Zolotarev's fifth and sixth problems

11/21/2020

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by   Evan S. Gawlik, et al.

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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 √(z) and (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.