  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.
