Integral Geometry and Convolution Equations

This book highlights new, previously unpublished results obtained in the last years in integral geometry and theory of convolution equations on bounded domains.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. In order to make this book as self-contained as possible, we have gathered all prerequisites needed in the first part. In addition, each part of the book ends with comments in which not only other investigations are documented but also open problems dealing with a broader perspective are posed. A great number of applications to various branches of mathematics are also considered, for example, applications to the theory of approximations, discrete geometry, harmonic analysis, measure-preserving transformations, harmonic functions. Some of the material in this book has been the subject of lectures delivered by the author for advanced students, doctors and professors of mathematical faculty in various universities and so this book should be of interest to the graduate students and researchers in this area.