A generalization of the Rudin-Carleson theorem

Proceedings of the American Mathematical Society

1980

The Bishop-Gamelin interpolation theorem asserts that given a compact Hausdorff space K, a closed subspace A of C(K), a positive continuous function p on K and a closed set F c K such that every measure in the annihilator of A vanishes on F, every function/_ G C(F) satisfying \f(s)\ < p(s) (s G F) extends to a function / G A satisfying \f(z)\ < p(z) (z G K). In the paper we consider a special case where the theorem is extended to the situation when the dominating function is nonnegative.

Section: ■^{Ttà-tr'mentioning

confidence: 99%

On Dominated Extensions in Function Algebras

Proceedings of the American Mathematical Society

1980

“…This simple dominated extension theorem is a special case of a more general theorem proved by E. Bishop [1]. See [l]-[3], [6], [7], [10] for such theorems in general spaces of continuous functions and see [7] for the most general dominated extension theorem in the disc algebra.…”

mentioning

confidence: 98%

Analytic extensions and selections

Trans. Amer. Math. Soc.

1979

Let G be a closed subset of the closed unit disc in C, let F be a closed subset of the unit circle of measure 0 and let 4> map G into the class of all open subsets of a complex Banach space X. Under suitable additional assumptions on $ we prove that given any continuous function/: F-*X satisfying_/(z) e closure(*(z)) (:E/nC) there exists a continuous function / from the closed unit disc into X, analytic in the open unit disc, which extends / and satisfies /(z) e i»(z) (:EG-F). This enables us to generalize and sharpen known dominated extension theorems for the disc algebra.

“…In known generalizations and applications of Bishop's theorem [1], [3], [4], [5], [6], [7], [9], [10], [11] the dominating functions q are always assumed to be positive on K. Our purpose here is to present a special case where the conclusion of Bishop's theorem can be strengthened to allow domination by nonnegative functions.…”

mentioning

confidence: 99%

On dominated extensions in function algebras

1980

Proc. Amer. Math. Soc.

The Bishop-Gamelin interpolation theorem asserts that given a compact Hausdorff space K, a closed subspace A of C ( K ) C(K) , a positive continuous function p on K and a closed set F ⊂ K F \subset K such that every measure in the annihilator of A vanishes on F, every function f ∈ C ( F ) f \in C(F) satisfying | f ( s ) | ⩽ p ( s ) ( s ∈ F ) |f(s)| \leqslant p(s)(s \in F) extends to a function f ~ ∈ A \tilde f \in A satisfying | f ~ ( z ) | ⩽ p ( z ) ( z ∈ K ) |\tilde f(z)| \leqslant p(z)(z \in K) . In the paper we consider a special case where the theorem is extended to the situation when the dominating function is nonnegative.