Suppose I say a function $f(x)$ is interpolated on the interval $[a,b]$ by a polynomial $p_n(x)$ whose degree does not exceed $n$. Also suppose further that $f$ is arbitrarily often differentiable on $[a, b]$ and that there exists $M$ such that $|f^{(k)}(x)| ≤ M$ for $k = 0,1,...$ and any $x ∈ [a,b]$.
Then with these assumptions is it possible to show ( without additional hypotheses about the location of interpolation points a≤x0
If anyone could give me any insight it would br great!