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Comments and Open Problems

In: Integral Geometry and Convolution Equations

Author

Listed:
  • V. V. Volchkov

    (Donetsk National University, Department of Mathematics)

Abstract

As has been already mentioned, D. Pompeiu was the first to consider equation (1.1). Pompeiu asserted [P2], [P4] that the disc possesses the Pompeiu property and even published an erroneous proof [P3]. (The error occurs on p. 268, formula (5)). The error was perpetuated by M. Nicolesco [N3], [N4], who sought to establish generalizations of Pompeiu’s result. Chakalov [C3] seems to have been the first to note explicitly that discs do not have the Pompeiu property. For the case of a square Pompeiu proved that the only solution of (1.1) tending to a limit at infinity is the zero function; for a simpler proof of a much more general result, see [C3]. Christov [C8], [C9] showed that Pompeiu’s requirement that f tend to a limit could be dropped and subsequently settled the corresponding problem for parallelograms [C10]. Somewhat earlier, Ilieff had dealt with the case of circular sectors [I1] and triangles [I2], [I3]. With the publication of [Z1] and [B42], almost twenty years later, the modern study of the Pompeiu problem may be said to have begun in earnest. For further background, see [Z3].

Suggested Citation

  • V. V. Volchkov, 2003. "Comments and Open Problems," Springer Books, in: Integral Geometry and Convolution Equations, chapter 0, pages 334-338, Springer.
    • Handle: RePEc:spr:sprchp:978-94-010-0023-9_26
    DOI: 10.1007/978-94-010-0023-9_26
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