Bivariate polynomial histopolation techniques on Padua, Fekete, and Leja triangles

Abstract This paper explores the reconstruction of a real-valued function f defined over a domain $$\varOmega \subset \mathbb {R}^2$$ Ω ⊂ R 2 using bivariate polynomials that satisfy triangular histopolation conditions. More precisely, we assume that only the averages of f over a given triangulation $$\mathcal {T}_N$$ T N of $$\varOmega $$ Ω are available and seek a bivariate polynomial that approximates f using a histopolation approach, potentially flanked by an additional regression technique. This methodology relies on the selection of a subset of triangles $$\mathcal {T}_M \subset \mathcal {T}_N$$ T M ⊂ T N for histopolation, ensuring both the solvability and the well-conditioning of the problem. The remaining triangles can potentially be used to enhance the accuracy of the polynomial approximation through a simultaneous regression. We will introduce histopolation and combined histopolation-regression methods using the Padua points, discrete Leja sequences, and approximate Fekete nodes. The proposed algorithms are implemented and evaluated through numerical experiments that demonstrate their effectiveness in function approximation.