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Explicit Multi-point Taylor Polynomial · arXivDesk

2106.11440Jun 21, 20219 pages, no figures. Keywords: multi-point Taylor polynomial, multi-point polynomial interpolation, Hermite interpolation, Osculatory interpolation

Explicit Multi-point Taylor Polynomial

Andrés Gómez Arias

Classical Analysis and ODEs

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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 $f$

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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.

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