On the approximation order of triangular Shepard interpolation (Q2794709)

By the Shepard interpolation, a bivariate function \(f:{\mathbb R}^2 \to {\mathbb R}\) can be reconstructed as a normalized blend of its values \(f(x_j)\) at scattered distinct points \(x_j \in {\mathbb R}^2\) \((j=1,\dots,n)\), using the inverse distance to the scattered points as point-based weight functions.NEWLINENEWLINEApplying a triangulation of the scattered points and triangle-based weight functions, the triangular Shepard operator is an extension of the classical Shepard operator, which interpolates all data and reproduces linear polynomials. The authors show that this triangular Shepard operator has quadratic approximation order, which is confirmed by numerical tests.