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 a 0 , a 1 , … , a n ∈ 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 a k 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.