Multinode rational operators for scattered data interpolation

In 1968 D. Shepard [1] introduces an approximation method for the interpolation of scattered data which consists in a weighted average of functional values at the data points. The method is easy to implement (indeed it is the fastest method for the interpolation of scattered data [2] but it reproduces exactly only constant polynomials and has flat spots in the neighbourhood of all data points. In 1983 F. Little [3] considers weighted average of local linear interpolants based on triples of data sites and takes as basis functions the normalization of the product of inverse distances from the points of the triples. This method overcomes the drawbacks of the Shepard method and, at the same time, maintains its features of simplicity of implementation and speed. In fact, the use of a searching technique to detect and select the nearest neighbor points [4] to determine the best local linear interpolant on compact triangulations [5], allows to consider the triangular Shepard method a fast meshfree method with an adequate order and a good accuracy of approximation. As Little suggests, his method can be generalized to higher dimensions and to sets of more than three points. In this talk we will discuss about some of these generalizations. (Joint work with Filomena Di Tommaso.)