Zur Abschätzung des Bestapproximationsfehlers bei der Approximation differenzierbarer Funktionen durch Polynome

Abstract

The problem attacked here is to find smaller factorsγ (n,ϱ),n > ϱ, for the inequality

$$\delta _n [f] \leqq \gamma (n,\varrho ) \cdot \mathop {\sup }\limits_{x \in [ - 1,1]} |f^{(\varrho )} (x)|$$

than are already known. Heref(^ϱ (x) denotes theϱ-th derivative off(x), andδ _n [f] is the error of the best Chebyshev-approximation off by algebraic polynomials of degree ≦n. A new approach to this problem is demonstrated and the results we got forn≦9,ϱ≦8 by the use of a computer are presented.