Comparison of the rates of rational and polynomial approximations of differentiable functions (Q920318)

For \(f\in C[a,b]\), let \(E_ n(f)\) and \(R_ n(f)\) be the best uniform approximation of \(f\) by polynomials of degree at most \(n\) and rational functions of degree at most \(n\), respectively. Let \(G\) be the set of all \(f\in C[a,b]\) such that \(R_ n(f)=o(E_ n(f))\). \textit{A. A. Gonchar} [Dokl. Akad. Nauk SSSR 128, 25--28 (1959; Zbl 0086.27102)] and \textit{E. P. Dolzhenko} [Math. Notes 1 (1967), 208--212 (1968); translation from Mat. Zametki 1, 313--320 (1967; Zbl 0201.07603)] posed the problem of describing the set \(G\). The author shows that \(G\) contains properly the set of all functions whose derivatives in the sense of Riemann-Liouville have discontinuities of the first kind.