Tangential boundary behavior in Hilbert spaces of analytic functions

Sarason's Hilbert space version of Carath\'eodory-Julia Theorem connects the non-tangential boundary behavior of functions in de Branges-Rovnyak space $H(b)$ with the existence of angular derivatives in the sense of Carath\'eodory for $b$, an analytic self-mapping of the unit disk. In this article, we continue the study of higher order extensions of this result that deal with derivatives of functions in $H(b)$, and we consider notions of approach regions more general than the non-tangential ones. Our main result generalizes the recent work of Duan-Li-Mashreghi on boundary behavior in model spaces to $H(b)$-spaces and to higher order derivatives, and we give a new self-contained proof of that result. It also generalizes earlier radial results of Fricain-Mashreghi. In relation to existence of angular derivatives, we show that in the classical Carath\'eodory-Julia Theorem one cannot replace the non-tangential approach region by any essentially larger region.