On the bivariate Shepard-Lidstone operators

We propose a new combination of the bivariate Shepard operators [10] by the three point Lidstone polynomials introduced in [12]. The new combination inherits both degree of exactness and Lidstone interpolation conditions at each node, which characterize the interpolation polynomial. These new operators …nd application to the scattered data interpolation problem when supplementary second order derivative data are given [22]. Numerical comparison with other well known combinations are presented. Key words: Combined Shepard operators, Lidstone interpolation, Functional approximation, Error analysis. Mathematical subject codes: 41A05, 41A25, 65D05 1 Introduction and position of the problem. Let N = fx1; :::;xNg be a set of N distinct points of R, in the following called nodes or sample points, and let f be a function de…ned on a domain D R containing N . The classical Shepard operators, …rst introduced in 1 Corresponding Author 2 This paper is co-…nanced with the support of the European Commission, European Social Fund and the Region of Calabria. The authors are solely responsible for this communication and the European Commission and the Region of Calabria are not responsible for any use that may be made of the information contained therein. Preprint submitted to Elsevier Science 18 July 2011