Pseudo-Taylor expansions and the Carathéodory–Fejér problem

We give a new solvability criterion for the boundary Carathéodory–Fejér problem: given a point x∈R and, a finite set of target values a0,a1,…,an∈C, to construct a function f in the Pick class such that the limit of f(k)(z)/k! as z→x nontangentially in the upper half-plane is ak for k=0,1,…,n. The criterion is in terms of positivity of an associated Hankel matrix. The proof is based on a reduction method due to Julia and Nevanlinna.