On a Theorem of Keldysh and Wiener

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 X°, the interior of X, was connected and X is the closure of X°, 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))

.

This research was supported by the National Science Foundation Grant # GP-8383.