The bivariate Shepard operator of Bernoulli type

The method of Shepard is an efficient method for interpolation of very large scattered data sets; unfortunately, it has poor reproduction qualities and high computational cost. In this paper we introduce a new operator which diminishes these drawbacks. This operator results from the combination of the Shepard operator with a new interpolation operator, recently proposed by Costabile and Dell'Accio, and generalizes to two variate functions the Shepard-Bernoulli operator introduced in [2]. We study this combined operator and give error bounds in terms of the modulus of continuity of high order and of the mesh length. We improve the accuracy and computational efficiency using a method introduced by Franke and Nielson.