Polynomial degree reduction in the $\mathcal{L}^2$-norm on a symmetric interval for the canonical basis

TL;DR

This paper presents a direct formula for optimal polynomial degree reduction in the -norm on symmetric intervals, improving computational efficiency and numerical stability over classical methods.

Contribution

It introduces a novel direct formula for polynomial degree reduction in the -norm, with proofs of efficiency and stability enhancements.

Findings

The formula is more computationally efficient than classical matrix methods.

The approach offers improved numerical stability.

Demonstrated through an illustrative example.

Abstract

In this paper, we develop a direct formula for determining the coefficients in the canonical basis of the best polynomial of degree $M$ that approximates a polynomial of degree $N>M$ on a symmetric interval for the $\mathcal{L}^2$-norm. We also formally prove that using the formula is more computationally efficient than using a classical matrix multiplication approach and we provide an example to illustrate that it is more numerically stable than the classical approach.