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Abstract

Let X be a compact set in the complex plane C. Let H(X) be the uniform closure on X of those real continuous functions each harmonic in a neighborhood of X. Let D(X) be the space of those real continuous functions on X harmonic on the interior of X. In 1941 M. V. Keldysh [4] showed that if , the interior of X, was connected and X is the closure of , then H(X) = D(X) if each regular point of dX was a point of stability for the Dirichlet problem on X. Much earlier Wiener [6] has proved that a boundary point of X was a regular point for the Dirichlet problem if and only if
$$ \sum\limits_{n = 1}^\infty {\frac{n}{{\log \frac{1}{{\gamma ({A_n}(x) - X^\circ )}}}} = \infty } $$
(1)
where A n (x) = {z:l/2n+1 ≦ |z-x|≧l/2n} and y denotes the logarithmic capacity. In [4] Keldysh observed that x would be a point of stability for the Dirichlet problem if and only if
$$ \sum\limits_{n = 1}^\infty {\frac{n}{{\log \frac{1}{{\gamma ({A_n}(x) - X)}}}} = \infty } $$
(2)
.

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Bibliography

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    E. Bishop, A minimal boundary for function algebras. Pacific. J. Math. 9 (1959), 629–642.CrossRefGoogle Scholar
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    M. Brelot, Sur un théorème de prolongement fonctionnel de Keldysh concernant le problème de Dirichlet. J. d’Analyse Math. 8 (1960), 273–288.CrossRefGoogle Scholar
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    P.C. Curtis, Jr., Peak points for algebras of analytic functions. J. Functional Analysis 3 (1969), 35–47.CrossRefGoogle Scholar
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    M. V. Keldysh, On the solubility and the stability of Dirichlet’s problem. Uspehi Mat. Nauk USSR 8 (1941), 171–231.Google Scholar
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    M. Tsuji, Potential Theory in Modern Function Theory. Maruzen, Tokyo 1959.Google Scholar
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    N. Wiener, Certain notions in potential theory. J. Math. Phys. 3 (1924), 24–51.Google Scholar

Copyright information

© Springer Basel AG 1969

Authors and Affiliations

  • Philip C. CurtisJr.
    • 1
  1. 1.University of CaliforniaLos AngelesUSA

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