Polynomial approximation of derivatives by the constrained mock-Chebyshev least squares operator

TL;DR

This paper introduces a new polynomial approximation method using constrained mock-Chebyshev least squares to improve derivative estimation and mitigate Runge's phenomenon, with theoretical analysis and numerical validation.

Contribution

It provides explicit error representations and a novel approach for approximating derivatives of smooth functions using the constrained mock-Chebyshev least squares operator.

Findings

Effective in approximating derivatives of smooth functions.

Theoretical error bounds and derivative estimates derived.

Numerical tests confirm improved accuracy over existing methods.

Abstract

The constrained mock-Chebyshev least squares operator is a linear approximation operator based on an equispaced grid of points. Like other polynomial or rational approximation methods, it was recently introduced in order to defeat the Runge phenomenon that occurs when using polynomial interpolation on large sets of equally spaced points. The idea is to improve the mock-Chebyshev subset interpolation, where the considered function $f$ is interpolated only on a proper subset of the uniform grid, formed by nodes that mimic the behavior of Chebyshev--Lobatto nodes. In the mock-Chebyshev subset interpolation all remaining nodes are discarded, while in the constrained mock-Chebyshev least squares interpolation they are used in a simultaneous regression, with the aim to further improving the accuracy of the approximation provided by the mock-Chebyshev subset interpolation. The goal of this paper is two-fold. We discuss some theoretical aspects of the constrained mock-Chebyshev least squares operator and present new results. In particular, we introduce explicit representations of the error and its derivatives. Moreover, for a sufficiently smooth function $f$ in $[-1,1]$, we present a method for approximating the successive derivatives of $f$ at a point $x\in [-1,1]$, based on the constrained mock-Chebyshev least squares operator and provide estimates for these approximations. Numerical tests demonstrate the effectiveness of the proposed method.