Vol. 70, No. 2, 1977

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Rational approximation and the growth of analytic capacity

Claes Fernström

Vol. 70 (1977), No. 2, 347–350

Let X be a compact set in the complex plane C. Denote by R(X) the closure in the supremum norm of the rational functions with poles off X and by A(X) the set of continuous functions, which are analytic on the interior of X. The analytic capacity of a set S is denoted by γ(S). For the definition of γ see below. Let Bz(δ) = {ζ C;|z ζ| < δ} and let ∂X denote the boundary of X. Vitushkin has proved that R(X) = A(X) if

lδim→0      δ     > 0 for all z ∈ ∂X.

Let ψ be a function from R+ to R+, where R+ = {x R;x 0}. We now ask the following questions. If limδ0ψ(δ) = 0, is it possible to find a compact set X such that R(X)A(X) and such that γ(Bz(δ) X) δψ(δ) for all z ∂X and for all δ, 0 < δ < δz? If the answer is yes, can the answer still be yes, if limδ0ψ(δ) = 0 is replaced by ---
limδ0ψ(δ) > 0? The answers of these questions can be found in Theorem 1 and Theorem 2.

Mathematical Subject Classification
Primary: 30A82, 30A82
Received: 31 August 1976
Revised: 20 January 1977
Published: 1 June 1977
Claes Fernström