Consider the interval $I=[-1,1]$ and the set of points $A=\{r_1, ..., r_n\}\subset I$. Suppose that there exists a $k$-times continuously differentiable $f\in C^k(I, \mathbb{R})$ of which we know its values in the points of $A$, $m_i=f(r_i)$ for $i=1, ..., n$. If $k<n$ and $\ell_i(t)$ is the $i$-th Lagrange polynomial for he grid $A$ what is the error
$$ e=\sup_{t\in I} \Big[ f(t)-\sum_{i=1}^n m_i \ell_i(t)\Big] $$ Moreover, what is the error of the $p$-th derivative?
$$ e_p = \sup_{t\in I} \Big[ \frac{d^p f}{d t^p}(t)-\sum_{i=1}^n m_i \frac{d^p \ell}{d t^p}(t)\Big] $$
Can anyone recommend some literature for this? Is there a result for an alternative to Lagrange polynomials?