I was tasked to interpolate function $f(x)=-x^5$ using nodes: $x_0=-2,\ x_1=0,\ x_2=2,\ x_3=-3,\ x_4=3,\ x_5=-4,\ x_6=4$
While it is obviously trivial, I have been told, that it can be done without any normal calculations at all and the interpolation polynomial would be just equal to the target function. Unfortunately I have no idea how to prove it. Can you give any tips on how to do that.
Does it have anything to do with the fact that the number of nodes is greater than the degree of $f$?
(it would also be great if you used only Lagrange's or Newton's interpolation methods)
It has everything to do with the fact that the number of nodes is greater than the degree of $f$.
The interpolating polynomial $L$ at $7$ points has degree at most $6$. Therefore, $L-f$ has degree at most $6$. Since $L-f$ has at least $7$ roots, it must be the zero polynomial. Thus $L=f$.