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

We give a new solvability criterion for the boundary Carath'{e}odory-Fej'{e}r problem: given a point $x\in \mathbb{R}$ and, a finite set of target values $a^0,a^1,\mathrm{.}\mathrm{.}\mathrm{.},a^n\in \mathbb{R}$, to construct a function $f$ in the Pick class such that the limit of $f^{(k)}(z)\mathrm{/}k!$ as $z \to x$nontangentiallyintheupperhalfplaneis$a^k$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.