Approximation by harmonic functions and the Dirichlet problem.

*(English)*Zbl 0759.41033
Approximation by solutions of partial differential equations, Proc. NATO Adv. Res. Workshop, Hanstholm/Den. 1991, NATO ASI Ser., Ser. C 365, 155-168 (1992).

Summary: [For the entire collection see Zbl 0744.00033.]

The purpose of this paper is to give a survey of several results on harmonic approximation in classical as well as abstract potential theory. Section 1 recalls basic notions of the theory of harmonic spaces representing a good framework for the subject under consideration. In Section 2 the question of approximation of harmonic functions by single layer potentials is investigated. Section 3 deals with pervasive function spaces and harmonic approximation. Finally, Section 4 is devoted to approximation of Perron-Wiener-Brelot solutions by solutions of the classical Dirichlet problem.

The purpose of this paper is to give a survey of several results on harmonic approximation in classical as well as abstract potential theory. Section 1 recalls basic notions of the theory of harmonic spaces representing a good framework for the subject under consideration. In Section 2 the question of approximation of harmonic functions by single layer potentials is investigated. Section 3 deals with pervasive function spaces and harmonic approximation. Finally, Section 4 is devoted to approximation of Perron-Wiener-Brelot solutions by solutions of the classical Dirichlet problem.

##### MSC:

41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |

31A05 | Harmonic, subharmonic, superharmonic functions in two dimensions |

##### Citations:

Zbl 0744.00033
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\textit{I. Netuka}, in: Approximation by solutions of partial differential equations. Proceedings of the NATO advanced research workshop on approximation by solutions of partial differential equations, quadrature formulae, and related topics, held in Hanstholm, Denmark, July 8-12, 1991. Dordrecht etc.: Kluwer Academic Publishers. 155--168 (1992; Zbl 0759.41033)

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