interpolation error using higher derivatives

Question

Given $x_{0},x_{1},x_{2}\in[a,b] $ each one different from the others,$f \in C^{4}[a,b]$ and $p\in\mathbb{P}_{3}$ so that $$p(x_{i})=f(x_{i}), i=0,1,2 $$ and $$p'(x_{1})=f'(x_{1})$$ prove that: $$\forall x \in [a,b] \hspace{0.2cm} \exists z \in (a,b) \hspace{0.2cm} f(x)-p(x)=\frac{1}{4!}(x-x_{0})(x-x_{1})^2(x-x_{2})f^{(4)}(z) $$ Normally, the interpolation error does not include the $(n+1)th$ derivative of $f$ (assuming here $n=3$). Any ideas on how to?

Answer 1

Just mimic the proof for the Lagrange interpolation case. Define $e(x)=f(x)-p(x)$ and, for $x\not\in\{x_0,x_1,x_2\}$, $$ R(t)=e(x)·(t-x_0)(t-x_1)^2(t-x_2)-e(t)·(x-x_0)(x-x_1)^2(x-x_2) $$ Then $R(t)$ has roots in $x,x_0,x_1,x_2$ with the root in $x_1$ a double root.

By Rolle's theorem, $R'(t)$ has (at least) $3$ roots inside the finite sub-intervals of the decomposition defined by $x,x_0,x_1,x_2$ and one root from the double root of $R$ in $x_1$. These $4$ roots of $R'$ lead by Rolle to at least $3$ roots of $R''$, $2$ roots of $R'''$, $1$ root $μ$ of $R^{(4)}$ inside the complex hull of $x,x_0,x_1,x_2\,$ and thus $$ 0=R^{(4)}(μ)=(f(x)−p(x))·4!-f^{(4)}(μ)(x−x_0)(x−x_1)^2(x−x_2) $$ which leads to the claim.