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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 

 

Real vs. complex rational Chebyshev approximation on an interval


Authors: Lloyd N. Trefethen and Martin H. Gutknecht
Journal: Trans. Amer. Math. Soc. 280 (1983), 555-561
MSC: Primary 41A25; Secondary 41A50
DOI: https://doi.org/10.1090/S0002-9947-1983-0716837-X
MathSciNet review: 716837
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Abstract: If $ f \in C[ - 1,1]$ is real-valued, let $ {E^{r}(f)}$ and $ {E^{c}(f)}$ be the errors in best approximation to $ f$ in the supremum norm by rational functions of type $ (m,n)$ with real and complex coefficients, respectively. It has recently been observed that $ {E^c}(f) < {E^r}(f)$ can occur for any $ n \geqslant 1$, but for no $ n \geqslant 1$ is it known whether $ {\gamma_{mn}} = \inf_f\,{E^c}(f)/{E^{r}(f)}$ is zero or strictly positive. Here we show that both are possible: $ {\gamma_{01}} > 0$, but $ {\gamma_{mn}} = 0$ for $ n \geqslant m + 3$. Related results are obtained for approximation on regions in the plane.


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DOI: https://doi.org/10.1090/S0002-9947-1983-0716837-X
Keywords: Chebyshev approximation, rational approximation
Article copyright: © Copyright 1983 American Mathematical Society