Integral Geometry and Convolution Equations

1. Front Matter

   Pages i-xii

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2. Preliminaries

   1. Sets and Mappings

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      Pages 1-4

   2. Some Classes of Functions

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      Pages 5-11

   3. Distributions

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      Pages 12-15

   4. Some Special Functions

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      Pages 16-25

   5. Some Results Related to Spherical Harmonics

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      Pages 26-36

   6. Fourier Transform and Related Questions

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      Pages 37-45

   7. Partial Differential Equations

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      Pages 46-48

   8. Radon Transform Over Hyperplanes

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      Pages 49-54

   9. Comments and Open Problems

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      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

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      Pages 57-99

   2. Averages Over Balls on Hyperbolic Spaces

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      Pages 100-121

   3. Functions with Zero Integrals Over Spherical Caps

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      Pages 122-136

   4. Comments and Open Problems

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      Pages 137-142

4. Convolution equation on domains in ℝ n

   1. One-Dimensional Case

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      Pages 143-168

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

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      Pages 169-190

   3. Behavior of Solutions of Convolution Equation at Infinity

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      Pages 191-200

   4. Systems of Convolution Equations

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      Pages 201-211

   5. Comments and Open Problems

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      Pages 212-213

5. Extremal versions of the Pompeiu problem

   1. Sets with the Pompeiu Property

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      Pages 214-225

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.

• Fourier transform
• convolution
• distribution
• harmonic analysis
• measure theory
• partial differential equation
• partial differential equations

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)

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

  V. V. Volchkov

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