A counterexample to a conjecture of Hasson
Let \(f\in C[-1,1]\) and \(\pi_ n\) be the class of algebraic polynomials of degree \(\leq n\). Define \(E_ n(f)=\min \{\| f-p\|:\) \(p\in \pi_ n\}\) and \(E_ n^{(1)}(f)=\min \{\| f-p\|:\) \(f\in \pi_ n\) and \(p_ n^{(1)}(0)=0\}\). In this paper the author constructs a counterexample to show that a conjecture of \textit{M. Hasson} [J. Approximation Theory 29, 103-115 (1980; Zbl 0456.41016)] that if \(f\in C[-1,1]\) and \(f^{(1)}\) does not exist at some point in \((-1,1)\) then (*) \(\limsup_{n}E_ n^{(1)}(f)/E_ n(f)<\infty,\) is false. Another problem that is posed here and now left open, is whether the result at (*) is true if `lim sup' is replaced by `lim inf'. The author gives another theorem as well in the paper that extends an earlier work of \textit{M. Hasson} and \textit{O. Shisha} [``Approximation Theory and Applications'' (Proc. Workshop, Technian-Israel Int. Haifor, 1981) pp. 311-317, Academic Press. N.Y. (1981)].
scientific article; zbMATH DE number 4140444