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

Published electronically:
February 9, 2001

MathSciNet review:
1838798

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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.**Thomas Bagby and P. M. Gauthier,*Approximation by harmonic functions on closed subsets of Riemann surfaces*, J. Analyse Math.**51**(1988), 259–284. MR**963157**, 10.1007/BF02791126**3.**T. Bagby and P. M. Gauthier,*Uniform approximation by global harmonic functions*, 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. 15–26. MR**1168705****4.**Thomas Bagby and Paul M. Gauthier,*Harmonic approximation on closed subsets of Riemannian manifolds*, Complex potential theory (Montreal, PQ, 1993) NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci., vol. 439, Kluwer Acad. Publ., Dordrecht, 1994, pp. 75–87. MR**1332959****5.**A. Boven and P. V. Paramonov,*Approximation by meromorphic and entire solutions of elliptic equations in Banach spaces of distributions*, Mat. Sb.**189**(1998), no. 4, 3–24 (Russian, with Russian summary); English transl., Sb. Math.**189**(1998), no. 3-4, 481–502. MR**1632406**, 10.1070/SM1998v189n04ABEH000303**6.**A. Bonilla and J. C. Fariña,*Meromorphic and holomorphic approximation in 𝐶^{𝑚}-norms*, J. Math. Anal. Appl.**181**(1994), no. 1, 132–149. MR**1257958**, 10.1006/jmaa.1994.1009**7.**A. Bonilla and J. C. Fariña,*𝐿𝑖𝑝𝛼 approximation on closed sets with 𝑙𝑖𝑝𝛼 extension*, Canad. Math. Bull.**38**(1995), no. 1, 23–33. MR**1319897**, 10.4153/CMB-1995-004-3**8.**A. Bonilla and J. C. Fariña,*Elliptic fusion lemma*, Math. Japon.**41**(1995), no. 2, 441–445. MR**1326977****9.**A. Bonilla and J. C. Fariña,*Uniform approximation by solutions of elliptic equations with continuous extension to the boundary*, Complex Variables Theory Appl.**28**(1995), no. 2, 111–120. MR**1700076****10.**A. Dufresnoy, P. M. Gauthier, and W. H. Ow,*Uniform approximation on closed sets by solutions of elliptic partial differential equations*, Complex Variables Theory Appl.**6**(1986), no. 2-4, 235–247. MR**871732****11.**J. Carlos Fariña,*Lipschitz approximation on closed sets*, J. Anal. Math.**57**(1991), 152–171. MR**1191745**, 10.1007/BF03041068**12.**Stephen J. Gardiner,*Superharmonic extension and harmonic approximation*, Ann. Inst. Fourier (Grenoble)**44**(1994), no. 1, 65–91 (English, with English and French summaries). MR**1262880****13.**Stephen J. Gardiner,*Harmonic approximation*, London Mathematical Society Lecture Note Series, vol. 221, Cambridge University Press, Cambridge, 1995. MR**1342298****14.**Stephen J. Gardiner,*Uniform harmonic approximation with continuous extension to the boundary*, J. Anal. Math.**68**(1996), 95–106. MR**1403252**, 10.1007/BF02790205**15.**Stephen J. Gardiner,*Decomposition of approximable harmonic functions*, Math. Ann.**308**(1997), no. 1, 175–185. MR**1446206**, 10.1007/s002080050071**16.**M. Goldstein and W. H. Ow,*Uniform harmonic approximation with continuous extension to the boundary*, Canad. J. Math.**40**(1988), no. 6, 1375–1388. MR**990103**, 10.4153/CJM-1988-061-3**17.**M. Goldstein and W. H. Ow,*A characterization of harmonic Arakelyan sets*, Proc. Amer. Math. Soc.**119**(1993), no. 3, 811–816. MR**1149971**, 10.1090/S0002-9939-1993-1149971-5**18.**G. M. Goluzin,*Geometric theory of functions of a complex variable*, Translations of Mathematical Monographs, Vol. 26, American Mathematical Society, Providence, R.I., 1969. MR**0247039****19.**Elgin H. Johnston,*The boundary modulus of continuity of harmonic functions*, Pacific J. Math.**90**(1980), no. 1, 87–98. MR**599322****20.**Joan Mateu and Joan Orobitg,*Lipschitz approximation by harmonic functions and some applications to spectral synthesis*, Indiana Univ. Math. J.**39**(1990), no. 3, 703–736. MR**1078735**, 10.1512/iumj.1990.39.39035**21.**Peter V. Paramonov and Joan Verdera,*Approximation by solutions of elliptic equations on closed subsets of Euclidean space*, Math. Scand.**74**(1994), no. 2, 249–259. MR**1298365****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.**Alice Roth,*Uniform and tangential approximations by meromorphic functions on closed sets*, Canad. J. Math.**28**(1976), no. 1, 104–111. MR**0470220****25.**Alice Roth,*Uniform approximation by meromorphic functions on closed sets with continuous extension into the boundary*, Canad. J. Math.**30**(1978), no. 6, 1243–1255. MR**511560**, 10.4153/CJM-1978-103-4**26.**Arne Stray,*Decomposition of approximable functions*, Ann. of Math. (2)**120**(1984), no. 2, 225–235. MR**763906**, 10.2307/2006941

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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