harmonic approximation on closed sets
Authors:
A. Bonilla and J. C. Fariña
Journal:
Proc. Amer. Math. Soc. 129 (2001), 27412752
MSC (2000):
Primary 31A05; Secondary 30E10
Published electronically:
February 9, 2001
MathSciNet review:
1838798
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Abstract: 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.
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 Arakeljan, N.: Uniform and tangential approximation by analytic functions, Transl. Amer. Math. Soc., 122 (1984), 8597.
 2.
 Bagby, T. and Gauthier, P. M.: Approximation by harmonic functions on closed subsets of Riemann surfaces, J. Analyse Math., 51 (1988), 259284. 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), 1526. MR 93g:31015
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 Bagby, T. and Gauthier, P. M.: Harmonic approximation on closed subsets of Riemannian manifolds, P. M. Gauthier (ed.), Complex Potential Theory, 7587 (1994), Kluwer Academic Publishers. MR 96k:31013
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 Boivin, A. and Paramonov, P.: Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions, Math. Sb., 189(4), (1998), 481502. MR 99i:41019
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 Bonilla, A and Fariña, J.C.: Meromorphic and Holomorphic Approximation in Norms, J. Math Anal. Appl. 181, (1994), 132149. MR 94m:30076
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 Bonilla, A and Fariña, J.C.: approximation and extension on closed sets, Canad. Math. Bull 38, (1995), 2333. MR 96e:30093
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 Bonilla, A and Fariña, J.C.: Elliptic fusion lemma, Math. Japon., 41 (1995), 441445. MR 96d:30047
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 Bonilla, A and Fariña, J.C.: Uniform approximation by solutions of elliptic equations with continuous extension to the boundary, Complex Variables, 28 (1995), 111120. MR 2000f:41028
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 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), 235247. MR 88b:35086
 11.
 Fariña, J. C.: Lip approximation on closed sets, J. Analyse Math., 57 (1991), 152171. MR 94e:30013
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 Gardiner, S. J.: Superharmonic extension and harmonic approximation, Ann. Inst. Fourier, 44 (1994), 6591. MR 95a:31006
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 Gardiner, S. J.: Harmonic approximation, Cambridge University Press, 1995. MR 96j:31001
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 Gardiner, S. J.: Harmonic approximation with continuous extension to the boundary, J. Analyse Math., 68 (1996), 95106. MR 98e:31004
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 Gardiner, S. J.: Decomposition of approximable harmonic functions, Math. Ann. 308 (1997), 175185. MR 98e:31005
 16.
 Goldstein, M. and Ow, W. H.: Uniform harmonic approximation with continuous extension to the boundary, Canad. J. Math., 40 (1988), 13751388. MR 90d:41033
 17.
 Goldstein, M. and Ow, W. H.: A characterization of harmonic Arakelyan sets, Proc. Amer. Math. Soc., 119 (1993), 811816. 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), 8798. 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), 703736. 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), 249259. MR 95i:41040
 22.
 Paramonov, P.: approximation by harmonic polynomial in compact sets in , Russian Acad. Sci. Sb. Math., 78 (1994), 231251.
 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), 104111. 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), 12431255. MR 80d:30037
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 Stray, A.: Decomposition of approximable functions, Annals of Math., 120 (1984), 225235. 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:
http://dx.doi.org/10.1090/S0002993901058683
PII:
S 00029939(01)058683
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
