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arXiv · 2011.10877

Zolotarev's fifth and sixth problems

Evan S. Gawlik·Yuji Nakatsukasa
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Abstract

In an influential 1877 paper, Zolotarev asked and answered four questions about polynomial and rational approximation. We ask and answer two questions: what are the best rational approximants $r$ and $s$ to $\sqrt{z}$ and $\mbox{sign}(z)$ on the unit circle (excluding certain arcs near the discontinuities), with the property that $|r(z)|=|s(z)|=1$ for $|z|=1$? We show that the solutions to these problems are related to Zolotarev's third and fourth problems in a nontrivial manner.

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Evan S. Gawlik, Yuji Nakatsukasa. 2020-11-21. Zolotarev's fifth and sixth problems. https://arxiv.org/abs/2011.10877

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