On a class of holomorphic functions representable by Carleman formulas in the disk from their values on the arc of the circle

Abstract

Let D be a unit disk and M be an open arc of the unit circle whose Lebesgue measure satisfies 0 < l(M) < 2π Our first result characterizes the restriction of the holomorphic functions f ∈ H(D), which are in the Hardy class H1 near the arc M and are denoted by N HM 1 (D), to the open arc M. This result is a direct consequence of the complete description of the space of holomorphic functions in the unit disk which are represented by the Carleman formulas on the open arc M. As an application of the above characterization, we present an extension theorem for a function f ∈ L1 (M) from any symmetric sub-arc L ⊂ M of the unit circle, such that L̄ ⊂ M, to a function f ∈ N H L1(D). © 2007 WILEY-VCH Verlag GmbH & Co. KGaA.