Tikhonov regularization for polynomial approximation problems in Gauss quadrature points

TL;DR

This paper introduces Tikhonov regularization into polynomial approximation on [-1,1] using Gauss quadrature points, effectively reducing noise impact and improving approximation stability.

Contribution

It develops a regularized approximation scheme with explicit barycentric formulas and error bounds, enhancing polynomial approximation robustness with noisy data.

Findings

Tikhonov regularization reduces the Lebesgue constant and noise-related errors.

Explicit barycentric formulas incorporate regularization via a multiplicative correction.

Numerical examples demonstrate improved accuracy with noisy or small data sets.

Abstract

This paper is concerned with the introduction of Tikhonov regularization into least squares approximation scheme on $[-1,1]$ by orthonormal polynomials, in order to handle noisy data. This scheme includes interpolation and hyperinterpolation as special cases. With Gauss quadrature points employed as nodes, coefficients of the approximation polynomial with respect to given basis are derived in an entry-wise closed form. Under interpolatory conditions, the solution to the regularized approximation problem is rewritten in forms of two kinds of barycentric interpolation formulae, by introducing only a multiplicative correction factor into both classical barycentric formulae. An $L_2$ error bound and a uniform error bound are derived, providing similar information that Tikhonov regularization is able to reduce the operator norm (Lebesgue constant) and the error term related to the level of noise, both by multiplying a correction factor which is less than one. Numerical examples show the benefits of Tikhonov regularization when data is noisy or data size is relatively small.