Zur Konvergenz der Ableitungen von Interpolationspolynomen

Letp _n denote the polynomial of degreen or less that interpolates a given smooth functionf at the ?eby?ev nodest _j ⁿ =cos(jπ/n), 0≤j≤n, and let ‖·‖ be the maximum norm inC[?1, 1]. It is proved that fork-th derivatives (2≤k≤n) estimates of the following type hold $$parallel f^{(k)} - p_n^{(k)} parallel leqslant c_k n^{k - 1} inf { parallel f^{(k)} - qparallel :q in Pi _{n - k} } .$$ In this relationc _k only depends onk andΠ _n?k denotes the space of polynomials up to degreen?k.