Sharpness of Timan's converse result for polynomial approximation (Q1180597)

The author proves a conjecture of M. Hasson which states that: suppose that \(\{a_n\}\) is a monotone increasing positive sequence with \(\sum \frac{1}{na_n}=\infty\). Then there exists a continuous function \(f\) which is not in \(C^{(r)}[-1,1]\) but for its best polynomial approximation \(E_n(f)\) by polynomials of degree at most \(n\) we have \[ E_n(f) = O(1/n^{2r}a_n). \] The result was proved by Hasson for the case \(r=1\). The proof of this conjecture also shows that a result of \textit{A. F. Timan} [Theory of approximation of functions of a real variable. Oxford etc.: Pergamon Press (1963; Zbl 0117.29001)] cannot be improved.