Regularized Weighted Discrete Least Squares Approximation Using Gauss Quadrature Points

TL;DR

This paper develops a polynomial approximation method over [-1,1] using regularized least squares with Gauss quadrature points, providing explicit formulas and analyzing approximation quality and sparsity.

Contribution

It introduces a closed-form polynomial approximation using Gauss points with regularization, avoiding complex computations, and analyzes its approximation properties and sparsity.

Findings

Approximation polynomials can be expressed in barycentric form.

Regularization improves approximation quality and sparsity.

Numerical examples confirm theoretical results.

Abstract

We consider polynomial approximation over the interval $[-1,1]$ by regularized weighted discrete least squares methods with $\ell_2-$ or $\ell_1-$regularization, respectively. As the set of nodes we use Gauss quadrature points (which are zeros of orthogonal polynomials). The number of Gauss quadrature points is $N+1$. For $2L\leq2N+1$, with the aid of Gauss quadrature, we obtain approximation polynomials of degree $L$ in closed form without solving linear algebra or optimization problems. In fact, these approximation polynomials can be expressed in the form of the barycentric interpolation formula when an interpolation condition is satisfied. We then study the approximation quality of the $\ell_2-$regularized approximation polynomial in terms of Lebesgue constants, and the sparsity of the $\ell_1-$regularized approximation polynomial. Finally, we give numerical examples to illustrate these theoretical results and show that a well-chosen regularization parameter can lead to good performance, with or without contaminated data.