harmonic approximation on closed sets

Authors:
A. Bonilla and J. C. Fariña

Journal:
Proc. Amer. Math. Soc. **129** (2001), 2741-2752

MSC (2000):
Primary 31A05; Secondary 30E10

DOI:
https://doi.org/10.1090/S0002-9939-01-05868-3

Published electronically:
February 9, 2001

MathSciNet review:
1838798

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

In this paper the harmonic approximation ( ) on relatively closed subsets of a domain in the complex plane is characterized under the same conditions given by S. Gardiner for the uniform case. Thus, the result of P. Paramonov on harmonic polynomial approximation for compact subsets is extended to closed sets. Moreover, the problem of uniform approximation with continuous extension to the boundary for harmonic functions and similar questions in harmonic approximation are also studied.

**1.**Arakeljan, N.:*Uniform and tangential approximation by analytic functions*, Transl. Amer. Math. Soc.,**122**(1984), 85-97.**2.**Bagby, T. and Gauthier, P. M.:*Approximation by harmonic functions on closed subsets of Riemann surfaces*, J. Analyse Math.,**51**(1988), 259-284. MR**89j:30064****3.**Bagby, T. and Gauthier, P. M.:*Uniform approximation by global harmonic functions*, in Approximation by solutions of partial differential equations, B. Fuglebe et al. (eds), Kluwer Academic Publishers (1992), 15-26. MR**93g:31015****4.**Bagby, T. and Gauthier, P. M.:*Harmonic approximation on closed subsets of Riemannian manifolds*, P. M. Gauthier (ed.), Complex Potential Theory, 75-87 (1994), Kluwer Academic Publishers. MR**96k:31013****5.**Boivin, A. and Paramonov, P.:*Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions*, Math. Sb.,**189**(4), (1998), 481-502. MR**99i:41019****6.**Bonilla, A and Fariña, J.C.:*Meromorphic and Holomorphic Approximation in**-Norms*, J. Math Anal. Appl.**181**, (1994), 132-149. MR**94m:30076****7.**Bonilla, A and Fariña, J.C.:*approximation and**extension on closed sets*, Canad. Math. Bull**38**, (1995), 23-33. MR**96e:30093****8.**Bonilla, A and Fariña, J.C.:*Elliptic fusion lemma*, Math. Japon.,**41**(1995), 441-445. MR**96d:30047****9.**Bonilla, A and Fariña, J.C.:*Uniform approximation by solutions of elliptic equations with continuous extension to the boundary*, Complex Variables,**28**(1995), 111-120. MR**2000f:41028****10.**Dufresnoy, A., Gauthier, P.M. and Ow, W.H.:*Uniform approximation on closed sets by solutions of elliptic partial differential equations*, Complex Variables,**6**(1986), 235-247. MR**88b:35086****11.**Fariña, J. C.:*Lip**approximation on closed sets*, J. Analyse Math.,**57**(1991), 152-171. MR**94e:30013****12.**Gardiner, S. J.:*Superharmonic extension and harmonic approximation*, Ann. Inst. Fourier,**44**(1994), 65-91. MR**95a:31006****13.**Gardiner, S. J.:*Harmonic approximation*, Cambridge University Press, 1995. MR**96j:31001****14.**Gardiner, S. J.:*Harmonic approximation with continuous extension to the boundary*, J. Analyse Math.,**68**(1996), 95-106. MR**98e:31004****15.**Gardiner, S. J.:*Decomposition of approximable harmonic functions*, Math. Ann.**308**(1997), 175-185. MR**98e:31005****16.**Goldstein, M. and Ow, W. H.:*Uniform harmonic approximation with continuous extension to the boundary*, Canad. J. Math.,**40**(1988), 1375-1388. MR**90d:41033****17.**Goldstein, M. and Ow, W. H.:*A characterization of harmonic Arakelyan sets*, Proc. Amer. Math. Soc.,**119**(1993), 811-816. MR**93m:31005****18.**Goluzin, G.M.,*Geometric theory of functions of a complex variable*, English transl. Amer. Math. Soc., Providence, R.I., 1969. MR**40:308****19.**Johnston, E. H.,*The boundary modulus of continuity of harmonic functions*, Pacific J. Math.**90**(1980), 87-98. MR**82a:31002****20.**Mateu, J. and Orobitg, J.:*Lipschitz approximation by harmonic functions and some applications to spectral sinthesis*, Indiana Univ. Math. J.,**39**(1990), 703-736. MR**92e:46052****21.**Paramonov, P. and Verdera, J.:*Approximation by solutions of elliptic equations on closed subsets of Euclidean space*, Math. Scand.,**74**(1994), 249-259. MR**95i:41040****22.**Paramonov, P.:*approximation by harmonic polynomial in compact sets in*, Russian Acad. Sci. Sb. Math.,**78**(1994), 231-251.**23.**Paramonov, P.:*Harmonic polynomial approximation on compact subsets of the plane*, Preprint Universitat Autonoma de Barcelona, 134, Setembre 1991.**24.**Roth, A.:*Uniform and tangential approximation by meromorphic functions on closed sets*, Canad. J. Math.,**28**(1976), 104-111. MR**57:9978****25.**Roth, A.:*Uniform approximation by meromorphic functions on closed sets with continuous extension into the boundary*, Canad. J. Math.,**30**(1978), 1243-1255. MR**80d:30037****26.**Stray, A.:*Decomposition of approximable functions*, Annals of Math.,**120**(1984), 225-235. MR**86b:30060**

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Additional Information

**A. Bonilla**

Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Canary Islands, Spain

Email:
abonilla@ull.es

**J. C. Fariña**

Affiliation:
Departamento de Análisis Matemático, Universidad de La Laguna, 38271 La Laguna (Tenerife), Canary Islands, Spain

Email:
jcfarina@ull.es

DOI:
https://doi.org/10.1090/S0002-9939-01-05868-3

Received by editor(s):
January 30, 2000

Published electronically:
February 9, 2001

Additional Notes:
This work was supported in part by Consejería de Educación, Gobierno Autónomo de Canarias, Proyecto PI 1999/105.

Communicated by:
J. Marshall Ash

Article copyright:
© Copyright 2001
American Mathematical Society