Explicit Multi-point Taylor Polynomial

TL;DR

This paper presents an explicit form of the multi-point Taylor polynomial that interpolates a function's derivatives at multiple points, providing a direct formula and proof of its properties, improving upon previous recursive methods.

Contribution

It introduces an explicit formula for the multi-point Taylor polynomial, offering a direct and proven expression that enhances previous recursive approaches.

Findings

Provides an explicit expression for the multi-point Taylor polynomial.

Proves the polynomial's interpolation properties for derivatives at multiple points.

Discusses advantages over recursive formulas in literature.

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 $P^{(n)}_{k,m}(a_i) = f^{(n)}(a_i)\quad \forall \, i = 1,...,m\quad \&\quad \forall \, n = 0,...,k$. A discussion regarding previous expressions presented in the literature, which mostly consisted in recursion formulas and not explicit formulas, is made.