations.- 8.7 Range of differential operators.- 8.8 Decreasing solutions of Maxwell's system.- 8.9 Symmetric differential forms.- 9 Notes.- Notes to Chapter 1.- Notes to Chapter 2.- Notes to Chapter 3.- Notes to Chapter 4.- Notes to Chapter 5.- Notes to Chapter 6.- Notes to Chapter 7.- Notes to Chapter 8.

This book covers facts and methods for the reconstruction of a function in a real affine or projective space from data of integrals, particularly over lines, planes, and spheres. Recent results stress explicit analytic methods. Coverage includes the relations between algebraic integral geometry and partial differential equations. The first half of the book includes the ray, the spherical mean transforms in the plane or in 3-space, and inversion from incomplete data.