On the constrained mock-Chebyshev least-squares

TL;DR

This paper enhances mock-Chebyshev polynomial interpolation by combining it with polynomial regression using simultaneous approximation theory, aiming to improve accuracy for analytic functions while addressing the Runge phenomenon.

Contribution

It introduces a method to select the regression degree and provides conditions to improve mock-Chebyshev interpolation accuracy in the uniform norm.

Findings

The combined approach reduces interpolation error for analytic functions.

Guidelines for choosing the regression degree improve approximation quality.

Numerical results confirm the effectiveness of the proposed method.

Abstract

The algebraic polynomial interpolation on uniformly distributed nodes is affected by the Runge phenomenon, also when the function to be interpolated is analytic. Among all techniques that have been proposed to defeat this phenomenon, there is the mock-Chebyshev interpolation which is an interpolation made on a subset of the given nodes whose elements mimic as well as possible the Chebyshev-Lobatto points. In this work we use the simultaneous approximation theory to combine the previous technique with a polynomial regression in order to increase the accuracy of the approximation of a given analytic function. We give indications on how to select the degree of the simultaneous regression in order to obtain polynomial approximant good in the uniform norm and provide a sufficient condition to improve, in that norm, the accuracy of the mock-Chebyshev interpolation with a simultaneous regression. Numerical results are provided.