Published online by Cambridge University Press: 17 April 2009
Suppose the qth derivative of a function f is Hölder continuous of index α, where 0 < α ≤ 1, on the interval [-1,1]. Suppose further that pn is any polynomial of degree at most n such that |τn(x)| = |f(x) - pn(x)| ≤ c {max ((1 - x2)½/n, 1/n2) } q+a [-1,1]. If
then it is shown that
‖τn‖β ≤ cn−q−α+β, 0 < β ≤ 1.