Another case of nonexistence in rational Chebyshev approximation with interpolatory constraints (Q1824787)

Let \(f\in C[0,1]\) and \(R_{n,m}(f)\) denote the class of rational functions \(r=p/q\), deg \(p\leq n\), deg \(q\leq m\) such that \(r(0)=f(0)\). The problem is does there exist \(r\in R_{n,m}(f)\) such that \(\| f- r^*\|_{\infty}=\inf \{\| f-r\|_{\infty}:\quad r\in R_{n,m}(f)\}.\) Negative results are given, in particular it is shown that given two sequences of positive integers \(\{n_ k\}\), \(\{m_ k\}\), \(k\in {\mathbb{N}}\), such that \(n_ k\to \infty\) as \(k\to \infty\), there exists \(f\in C[0,1]\) such that for infinitely many k, f has no best approximation in \(R_{n_ km_ k}(f)\).