The main result of this paper is a new remainder formula for divided difference expansions of the form based on relatively elementary properties of the symmetric polynomials &sgr;j = &sgr;&tgr;0 + ... + &tgr;n &dgr;0 ... &dgr;n. This leads to a general proof of the error bound | Rp | <= Chn-p || f(p) || that, until now, has been established only for uniform meshes and a few other special cases. Specializing to p=n+2 leads to finite difference schemes on nonuniform meshes with O(h2) truncation error.
Although this is basically a technical result, the paper is well written and the results should be accessible to anyone with a good grasp of elementary approximation theory.
