The Reflection Principle for Banach Space-Valued Analytic Functions

Extract

We give sufficient conditions for the continuation of an analytic function with values in a Branch space. For analytic functions taking complex numbers as values, the principle is known as the Schwarz Reflection Principle.

A function defined on a domain of the complex plane with values in a Banach space X is analytic if it possesses at each point Z₀ of the domain a convergent power series in z, with coefficients in X.

THEOREM. Let D be a domain in the upper half-plane, and E a regular subset of the boundary of D. Suppose that E is an interval of the real axis (a,b). Let f be an analytic function defined on D, continuous up to E, taking values in a Banach space X. Let the image of D under f be Ω, and let Γ be the part of the boundary of Ω which is the image of E under f.