Polynomial histopolation on mock-Chebyshev segments

TL;DR

This work addresses the ill-conditioning of polynomial interpolation on equispaced segments by extending mock-Chebyshev strategies to histopolation, where data are segment means rather than point evaluations. It introduces three segmental mock-Chebyshev methods—concatenated, quasi-nodal, and constrained—to produce stable histopolants on equispaced partitions and analyzes the growth of segmental Lebesgue constants, establishing quasi-optimal logarithmic growth in two methods. Theoretical stability results show that perturbations of segments preserve unisolvence and bound the interpolation operator, while numerical experiments demonstrate substantial accuracy gains over uniform segment interpolation and reveal nuanced behavior of the constrained method. The proposed techniques offer robust, data-efficient tools for function approximation when only average information over segments is available, with implications for numerical analysis and applied computations involving integral data.

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.

Figures (7)

• Figure 1: The Lebesgue constant $\Lambda_m(\mathcal{S}_{m}^{\mathrm{MC}})$ of the mock-Chebyshev segments $\mathcal{S}_{m}^{\mathrm{MC}}$ depending on the selection of the equidistant grid size $n$. The Lebesgue constant $\Lambda_m(\mathcal{S}_{m}^{\mathrm{CL}})$ for the Chebyshev segments is highlighted in yellow
• Figure 2: Cartesian grid of $n+1=51$ equispaced nodes ($\bullet$), mock-Chebyshev nodes ${X}_{m+1}^{\mathrm{MC}}$ of size $m+1=16$ ($\bullet$), the black line highlights the mock-Chebyshev segment $s_4^{\mathrm{MC}}=[x_3^{\mathrm{MC}},x_4^{\mathrm{MC}}]$. The mock-Chebyshev segments $\mathcal{S}_{m}^{\mathrm{MC}}$ partition the interval $I$ and have the nodes ${X}_{m+1}^{\mathrm{MC}}$ as end-points.
• Figure 3: Cartesian grid with $n+1=51$ equidistant nodes ($\bullet$) and the Chebyshev nodes $X_{m}^{\mathrm{CF}}$ of first kind of order $m=16$ ($\blacklozenge$). The mock-Chebyshev segments $\mathcal{S}_{m}^{\mathrm{MCF}}$ are visualized with black lines and are characterized as elements of the uniform segments $\mathcal{S}_{n}^{\mathrm{eq}}$ that contain a Chebyshev node.
• Figure 4: Segmental Lebesgue constant $\Lambda_m (\mathcal{S}_{m}^{\mathrm{MCF}})$ for the quasi-nodal segments $\mathcal{S}_{m}^{\mathrm{MCF}}$ depending on the selection of the equidistant grid size $n$.
• Figure 5: Trend of the condition number of the Vandermonde matrix associated by the interpolation on equispaced segments (magenta line) with that obtained through the concatenated mock-Chebyshev method (blue line), the quasi-nodal mock-Chebyshev method (black line) and the trend of the KKT matrix relative to the constrained mock-Chebyshev method (red line) based on a grid of $n+1=51:50:1001$ uniform nodes.

Theorems & Definitions (21)

• proposition thmcounterproposition
• proof
• proposition thmcounterproposition: Stability related to segmental norming sets
• proof
• remark thmcounterremark
• theorem 1: Stability of the segmental Lebesgue constant
• proof
• remark thmcounterremark
• lemma thmcounterlemma
• proof