On a Class of Holomorphic Functions Representable by Carleman Formulas in Some Class of Bounded, Simply Connected Domains From Their Values on an Analytic Arc

摘要

Let ${cal U}$ be a bounded, simply connected domain with Jordan rectifiable boundary and let $ M subset partial {cal U}$ be an open analytic arc whose Lebesgue measure satisfies $ 0 < m(M) < m(partial {cal U})$ . Our result gives a complete description of the class of holomorphic functions in ${cal U}$ which are represented by the Carleman formulas on the open arc M, when $partial {cal U}$ is almost regular with respect to M (Section 2). That is, we give a type of integral representation formulas for functions holomorphic in a domain ${cal U}$ by its values on a part M of the boundary $partial {cal U}$ . This class is denoted by ${cal NH}^1_M({cal U})$ .