Multilevel quasi-interpolation

Abstract

Abstract Quasi-interpolation methods are well-established tools in multivariate approximation. They are efficient as they do not require, in contrast to interpolation, the solution of a linear system. Quasi-interpolations are often first studied on infinite grids. Here, it is usually required that the quasi-interpolation operator reproduces polynomials of a certain degree exactly. This degree corresponds to the approximation order of the quasi-interpolation process. Unfortunately, if a radial, compactly supported kernel is employed for building the quasi-interpolation operator, it is well known that polynomial reproduction is impossible in two or more dimensions. As such operators are numerically appealing and are frequently used in particle methods, we will, in this paper, look at such quasi-interpolation operators that do not reproduce polynomials and show that they lead, when employed in a multilevel scheme, to an efficient and converging approximation method.