Let E be a closed subset of
the compact Hausdorff X and let A be a closed separating subspace of Cc(X). Let ρ
be a dominator (strictly positive, l.s.c.) defined on X × T, T the unit circle in C.
Conditions, formulated in terms of boundary measures, are discussed for approximate
and exact solutions to the problem of finding ρ-dominated extensions in A of
functions g ∈ (A|E)− satisfying re tg (x) ≦ ρ(x,t) on E × T. Various interpolation
theorems of Rudin-Carleson type for superharmonic dominators are incorporated into
this framework.