Mergelyan pairs for harmonic functions

Author:
Stephen J. Gardiner

Journal:
Proc. Amer. Math. Soc. **126** (1998), 2699-2703

MSC (1991):
Primary 31B05; Secondary 41A28

DOI:
https://doi.org/10.1090/S0002-9939-98-04334-2

MathSciNet review:
1451804

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Abstract | References | Similar Articles | Additional Information

Abstract: Let $\Omega \subseteq \mathbb R^n$ be open and $E\subseteq \Omega$ be a bounded set which is closed relative to $\Omega$. We characterize those pairs $(\Omega ,E)$ such that, for each harmonic function $h$ on $\Omega$ which is uniformly continuous on $E$, there is a sequence of harmonic polynomials which converges to $h$ uniformly on $E$. As an immediate corollary we obtain a characterization of Mergelyan pairs for harmonic functions.

- A. Bonilla, F. Pérez-González and R. Trujillo-González,
*Mergelyan sets for certain classes of harmonic functions*, Complex Variables**31**(1996), 9–18;*Correction*, to appear. - Leon Brown and Allen L. Shields,
*Approximation by analytic functions uniformly continuous on a set*, Duke Math. J.**42**(1975), 71–81. MR**367216** - Chen Huaihui and P. M. Gauthier,
*A maximum principle for subharmonic and plurisubharmonic functions*, Canad. Math. Bull.**35**(1992), no. 1, 34–39. MR**1157461**, DOI https://doi.org/10.4153/CMB-1992-005-3 - A. Debiard and B. Gaveau,
*Potentiel fin et algèbres de fonctions analytiques. I*, J. Functional Analysis**16**(1974), 289–304 (French). MR**0380426**, DOI https://doi.org/10.1016/0022-1236%2874%2990075-5 - J. Deny,
*Un théorème sur les ensembles effilés*, Ann. Univ. Grenoble Sect. Sci. Math. Phys.**23**(1948), 139–142. - J. L. Doob,
*Classical potential theory and its probabilistic counterpart*, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 262, Springer-Verlag, New York, 1984. MR**731258** - Bent Fuglede,
*Finely harmonic functions*, Lecture Notes in Mathematics, Vol. 289, Springer-Verlag, Berlin-New York, 1972. MR**0450590** - Stephen J. Gardiner,
*Harmonic approximation*, London Mathematical Society Lecture Note Series, vol. 221, Cambridge University Press, Cambridge, 1995. MR**1342298** - S. J. Gardiner,
*Uniform harmonic approximation with continuous extension to the boundary*, J. Analyse Math.**68**(1996), 95–106. - S. J. Gardiner,
*Decomposition of approximable harmonic functions*, Math. Ann.**308**(1997), 175–185. - Lester L. Helms,
*Introduction to potential theory*, Robert E. Krieger Publishing Co., Huntington, N.Y., 1975. Reprint of the 1969 edition; Pure and Applied Mathematics, Vol. XXII. MR**0460666** - A. Stray,
*Characterization of Mergelyan sets*, Proc. Amer. Math. Soc.**44**(1974), 347–352. MR**361097**, DOI https://doi.org/10.1090/S0002-9939-1974-0361097-5

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Additional Information

**Stephen J. Gardiner**

Affiliation:
Department of Mathematics, University College Dublin, Dublin 4, Ireland

MR Author ID:
71385

ORCID:
0000-0002-4207-8370

Email:
stephen.gardiner@ucd.ie

Received by editor(s):
October 21, 1996

Received by editor(s) in revised form:
February 3, 1997

Communicated by:
Albert Baernstein II

Article copyright:
© Copyright 1998
American Mathematical Society