On the asymptotic distribution of oscillation points in rational approximation
Let \(R_{m,n}\) denote the set of algebraic rational functions of degree \((m,n)\) on \([-1,1]\). For \(f\in C[-1,1]\) the distribution of the oscillation points of the best uniform rational approximation to \(f\) from \(R_{m,n}\) is studied. For \(n=0\), \(m\to\infty\), Kadec proved that one gets uniform distribution of the oscillatory points (with respect to Chebyshev measure). This result was shown recently by Borwein, Grothmann, Kroó and Saff to still hold if \(n=n(m)\) with \(n=o(m)\); whereas, for \(n=m-1\), it was shown to no longer be true. In this paper, it is shown that Kadec's result also holds for \(n=[cm]\) with \(0<c<1\), which had remained open in the previous paper.
scientific article; zbMATH DE number 536535