I have a question about interpolation. I think that question is a theorem, but I don´t find nothing about that. Anyone can help me?
Show that, if $g$ is the polynomial of degree $m<n$ that interpolate $f(x)$ in $(n+1)$ points $x_0,x_1,...,x_n$ so, the column of order $m$ of the table of difference divided of $f(x)$ is composed to $(n+1-m)$ values equals at $f[x_0,x_1,...,x_m]$.
How can I do to solve that?
Each difference stage reduces the degree by 1. $f[x_0,x_1]$ has degree $m-1$ and by iterating $f[x_0,x_1,...,x_m]$ has degree $0$, and thus is a constant.