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This book highlights new, previously unpublished results obtained in the last years in integral geometry and theory of convolution equations on bounded domains.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.

From the reviews: "The book under review reflects the modern state of the results and is mainly based on the results of the author. ... is written in a very clear manner and should be useful both for experts in the field and for postgraduate students. The wide list of citations includes more than 250 items." (Nikolai K. Karapetyants, Zentralblatt MATH, Vol. 1043 (18), 2004) "The book is devoted to the problem of injectivity for convolution operators of geometric nature. ... A survey of works in the area by other authors is presented as well. The monograph contains a collection of interesting and original results ... . The book will be of interest for specialists in analysis, in particular, in harmonic analysis, spectral theory, invariant function spaces and integral equations. It may serve as a source for further research in the area." (Mark Agranovsky, Mathematical Reviews, Issue 2005 e)

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."

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.

Preface. 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 Rn. 2.Averages over Balls on Hyperbolic Spaces. 3. Functions with Zero Integrals over Spherical Caps. 4. Comments and Open Problems. 3: Convolution Equation Domains in Rn. 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. Bibliography. Author Index. Subject Index. Basic Notations.