Explicit Multi-point Taylor Polynomial

Abstract

The multi-point Taylor polynomial, which is the general, unique and of minimum degree ($mk+m-1$) polynomial $P_{k,m}(x)$ which interpolates a function's derivatives in multiple points is presented in its explicit form. A proof that this expression satisfies the multi-point Taylor polynomial's defining property is given. Namely, it is proven that for a k-differentiable function $f$ and a set of different m-points $\{a_1,...,a_m\}$, this polynomial satisfies $P_{k,m}^{(n)}(a_i)=f^{(n)}(a_i)\forall i=1,...,m\&\forall n=0,...,k$. A discussion regarding previous expressions presented in the literature, which mostly consisted in recursion formulas and not explicit formulas, is made.

9 pages, no figures. Keywords: multi-point Taylor polynomial, multi-point polynomial interpolation, Hermite interpolation, Osculatory interpolation