On Chebyshev-Markov rational functions over several intervals. (English) Zbl 0926.41012
The following extremal problem
\[
\min_{c_1,c_2,\dots,c_n\in R} \bigg\| \frac {x^n+ c_1x^{n-1}+\dots +c_n} {\omega_n(x)} \bigg\| _{C(K)}
\]
where \(K\) is a compact subset of \(\mathbb R\) and \(\omega_n(x)\) is a fixed real polynomial of degree less than \(n\) and positive on \(K\) is studied in this paper. The rational solutions of the given problem are Chebyshev-Markov rational functions. There is found a parametric representation of Chebyshev-Markov rational functions for \(K=[b_1,b_2]\cup [b_3,b_4]\cup \dots \cup [b_{2p-1},b_{2p}]\), \(-\infty < b_1\leq b_2<\dots < b_{2p-1}\leq b_{2p}<\infty \) in terms of Schottky-Burnside automorphic functions.
Reviewer: J.Kofroň (Praha)
MSC:
| 41A20 | Approximation by rational functions |
Keywords:
Chebyshev-Markov rational functions; extremal problem; Shottky-Burnside automorphic functionsReferences:
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