Polynomial histopolation on mock-Chebyshev segments

TL;DR

This paper extends mock-Chebyshev techniques to histopolation, a segmental averaging polynomial method, analyzing their stability and demonstrating quasi-optimal growth of Lebesgue constants through theoretical and numerical evaluations.

Contribution

It introduces three mock-Chebyshev approaches for histopolation and analyzes their stability via Lebesgue constants, advancing polynomial approximation methods.

Findings

Segmental mock-Chebyshev approaches have logarithmic Lebesgue constant growth.

The methods demonstrate high stability and accuracy in numerical experiments.

Theoretical analysis confirms quasi-optimal conditioning of the proposed techniques.

Abstract

In computational practice, we often encounter situations where only measurements at equally spaced points are available. Using standard polynomial interpolation in such cases can lead to highly inaccurate results due to numerical ill-conditioning of the problem. Several techniques have been developed to mitigate this issue, such as the mock-Chebyshev subset interpolation and the constrained mock-Chebyshev least-squares approximation. The high accuracy and the numerical stability achieved by these techniques motivate us to extend these methods to histopolation, a polynomial interpolation method based on segmental function averages. While classical polynomial interpolation relies on function evaluations at specific nodes, histopolation leverages averages of the function over subintervals. In this work, we introduce three types of mock-Chebyshev approaches for segmental interpolation and theoretically analyse the stability of their Lebesgue constants, which measure the numerical conditioning of the histopolation problem under small perturbations of the segments. We demonstrate that these segmental mock-Chebyshev approaches yield a quasi-optimal logarithmic growth of the Lebesgue constant in relevant scenarios. Additionally, we compare the performance of these new approximation techniques through various numerical experiments.