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Offbeat Integral Geometry on Symmetric Spaces

  • Book
  • © 2013

Overview

Authors:
  1. Valery V. Volchkov

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

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  2. Vitaly V. Volchkov

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

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    You can also search for this author in PubMed   Google Scholar

  • Covers over twenty years of extensive research on local aspects of integral geometry on symmetric spaces and the Heisenberg group in one condensed text

  • Highlights significant and previously unpublished results

  • Includes numerous new problems

  • Includes supplementary material: sn.pub/extras

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Table of contents (14 chapters)

  1. Front Matter

    Pages i-x
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  2. Analysis on Symmetric Spaces

    1. Front Matter

      Pages 1-2
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    2. Preliminaries

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 3-42

    3. The Euclidean Case

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 43-83

    4. Symmetric Spaces of the Non-compact Type

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 85-110

    5. Analogies for Compact Two-point Homogeneous Spaces

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 111-134

    6. The Phase Space Associated to the Heisenberg Group

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 135-156

  3. Offbeat Integral Geometry

    1. Front Matter

      Pages 157-158
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    2. Functions with Zero Ball Means on Euclidean Space

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 159-228

    3. Two-radii Theorems in Symmetric Spaces

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 229-264

    4. The Problem of Findinga Function from Its Ball Means

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 265-307

    5. Sets with the Pompeiu Property

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 309-357

    6. Functions with Zero Integrals over Polytopes

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 359-391

    7. Ellipsoidal Means

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 393-433

    8. The Pompeiu Property on a Sphere

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 435-496

    9. The Pompeiu Transform on Symmetric Spaces and Groups

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 497-526

    10. Pompeiu Transforms on Manifolds

      • Valery V. Volchkov, Vitaly V. Volchkov
      Pages 527-555

  4. Back Matter

    Pages 557-592
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Keywords

About this book

The book demonstrates the development of integral geometry on domains of homogeneous spaces since 1990. It covers a wide range of topics, including analysis on multidimensional Euclidean domains and Riemannian symmetric spaces of arbitrary ranks as well as recent work on phase space and the Heisenberg group. The book includes many significant recent results, some of them hitherto unpublished, among which can be pointed out uniqueness theorems for various classes of functions, far-reaching generalizations of the two-radii problem, the modern versions of the Pompeiu problem, and explicit reconstruction formulae in problems of integral geometry. These results are intriguing and useful in various fields of contemporary mathematics. The proofs given are “minimal” in the sense that they involve only those concepts and facts which are indispensable for the essence of the subject. Each chapter provides a historical perspective on the results presented and includes many interesting open problems. Readers will find this book relevant to harmonic analysis on homogeneous spaces, invariant spaces theory, integral transforms on symmetric spaces and the Heisenberg group, integral equations, special functions, and transmutation operators theory.

Reviews

From the reviews:

“The book presents a broad survey of the results in an interesting area, where the harmonic analysis and differential and integral equations interact in a fruitful way. The authors themselves have contributed a lot to the subject. The book will be interesting to those working in the border areas between analysis, integral geometry and Lie groups representation theory.” (Mark Agranovsky, Mathematical Reviews, March, 2014)

Authors and Affiliations

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

    Valery V. Volchkov, Vitaly V. Volchkov

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