Integral geometry deals with the problem of determining functions by their integrals over given families of sets. These integrals de?ne the corresponding integraltransformandoneofthemainquestionsinintegralgeometryaskswhen this transform is injective. On the other hand, when we work with complex measures or forms, operators appear whose kernels are non-trivial but which describe important classes of functions. Most of the questions arising here relate, in one way or another, to the convolution equations. Some of the well known publications in this ?eld include the works by J. Radon, F. John, J. Delsarte, L. Zalcman, C. A. Berenstein, M. L. Agranovsky and recent monographs by L. H ̈ ormander and S. Helgason. Until recently research in this area was carried out mostly using the technique of the Fourier transform and corresponding methods of complex analysis. In recent years the present author has worked out an essentially di?erent methodology based on the description of various function spaces in terms of - pansions in special functions, which has enabled him to establish best possible results in several well known problems. Front Matter....Pages i-xii Sets and Mappings....Pages 1-4 Some Classes of Functions....Pages 5-11 Distributions....Pages 12-15 Some Special Functions....Pages 16-25 Some Results Related to Spherical Harmonics....Pages 26-36 Fourier Transform and Related Questions....Pages 37-45 Partial Differential Equations....Pages 46-48 Radon Transform Over Hyperplanes....Pages 49-54 Comments and Open Problems....Pages 55-56 Functions with Zero Averages Over Balls on Subsets of the Space R n ....Pages 57-99 Averages Over Balls on Hyperbolic Spaces....Pages 100-121 Functions with Zero Integrals Over Spherical Caps....Pages 122-136 Comments and Open Problems....Pages 137-142 One-Dimensional Case....Pages 143-168 General Solution of Convolution Equation in Domains with Spherical Symmetry....Pages 169-190 Behavior of Solutions of Convolution Equation at Infinity....Pages 191-200 Systems of Convolution Equations....Pages 201-211 Comments and Open Problems....Pages 212-213 Sets with the Pompeiu Property....Pages 214-225 Functions with Vanishing Integrals Over Parallelepipeds....Pages 226-249 Polyhedra with Local Pompeiu Property....Pages 250-270 Functions with Vanishing Integrals over Ellipsoids....Pages 271-302 Other Sets with Local Pompeiu Property....Pages 303-310 The ‘Three Squares’ Problem and Related Questions....Pages 311-319 Injectivity Sets of the Pompeiu Transform....Pages 320-333 Comments and Open Problems....Pages 334-338 Injectivity Sets for Spherical Radon Transform....Pages 339-358 Some Questions of Approximation Theory....Pages 359-365 Gap Theorems....Pages 366-377 Morera Type Theorems....Pages 378-389 Mean Value Characterization of Various Classes of Functions....Pages 390-407 Applications to Partial Differential Equations....Pages 408-415 Some Questions of Measure Theory....Pages 416-419 Functions with Zero Integrals in Problems of the Discrete Geometry....Pages 420-426 Comments and Open Problems....Pages 427-429 Back Matter....Pages 430-454 This book highlights new, previously unpublished results obtained in the last years in integral geometry and theory of convolution equations on bounded domains. All results included here are definitive and include for example the definitive version of the two-radii theorem, the solution of the support problem for ball mean values, the extreme variants of the Pompeiu problem, the definitive versions of uniqueness theorems for multiple trigonometric series with gaps.
