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Integral Geometry and Convolution Equations

  • V. V. Volchkov

Table of contents

  1. Front Matter
    Pages i-xii
  2. Preliminaries

    1. V. V. Volchkov
      Pages 1-4
    2. V. V. Volchkov
      Pages 5-11
    3. V. V. Volchkov
      Pages 12-15
    4. V. V. Volchkov
      Pages 16-25
    5. V. V. Volchkov
      Pages 26-36
    6. V. V. Volchkov
      Pages 37-45
    7. V. V. Volchkov
      Pages 46-48
    8. V. V. Volchkov
      Pages 49-54
    9. V. V. Volchkov
      Pages 55-56
  3. Functions with zero integrals over balls of a fixed radius

  4. Convolution equation on domains in ℝ n

    1. V. V. Volchkov
      Pages 143-168
    2. V. V. Volchkov
      Pages 201-211
    3. V. V. Volchkov
      Pages 212-213
  5. Extremal versions of the Pompeiu problem

    1. V. V. Volchkov
      Pages 214-225
    2. V. V. Volchkov
      Pages 250-270
    3. V. V. Volchkov
      Pages 303-310
    4. V. V. Volchkov
      Pages 320-333
    5. V. V. Volchkov
      Pages 334-338
  6. First applications and related questions

    1. V. V. Volchkov
      Pages 339-358
    2. V. V. Volchkov
      Pages 359-365
    3. V. V. Volchkov
      Pages 366-377
    4. V. V. Volchkov
      Pages 378-389
    5. V. V. Volchkov
      Pages 408-415
    6. V. V. Volchkov
      Pages 416-419
    7. V. V. Volchkov
      Pages 427-429
  7. Back Matter
    Pages 430-454

About this book

Introduction

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.

Keywords

Fourier transform convolution distribution harmonic analysis measure theory partial differential equation

Authors and affiliations

  • V. V. Volchkov
    • 1
  1. 1.Department of MathematicsDonetsk National UniversityDonetskUkraine

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