A general Rudin-Carleson theorem

Abstract

which is analytic on D -L such that F(z) =f(z) for all z in S. It is the purpose of this paper to generalize this theoren'r- Before stating the generalization, we remark that the Rudin-Carleson theorem is closely related to a theorem of F. and M. Riesz, which states that any (finite, complex-valued, Baire) measure on L which is orthogonal to all continuous functions F on D which are analytic on D-L is absolutely continuous with respect to Lebesgue measure dO on L. The proofs of the two theorems show that the results of RudinCarleson and of F. and M. Riesz are closely related. We shall state an abstract theorem which shows that the Rudin-Carleson theorem is a direct consequence of the F. and M. Riesz theorem. This abstract theoremi will permit an automatic generalization of the RudinCarleson result to any situation to which the F. and M. Riesz result can be generalized. The theorem to be proved reads as follows: THEOREM 1. Let C(X) be the uniform.ly-normed Bw-nach space of all continuous complex-valued functions on a compact Hausdorff space X. Let B be a closed subspace of C(X). Let B' consist of all (finite, complexvalued, Baire) measures ,u on X such that ffd,u =0 for all f in B. Let ,u be the regular Borel extension of the Baire measure ,u. Let S be a closed subset of X with the property that ,u(T) = 0 for every Borel subset T of S and every L,u in B'. Let f be a continuous complex-valued function on S and A a positive function on X such that jf(x) I <A(x) for all x in S. Then there exists F in B with I F(x) I <A(x) for all x in X and F(x) =f(x) for all x in S. If X is taken to be the set L defined above and B is taken to be those functions in C(L) which are restrictions to L of functions in

Locations

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