Generalized Taylor's Theorem

TL;DR

This paper introduces a generalized version of Taylor's theorem that allows for function expansion around multiple points with specified multiplicities, enhancing approximation and interpolation techniques in numerical analysis.

Contribution

It extends Taylor's theorem using the Euclidean algorithm to include multiple points and provides a remainder expression, also covering rational approximation.

Findings

Enables multi-point function expansion with multiplicities

Provides a simple remainder expression for the generalized theorem

Includes a theorem for rational approximation

Abstract

The Euclidean algorithm makes possible a simple but powerful generalization of Taylor's theorem. Instead of expanding a function in a series around a single point, one spreads out the spectrum to include any number of points with given multiplicities. Taken together with a simple expression for the remainder, this theorem becomes a powerful tool for approximation and interpolation in numerical analysis. We also have a corresponding theorem for rational approximation.