Vol. 69, No. 2, 1977

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The range of analytic extensions

Josip Globevnik

Vol. 69 (1977), No. 2, 365–384
Abstract

Denote by Δ, Δ, Δ the open unit disc in C, its closure and its boundary, respectively. Let X be a complex Banach space and denote by 𝒜(X) the class of all non-empty sets P X having the following property: given any closed set F Δ of measure 0 and any continuous function f : F P there exists a continuous extension f : Δ X of f, analytic on Δ and satisfying f(Δ F) Int P.

Theorem. P ∈𝒜(X) if and only if Int P is connected, locally connected at every point of P and satisfies P closure (Int P).

Theorem. If P C consists of more than one point then P ∈𝒜(C) if and only if given any F and f as above there exists a continuous extension f : Δ C of f, analytic on Δ and satisfying f(Δ) P.

This generalizes a theorem of Rudin which asserts that such f exists if P C is homeomorphic to Δ.

Theorem. If P ∈𝒜(X) then given any relatively open set B Δ, any relatively closed set F B of measure 0 and any continuous function f : F P there exists a continuous extension f : Δ B X of f, analytic on Δ and satisfying f((Δ B) F) Int P.

Mathematical Subject Classification 2000
Primary: 46J15
Secondary: 30A72
Milestones
Received: 1 July 1976
Revised: 27 October 1976
Published: 1 April 1977
Authors
Josip Globevnik