The boundary Carathéodory–Fejér interpolation 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, to construct a function f in the Pick class such that the first few derivatives of f take on the prescribed target values at x. We also derive a linear fractional parametrization of the set of solutions of the interpolation problem with real target values. The proofs are based on a reduction method due to Julia and Nevanlinna.