Function reconstruction using a Jacobi-weighted quadratic enriched histopolation method

Reconstructing functions based on their integral data is a fundamental task in computational science and engineering. Histopolation techniques provide a natural framework for this task, as they are specifically designed to reconstruct functions from averaged data. We introduce a new family of Jacobi-weighted enriched histopolation schemes that extend classical edge-based local histopolation by incorporating additional weighted linear functionals. The method combines quadratic polynomial enrichment with weighted linear functionals, enhancing its ability to capture complex features. We establish unisolvence of the enriched scheme and demonstrate, through extensive numerical experiments, its significant accuracy gains over standard histopolation methods. These results establish weighted enriched histopolation as a powerful and versatile tool for high-fidelity function reconstruction in computational science.