On the extension of a theorem of W. Dickmeis et al (Q582518)

The paper is concerned with counterexample type theorems to approximation problems with rates, in the sense of \textit{W. Dickmeis}, \textit{R. J. Nessel} and \textit{E. van Wickeren} [Math. Nachr. 117, 263-271 (1984; Zbl 0595.42005)]. By extending a theorem proved by these authors, the author of the present paper obtains as a corollary a positive answer to a conjecture of \textit{M. Hasson} [J. Approximation Theory 29, 91-102 (1980; Zbl 0456.41008)]: for \(N\geq 1\), there exists \(f\in C^{2N}[-1,1]\), such that \(\overline{\lim}_{n\to \infty}\| f^{(k)}-P_ n^{(k)}(f)\|_{C[-1,1]}=\infty,\) where \(P_ n(f)\) denotes the nth polynomial of best approximation to f.