Comments and Open Problems

V. V. Volchkov
Additional contact information
V. V. Volchkov: Donetsk National University, Department of Mathematics

Chapter Chapter 8 in Integral Geometry and Convolution Equations, 2003, pp 334-338 from Springer

Abstract: 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].

Keywords: Trigonometric Polynomial; Circular Sector; Zero Function; Modern Study; Convolution Equation (search for similar items in EconPapers)
Date: 2003
References: Add references at CitEc
Citations:

There are no downloads for this item, see the EconPapers FAQ for hints about obtaining it.

Related works:
This item may be available elsewhere in EconPapers: Search for items with the same title.

Export reference: BibTeX RIS (EndNote, ProCite, RefMan) HTML/Text

Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-94-010-0023-9_26

Ordering information: This item can be ordered from
http://www.springer.com/9789401000239

DOI: 10.1007/978-94-010-0023-9_26

Access Statistics for this chapter

More chapters in Springer Books from Springer
Bibliographic data for series maintained by Sonal Shukla () and Springer Nature Abstracting and Indexing ().