Integral Geometry and Convolution Equations

1. Preliminaries.- 1 Sets and Mappings.- 2 Some Classes of Functions.- 3 Distributions.- 4 Some Special Functions.- 5 Some Results Related to Spherical Harmonics.- 6 Fourier Transform and Related Questions.- 7 Partial Differential Equations.- 8 Radon Transform Over Hyperplanes.- 9 Comments And Open Problems.- 2. Functions with zero integrals over balls of a fixed radius.- 1 Functions With Zero Averages Over Balls on Subsets of The Space ?n.- 2 Averages Over Balls on Hyperbolic Spaces.- 3 Functions With Zero Integrals Over Spherical Caps.- 4 Comments And Open Problems.- 3. Convolution equation on domains in ?n.- 1 One-Dimensional Case.- 2 General Solution Of Convolution Equation In Domains With Spherical Symmetry.- 3 Behavior of Solutions of Convolution Equation at Infinity.- 4 Systems Of Convolution Equations.- 5 Comments And Open Problems.- 4. Extremal versions of the Pompeiu problem.- 1 Sets With The Pompeiu Property.- 2 Functions With Vanishing Integrals Over Parallelepipeds.- 3 Polyhedra With Local Pompeiu Property.- 4 Functions with Vanishing Integrals over Ellipsoids.- 5 Other Sets With Local Pompeiu Property.- 6 The ‘Three Squares’ Problem And Related Questions.- 7 Injectivity Sets of the Pompeiu Transform.- 8 Comments and Open Problems.- 5. First applications and related questions.- 1 Injectivity Sets For Spherical Radon Transform.- 2 Some Questions Of Approximation Theory.- 3 Gap Theorems.- 4 Morera Type Theorems.- 5 Mean Value Characterization of Various Classes of Functions.- 6 Applications To Partial Differential Equations.- 7 Some Questions Of Measure Theory.- 8 Functions With Zero Integrals In Problems Of The Discrete Geometry.- 9 Comments And Open Problems.- Author Index.- Basic Notations.