I saw a previous thread that was related, but it seemed to be missing some information, and it was never solved.

I know that for Hermite interpolation, every textbook has

$\displaystyle

K_{2n+1}(x) = \sum_{j=0}^n f(x_j) H_{n,j}(x) + \sum_{j=0}^n f'(x_j) \hat H_{n,j}(x)

$

where

$\displaystyle

H_{n,j}(x) = [1 - 2(x-x_j)L'_{n,j}(x_j)]L^2_{n,j}(x)

$

and

$\displaystyle

\hat H_{n,j}(x) = (x-x_j)L^2_{n,j}(x)

$

But how do I make it to the next step? What's the 3rd term, if I'm looking for $\displaystyle \tilde H_{n,j}(x)$ such that

$\displaystyle

\tilde H''_{n,j}(x_i) = \begin{cases}

1 &\text{ if } i=j\\

0 &\text{ otherwise}

\end{cases}

$

I would love to know if you could offer any help. I'm stumped...