Two interpolation operators on irregularly distributed data in inner product spaces

Abstract

Two interpolation operators in inner product spaces for irregularly distributed data are compared. The first is a well-known polynomial operator, which in a certain sense generalizes the classical Lagrange interpolation polynomial. The second can be obtained by modifying the first so as to get a partition-of-unity interpolant. Numerical tests and considerations on errors show that the two operators have very different approximation performances, and that by suitable modifications both can provide acceptable results, working in particular from ${\mathbb{R}}^m$ to ${\mathbb{R}}^n$ and from $\mathcal{C}[-\pi ,\pi ]$ to $\mathbb{R}$.