 Hermite interpolation error formula

Let $x_0<\cdots <x_n$ be distinct interpolation nodes, let $f\in C^{2n+2}[x_0,x_n]$, and let $H$ be the Hermite interpolating polynomial matching the values and first derivatives of $f$ at all nodes. For each $x$ in the interval there is an interior point $\xi$ for which the displayed formula holds. At a node the formula is immediate. Otherwise put $e=f-H$, $\pi (t)={\prod }_i(t-x_i)$ and $\phi (t)=e(t)-e(x)\pi (t{)}^2/\pi (x{)}^2$. This function vanishes at the $n+1$ nodes and at $x$, while its derivative vanishes at every node. Rolle's theorem gives $n+1$ additional zeros of its derivative between successive zeros of $\phi$. Repeating the theorem therefore gives ${\phi }^{(2n+2)}(\xi )=0$. Since $H^{(2n+2)}=0$ and $({\pi }^2{)}^{(2n+2)}=(2n+2)!$, rearrangement yields the formula. The extra point $x$ is a zero of $\phi$, not necessarily of its derivative.

For two nodes $a,b$, this gives the uniform bound $\Vert f-H{\Vert }_{\infty }\le \Vert f^{(4)}{\Vert }_{\infty }(b-a{)}^4/384$, since $(x-a{)}^2(b-x{)}^2\le (b-a{)}^4/16$. The squared nodal factors encode simultaneous matching of values and derivatives.