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arXiv · 1101.1251

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

Jim Agler·Zinaida A. Lykova·N. J. Young
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Abstract

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 $a^0,a^1,...,a^n \in \mathbb{R}$, to construct a function $f$ in the Pick class such that the limit of $f^{(k)}(z)/k!$ as $z \to x$ 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.

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Jim Agler, Zinaida A. Lykova, N. J. Young. 2011-01-06. Pseudo-Taylor expansions and the Carath\'{e}odory-Fej\'{e}r problem. https://doi.org/10.1016/j.jmaa.2011.08.001.2011

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