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A Bibliographic Survey of the Pompeiu Problem

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Approximation by Solutions of Partial Differential Equations

Part of the book series: NATO ASI Series ((ASIC,volume 365))

Abstract

A compact set K ⊂ ℝn is said to have the Pompeiu property (PP) if the only function \( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbfv3ySLgzGueE0jxyaibaieYlf9ir % Veeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqaq-Jfr % VkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGa % ciaacaqabeaadaqaaqaaaOqaaiabdAgaMbaa!3272! f \)∊C(ℝn) satisfying for all rigid motions σ of ℝnis\( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbfv3ySLgzGueE0jxyaibaieYlf9ir % Veeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqaq-Jfr % VkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGa % ciaacaqabeaadaqaaqaaaOqaaiabdAgaMbaa!3272! f \) ≡ 0. A collection K, of compact sets in ℝn has (PP) if whenever \( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbfv3ySLgzGueE0jxyaibaieYlf9ir % Veeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqaq-Jfr % VkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGa % ciaacaqabeaadaqaaqaaaOqaaiabdAgaMbaa!3272! f \) ∊ C(ℝn) and (1) holds for all K ∊ K (and all rigid motions σ) it follows that \( % MathType!MTEF!2!1!+- % feaagaart1ev2aaatCvAUfKttLearuavP1wzZbItLDhis9wBH5garm % Wu51MyVXgaruWqVvNCPvMCaebbfv3ySLgzGueE0jxyaibaieYlf9ir % Veeu0dXdh9vqqj-hEeeu0xXdbba9frFj0-OqFfea0dXdd9vqaq-Jfr % VkFHe9pgea0dXdar-Jb9hs0dXdbPYxe9vr0-vr0-vqpWqaaeaabiGa % ciaacaqabeaadaqaaqaaaOqaaiabdAgaMbaa!3272! f \) ≡ 0.

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Authors and Affiliations

  1. Department of Mathematics and Computer Science, Bar-Ilan University, Ramat-Gan, 52900, Israel

    L. Zalcman

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  1. L. Zalcman
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Editors and Affiliations

  1. Department of Mathematics, University of Copenhagen, Copenhagen, Denmark

    B. Fuglede

  2. Department of Mathematics, Arizona State University, Tempe, Arizona, USA

    M. Goldstein

  3. Department of Mathematics, University of Duisburg, Duisburg, Germany

    W. Haussmann  & L. Rogge  & 

  4. Department of Mathematics, University of York, Heslington, York, UK

    W. K. Hayman

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Zalcman, L. (1992). A Bibliographic Survey of the Pompeiu Problem. In: Fuglede, B., Goldstein, M., Haussmann, W., Hayman, W.K., Rogge, L. (eds) Approximation by Solutions of Partial Differential Equations. NATO ASI Series, vol 365. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-2436-2_17

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  • DOI: https://doi.org/10.1007/978-94-011-2436-2_17

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