On the Divergence Phenomenon in Hermite–Fejér Interpolation

Generalizing results of L. Brutman and I. Gopengauz (1999, Constr. Approx.15, 611–617), we show that for any nonconstant entire function f and any interpolation scheme on [−1, 1], the associated Hermite–Fejér interpolating polynomials diverge on any infinite subset of [−1, 1]. Moreover, it turns out that even for the locally uniform convergence on the open interval ]−1, 1[ it is necessary that the interpolation scheme converges to the arcsine distribution.