Let $x_1,…,x_n$ be distinct points in $[a,b]$ and $l_i(x):=\prod_{k\ne i}\dfrac {x-x_k}{x_i-x_k} $. Let $w_i=\int_a^b l_i(x)dx$. For every $f \in C[a,b]$, let $I_n(f):=\sum_{i=1}^n w_i f(x_i)$.
If $I_n(P)=\int_a^b P(t) dt$ for every polynomial $P(x)$ of degree $\le m$, then how to prove that $m \le 2n-1$ ?
(Please do not quote any big theorems, I am trying to prove that any numerical intergation formula that comes from interpolation at $n$-points, has degree of precision at most $2n-1$)
Best Answer
Consider $f(x)=(x-x_1)^2(x-x_2)^2\cdots (x-x_n)^2$.
This $f$ is nonnegative everywhere and positive at all but finitely many points, so its integral over any nontrivial interval is positive. On the other hand, any integration rule based on the values at the $n$ points $x_1,x_2,\dots,x_n$ must return zero, so it can't integrate $f$ exactly. That's a polynomial of degree $2n$, and thus no linear combination of values at $n$ points can find the integral accurately for all polynomials of degree up to $2n$.