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 nonempty 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.
