The boundary analog of the Carath\'eodory-Schur interpolation problem

Abstract

Characterization of Schur-class functions (analytic and bounded by one in modulus on the open unit disk) in terms of their Taylor coefficients at the origin is due to I. Schur. We present a boundary analog of this result: necessary and sufficient conditions are given for the existence of a Schur-class function with the prescribed nontangential boundary expansion $f(z)=s_0+s_1(z-t_0)+\ldots+s_N(z-t_0)^N+o(|z-t_0|^N)$ at a given point $t_0$ on the unit circle.