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.

翻译：在计算实践中，我们常遇到仅能获得等间距点处测量值的情况。此时使用标准多项式插值会因问题的数值病态性而导致结果极不准确。为缓解此问题，已发展出多种技术，如模拟切比雪夫子集插值与约束模拟切比雪夫最小二乘逼近。这些技术所实现的高精度与数值稳定性促使我们将此类方法推广至直方图插值——一种基于区间函数平均值的多项式插值方法。经典多项式插值依赖于特定节点处的函数求值，而直方图插值则利用函数在子区间上的平均值。本文针对区间插值问题引入三类模拟切比雪夫方法，并从理论上分析其勒贝格常数的稳定性——该常数度量了在区间微小扰动下直方图插值问题的数值条件数。我们证明这些区间模拟切比雪夫方法在相关场景下能实现勒贝格常数的拟最优对数增长。此外，我们通过多组数值实验对比了这些新型逼近技术的性能表现。