With the Lagrange functions, why do we drop the greyed-out terms?
Note that the crossing-out of the ‘greyed’ terms is not, mathematically, correct. They do not cancel as such. I’ve just put them in gray to show the pattern, which can be hard to remember. But if you look at the Lagrange formula, you’ll find every term of the form “(t-t_(k+n))” present *except* for t-t_k. In other words, the term for which t=t_k is simply not present (the “middle” term in those big equations). Try some of the exercises and hopefully you’ll see what I mean with respect to the pattern of expansion terms. As for a reason *why* this term is not present – well, if you look at the equation and put t=t_k, the whole L_k(t) works out to be one. This in turn means the interpolation is guaranteed to pass through the sample points (as an aside, this is unlike some other interpolations like Bezier curves, which do not pass through every point.
