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We give a new solvability criterion for the boundaryCarath\'{e}odory-Fej\'{e}r problem: given a point $x \in \mathbb{R}$ and, afinite set of target values $a^0,a^1,...,a^n \in \mathbb{R}$, to construct afunction $f$ in the Pick class such that the limit of $f^{(k)}(z)/k!$ as $z \tox$ nontangentially in the upper half plane is $a^k$ for $k= 0,1,...,n$. Thecriterion is in terms of positivity of an associated Hankel matrix. The proofis based on a reduction method due to Julia and Nevanlinna.

Pseudo-Taylor expansions and the Carath\'{e}odory-Fej\'{e}r problem