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Global approximation in harmonic spaces


Authors: Stephen J. Gardiner, Myron Goldstein and Kohur GowriSankaran
Journal: Proc. Amer. Math. Soc. 122 (1994), 213-221
MSC: Primary 41A30; Secondary 31D05
DOI: https://doi.org/10.1090/S0002-9939-1994-1203986-8
MathSciNet review: 1203986
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Abstract: This paper characterizes, in terms of thinness, compact sets K in a suitable harmonic space $ \Omega $ which have the following property: functions which are harmonic (resp. continuous and superharmonic) on a neighbourhood of K can be uniformly approximated on K by functions which are harmonic (resp. continuous and superharmonic) on $ \Omega $. The corresponding problems of approximating functions which are continuous on K and harmonic (resp. superharmonic) on the interior $ \ring{K}$ are also solved.


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  • [BG] T. Bagby and P. M. Gauthier, Uniform approximation by global harmonic functions, NATO Adv. Study Inst. Ser. C: Math. Phys. Sci., vol. 365, Kluwer Acad., Dordrecht, 1992, pp. 15-26. MR 1168705 (93g:31015)
  • [BH] J. Bliedtner and W. Hansen, Simplicial cones in potential theory II (approximation theorems), Invent. Math. 46 (1978), 255-275. MR 0492345 (58:11473)
  • [Bre] M. Brelot, Lectures on potential theory, Tata Institute, Bombay, 1960. MR 0118980 (22:9749)
  • [CC] C. Constantinescu and A. Cornea, Potential theory on harmonic spaces, Springer, Berlin, 1972. MR 0419799 (54:7817)
  • [Con] J. B. Conway, Functions of one complex variable, second edition, Springer, New York, 1978. MR 503901 (80c:30003)
  • [Gar] S. J. Gardiner, Superharmonic extension and harmonic approximation, Ann. Inst. Fourier (Grenoble) (to appear). MR 1262880 (95a:31006)
  • [Han] W. Hansen, Harmonic and superharmonic functions on compact sets, Illinois J. Math. 29 (1985), 103-107. MR 769760 (86b:31019)
  • [He1] R. M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415-571. MR 0139756 (25:3186)
  • [He2] -, Quelques propriétés des fonctions surharmoniques associées à une équation uniformément elliptique de la forme $ Lu = - \sum\nolimits_i {(\partial /\partial {x_i})(} \sum\nolimits_j {{a_{ij}}\partial u/\partial {x_j})} = 0$, Ann. Inst. Fourier (Grenoble) 15 (1965), 215-224.
  • [Mal] B. Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1956), 271-355. MR 0086990 (19:280a)
  • [Mer] S. N. Mergelyan, On the representation of functions by series of polynomials on closed sets, Dokl. Akad. Nauk. SSR 78 (1951), 405-408; English transl. in Amer. Math. Soc. Transl. Ser. 1, vol. 85, Amer. Math. Soc., Providence, RI, 1953.
  • [Pra] A. de la Pradelle, Approximation et caractère de quasi-analyticité dans la théorie axiomatique des fonctions harmoniques, Ann. Inst. Fourier (Grenoble) 17 (1967), 383-399. MR 0227456 (37:3040)
  • [Rud] W. Rudin, Real and complex analysis, third edition, McGraw-Hill, New York, 1987. MR 924157 (88k:00002)
  • [Run] C. Runge, Zur Theorie der eindeutigen analytischen Funktionen, Acta Math. 6 (1885), 228-244. MR 1554664

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DOI: https://doi.org/10.1090/S0002-9939-1994-1203986-8
Article copyright: © Copyright 1994 American Mathematical Society

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