Ebook: Integral Geometry and Convolution Equations

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.

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This book highlights new, previously unpublished results obtained in the last years in integral geometry and theory of convolution equations on bounded domains.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

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This book highlights new, previously unpublished results obtained in the last years in integral geometry and theory of convolution equations on bounded domains.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.
Content:
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
Averages Over Balls on Hyperbolic Spaces....Pages 57-99
Functions with Zero Integrals Over Spherical Caps....Pages 100-121
Comments and Open Problems....Pages 122-136
One-Dimensional Case....Pages 137-142
General Solution of Convolution Equation in Domains with Spherical Symmetry....Pages 143-168
Behavior of Solutions of Convolution Equation at Infinity....Pages 169-190
Systems of Convolution Equations....Pages 191-200
Comments and Open Problems....Pages 201-211
Sets with the Pompeiu Property....Pages 212-213
Functions with Vanishing Integrals Over Parallelepipeds....Pages 214-225
Polyhedra with Local Pompeiu Property....Pages 226-249
Functions with Vanishing Integrals over Ellipsoids....Pages 250-270
Other Sets with Local Pompeiu Property....Pages 271-302
The ‘Three Squares’ Problem and Related Questions....Pages 303-310
Injectivity Sets of the Pompeiu Transform....Pages 311-319
Comments and Open Problems....Pages 320-333
Injectivity Sets for Spherical Radon Transform....Pages 334-338
Some Questions of Approximation Theory....Pages 339-358
Gap Theorems....Pages 359-365
Morera Type Theorems....Pages 366-377
Mean Value Characterization of Various Classes of Functions....Pages 378-389
Applications to Partial Differential Equations....Pages 390-407
Some Questions of Measure Theory....Pages 408-415
Functions with Zero Integrals in Problems of the Discrete Geometry....Pages 416-419
Comments and Open Problems....Pages 420-426
Back Matter....Pages 427-429
....Pages 430-454