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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 $$f$$.

MSC:
 41A20 Approximation by rational functions 30E10 Approximation in the complex plane 30D50 Blaschke products, etc. (MSC2000)
Keywords:
rational approximation; CF method
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