Boundary behavior of functions in the de Branges--Rovnyak spaces

Abstract

This paper deals with the boundary behavior of functions in the de Branges--Rovnyak spaces. First, we give a criterion for the existence of radial limits for the derivatives of functions in the de Branges--Rovnyak spaces. This criterion generalizes a result of Ahern-Clark. Then we prove that the continuity of all functions in a de Branges--Rovnyak space on an open arc $I$ of the boundary is enough to ensure the analyticity of these functions on $I$. We use this property in a question related to Bernstein's inequality