Approximating potential integrals by cardinal basis interpolants on multivariate scattered data

A multivariate interpolation operator on scattered data, expressed as a convex combination of cardinal basis functions depending on the inverse (s − 2)-power of the Euclidean distance in (s ≥ 3) is proposed to give numerical approximations of the integral representing the potential function of the Newtonian field generated by a continuous mass distribution. The operator can be used to interpolate the mass density or directly the potential function, as well as to remap them on a regular grid or a convenient point set. Considerations on the Newtonian potential energy of a system of mass points permit us to introduce quite naturally the operator and to prove some remarkable properties; then the application to the continuous case is considered. Computational performances and possible applications of the operator are outlined.