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The interaction of alternation points and poles in rational approximation. (English) Zbl 1072.41005
The author investigates the interrelation of alternation points for the minimal error function and poles of best Chebyshev approximants if uniform approximation on the interval $[-1, 1]$ by rational functions of degree $(n(s), m(s))$ is considered, $s \in N$. He shows that at least for a subsequence $\Lambda \subset N$, the asymptotic behaviour of the alternation points of the degree $(n(s), m(s))$, $s \in\Lambda$ is completely determined by the location of the poles of the best approximants and vice versa, if $m(s) \leq n(s)$ or $m(s) - n(s) = o(s/\log s)$ as $s \to\infty$.

41A20Approximation by rational functions
Full Text: DOI
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