On the divergence of certain Hermite-Fejér interpolation

Summary

In his paper [1]P. Turán discovers the interesting behaviour of Hermite-Fejér interpolation (based on the Čebyšev roots) not describing the derivative values at “exceptional” nodes {η_n} ^∞_n=1 . Answering to his question we construct such exceptional node-sequence for which the mentioned process is bounded for bounded functions whenever −1<x<1 but does not converge for a suitable continuous function at any point of the whole interval [−1, 1].