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Integral Geometry and Convolution Equations

  • Book
  • © 2003

Overview

Authors:
  1. V. V. Volchkov

    1. Department of Mathematics, Donetsk National University, Donetsk, Ukraine

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Table of contents (35 chapters)

  1. Front Matter

    Pages i-xii
    Download chapter PDF

  2. Preliminaries

    1. Sets and Mappings

      • V. V. Volchkov
      Pages 1-4

    2. Some Classes of Functions

      • V. V. Volchkov
      Pages 5-11

    3. Distributions

      • V. V. Volchkov
      Pages 12-15

    4. Some Special Functions

      • V. V. Volchkov
      Pages 16-25

    5. Some Results Related to Spherical Harmonics

      • V. V. Volchkov
      Pages 26-36

    6. Fourier Transform and Related Questions

      • V. V. Volchkov
      Pages 37-45

    7. Partial Differential Equations

      • V. V. Volchkov
      Pages 46-48

    8. Radon Transform Over Hyperplanes

      • V. V. Volchkov
      Pages 49-54

    9. Comments and Open Problems

      • V. V. Volchkov
      Pages 55-56

  3. Functions with zero integrals over balls of a fixed radius

    1. Functions with Zero Averages Over Balls on Subsets of the Space ℝn

      • V. V. Volchkov
      Pages 57-99

    2. Averages Over Balls on Hyperbolic Spaces

      • V. V. Volchkov
      Pages 100-121

    3. Functions with Zero Integrals Over Spherical Caps

      • V. V. Volchkov
      Pages 122-136

    4. Comments and Open Problems

      • V. V. Volchkov
      Pages 137-142

  4. Convolution equation on domains in ℝ n

    1. One-Dimensional Case

      • V. V. Volchkov
      Pages 143-168

    2. General Solution of Convolution Equation in Domains with Spherical Symmetry

      • V. V. Volchkov
      Pages 169-190

    3. Behavior of Solutions of Convolution Equation at Infinity

      • V. V. Volchkov
      Pages 191-200

    4. Systems of Convolution Equations

      • V. V. Volchkov
      Pages 201-211

    5. Comments and Open Problems

      • V. V. Volchkov
      Pages 212-213

  5. Extremal versions of the Pompeiu problem

    1. Sets with the Pompeiu Property

      • V. V. Volchkov
      Pages 214-225
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Keywords

About this book

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.

Reviews

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)

Authors and Affiliations

  • Department of Mathematics, Donetsk National University, Donetsk, Ukraine

    V. V. Volchkov

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