Bivariate Shepard–Bernoulli operators

Abstract

Abstract. In this paper we extend the Shepard-Bernoulli operators introduced in [6] to the bivariate case. These new interpolation operators are realized by using local support basis functions introduced in [23] instead of classical Shepard basis functions and the bivariate three point extension [13] of the generalized Taylor polynomial introduced by F. Costabile in [11]. The new operators do not require either the use of special partitions of the node convex hull or special structured data as in [8]. We deeply study their approximation properties and provide an application to the scattered data interpolation problem; the numerical results show that this new approach is comparable with the other well known bivariate schemes QSHEP2D and CSHEP2D by Renka [34,35]. The problemLet N = {x 1 , ..., x N } be a set of N distinct points (called nodes or sample points) of R s , s ∈ N, and let f be a function defined on a domain D containing N . The classical Shepard operators (first introduced in [37] in the particular case s = 2) are defined bywhere the weight functions A µ,i (x) in barycentric form areand |·| denotes the Euclidean norm in R s . The interpolation operator S N,µ [·] is stable, in the sense thatbut for µ > 1 the interpolating function S N,µ [f ] (x) has flat spots in the neighborhood of all data points. Moreover, the degree of exactness of the operator S N,µ [·] is 0, i.e. if it is restricted to the polynomial space P