Global approximation in harmonic spaces
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- by Stephen J. Gardiner, Myron Goldstein and Kohur GowriSankaran PDF
- Proc. Amer. Math. Soc. 122 (1994), 213-221 Request permission
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 $\mathring {K}$ are also solved.References
- 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
- J. Bliedtner and W. Hansen, Simplicial cones in potential theory. II. Approximation theorems, Invent. Math. 46 (1978), no. 3, 255–275. MR 492345, DOI 10.1007/BF01390278
- M. Brelot, Lectures on potential theory, Lectures on Mathematics, vol. 19, Tata Institute of Fundamental Research, Bombay, 1960. Notes by K. N. Gowrisankaran and M. K. Venkatesha Murthy. MR 0118980
- Corneliu Constantinescu and Aurel Cornea, Potential theory on harmonic spaces, Die Grundlehren der mathematischen Wissenschaften, Band 158, Springer-Verlag, New York-Heidelberg, 1972. With a preface by H. Bauer. MR 0419799
- John B. Conway, Functions of one complex variable, 2nd ed., Graduate Texts in Mathematics, vol. 11, Springer-Verlag, New York-Berlin, 1978. MR 503901
- 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
- W. Hansen, Harmonic and superharmonic functions on compact sets, Illinois J. Math. 29 (1985), no. 1, 103–107. MR 769760
- R.-M. Hervé, Recherches axiomatiques sur la théorie des fonctions surharmoniques et du potentiel, Ann. Inst. Fourier (Grenoble) 12 (1962), 415–571 (French). MR 139756 —, 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.
- Bernard Malgrange, Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution, Ann. Inst. Fourier (Grenoble) 6 (1955/56), 271–355 (French). MR 86990 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.
- Arnaud de la Pradelle, Approximation et caractère de quasi-analyticité dans la théorie axiomatique des fonctions harmoniques, Ann. Inst. Fourier (Grenoble) 17 (1967), no. fasc. 1, 383–399 (French). MR 227456
- Walter Rudin, Real and complex analysis, 3rd ed., McGraw-Hill Book Co., New York, 1987. MR 924157
- C. Runge, Zur Theorie der Eindeutigen Analytischen Functionen, Acta Math. 6 (1885), no. 1, 229–244 (German). MR 1554664, DOI 10.1007/BF02400416
Additional Information
- © Copyright 1994 American Mathematical Society
- 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