A Polynomial Approximation for Arbitrary Functions

TL;DR

This paper introduces a Legendre polynomial expansion method for approximating arbitrary functions, demonstrating faster convergence and smaller errors compared to Taylor series, especially for global accuracy.

Contribution

It presents a novel polynomial approximation technique using Legendre polynomials, showing superior convergence and accuracy over traditional Taylor expansions.

Findings

Legendre series coefficients decay faster than Taylor coefficients.

Legendre approximation achieves at least ten times smaller error.

Numerical experiments confirm the theoretical advantages.

Abstract

We describe an expansion of Legendre polynomials, analogous to the Taylor expansion, to approximate arbitrary functions. We show that the polynomial coefficients in Legendre expansion, therefore the whole series, converge to zero much more rapidly compared to the Taylor expansion of the same order. Furthermore, using numerical analysis using sixth-order polynomial expansion, we demonstrate that the Legendre polynomial approximation yields an error at least an order of magnitude smaller than the analogous Taylor series approximation. This strongly suggests that Legendre expansions, instead of Taylor expansions, should be used when global accuracy is important.