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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Real vs. complex rational Chebyshev approximation on an interval
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by Lloyd N. Trefethen and Martin H. Gutknecht PDF
Trans. Amer. Math. Soc. 280 (1983), 555-561 Request permission

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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Additional Information
  • © Copyright 1983 American Mathematical Society
  • 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