A method for rational Chebyshev approximation of rational functions on the unit disk and on the unit interval (Q5934373)

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 \(f\).