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A method for rational Chebyshev approximation of rational functions on the unit disk and on the unit interval. (English) Zbl 1172.41303
Let \(R_{MN}\) denote the space of rational functions with complex coefficients, with degree of numerator and denominator at most \(M\) and \(N\), respectively. The problem under consideration is: given a function \(f\in R_{MN}\) analytic on the unit disk, find a best approximation \(r^\star_{mn}\in R_{mn}\) with \(m<M\) or \(n<N\), so that \[ \delta=\| f-r^\star_{mn}\| =\min_{r\in R_{mn}}\| f-r\| , \] where \(\| \cdot\| \) denotes the supremum norm on the unit disk. It is well known that a best approximant always exist but is not unique in general. The author presents a method for construction of a best approximant for the following four cases: 1) \(m=M-1\) and \(n\geq N-1\), 2) \(m\geq M-1\) and \(n=N-1\), 3) \(m\geq N-1=n\) and 4) \(M-1\leq m\leq N-1=n\). The value of the deviation \(\delta\) is given explicitely in terms of coefficients of \(f\).

41A20 Approximation by rational functions
30E10 Approximation in the complex plane
30D50 Blaschke products, etc. (MSC2000)
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