Univalent functions and the Pompeiu problem

Authors:
Nicola Garofalo and Fausto Segàla

Journal:
Trans. Amer. Math. Soc. **346** (1994), 137-146

MSC:
Primary 30E15; Secondary 35N05

DOI:
https://doi.org/10.1090/S0002-9947-1994-1250819-4

MathSciNet review:
1250819

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we prove a result on the Pompeiu problem. If the Schwarz function of the boundary of a simply-connected domain extends meromorphically into a certain portion of with a pole at some point , then has the Pompeiu property unless is a Möbius transformation, in which case is a disk.

**[B]**Carlos Alberto Berenstein,*An inverse spectral theorem and its relation to the Pompeiu problem*, J. Analyse Math.**37**(1980), 128–144. MR**583635**, https://doi.org/10.1007/BF02797683**[BK]**Leon Brown and Jean-Pierre Kahane,*A note on the Pompeiu problem for convex domains*, Math. Ann.**259**(1982), no. 1, 107–110. MR**656655**, https://doi.org/10.1007/BF01456832**[BST]**Leon Brown, Bertram M. Schreiber, and B. Alan Taylor,*Spectral synthesis and the Pompeiu problem*, Ann. Inst. Fourier (Grenoble)**23**(1973), no. 3, 125–154 (English, with French summary). MR**0352492****[C]**L. Tchakaloff,*Sur un problème de D. Pompéiu*, Annuaire [Godišnik] Univ. Sofia. Fac. Phys.-Math. Livre 1.**40**(1944), 1–14 (Bulgarian, with French summary). MR**0031980****[E1]**Peter Ebenfelt,*Some results on the Pompeiu problem*, Ann. Acad. Sci. Fenn. Ser. A I Math.**18**(1993), no. 2, 323–341. MR**1234737****[E2]**-,*Propagation of singularities from singular and infinite points in certain complex-analytic Cauchy problems and an application to the Pompeiu problem*, preprint, 1993.**[GS1]**Nicola Garofalo and Fausto Segàla,*Asymptotic expansions for a class of Fourier integrals and applications to the Pompeiu problem*, J. Analyse Math.**56**(1991), 1–28. MR**1243097**, https://doi.org/10.1007/BF02820458**[GS2]**Nicola Garofalo and Fausto Segàla,*New results on the Pompeiu problem*, Trans. Amer. Math. Soc.**325**(1991), no. 1, 273–286. MR**994165**, https://doi.org/10.1090/S0002-9947-1991-0994165-X**[GS3]**-,*Another step toward the solution of the Pompeiu in the plane*, Comm. Partial Differential Equations (to appear).**[L]**N. N. Lebedev,*Special functions and their applications*, Dover Publications, Inc., New York, 1972. Revised edition, translated from the Russian and edited by Richard A. Silverman; Unabridged and corrected republication. MR**0350075****[S]**Glenn Schober,*Univalent functions—selected topics*, Lecture Notes in Mathematics, Vol. 478, Springer-Verlag, Berlin-New York, 1975. MR**0507770****[W]**G. N. Watson,*A treatise on the theory of Bessel functions*, Cambridge Mathematical Library, Cambridge University Press, Cambridge, 1995. Reprint of the second (1944) edition. MR**1349110****[Z]**L. Zalcman,*A bibliographic survey of the Pompeiu problem*, Approximation by solutions of partial differential equations (Hanstholm, 1991) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 365, Kluwer Acad. Publ., Dordrecht, 1992, pp. 185–194. MR**1168719**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC:
30E15,
35N05

Retrieve articles in all journals with MSC: 30E15, 35N05

Additional Information

DOI:
https://doi.org/10.1090/S0002-9947-1994-1250819-4

Article copyright:
© Copyright 1994
American Mathematical Society