The boundary Carath\'{e}odory-Fej\'{e}r interpolation problem

TL;DR

This paper provides an elementary proof for the solvability of the boundary Carathéodory-Fejér interpolation problem, characterizing when a Pick class function can match prescribed derivatives at a real point.

Contribution

It introduces a new elementary proof for the solvability criterion and offers a linear fractional parametrization of all solutions to the problem.

Findings

Established a solvability criterion for the boundary Carathéodory-Fejér problem.

Derived a linear fractional parametrization of the solution set.

Utilized a reduction method based on Julia and Nevanlinna techniques.

Abstract

We give an elementary proof of a solvability criterion for the {\em boundary Carath\'{e}odory-Fej\'{e}r problem}: given a point $x \in \R$ 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. The proofs are based on a reduction method due to Julia and Nevanlinna.