The Rudin-Carleson theorem for vector-valued functions

Abstract:

The following generalization of the Rudin-Carleson theorem is proved. Let $X$ be a complex Banach space and let $f:F \to X$ be a continuous function, where $F$ is a closed subset of the unit circle in $C$ of Lebesgue measure zero. There exists a continuous function $g$ from the closed unit disc to $X$ which is analytic on the open unit disc and satisfies (i) $g|F = f$, (ii) ${\max _{|z| \leqslant 1}}||g(z)|| = {\max _{z\epsilon F}}||f(z)||$.