Approximating potential integrals by cardinal basis interpolants on multivariate scattered data (Q1609095)

The paper is concerned with the numerical approximation of the integral representation the potential function of the Newtonian field generated by a continuous mass distribution \[ u(x)= {1\over s- 2} \int_V {\mu(u)\over d^{s- 2}(x, u)} du,\quad s= 3,4,\dots, \] where \(d(\cdot,\cdot)\) denotes the Euclidean distance. The proposed approximations are based on the multivariate interpolation operator \[ \Phi(x; f,\Delta_n)= \sum^n_{j=1} f(x_j) {\prod^n_{\substack{ k=1\\ k\neq j}} d^{s-2}(x, x_k)\over \sum^n_{i=1} \prod^n_{\substack{ k=1\\ k\neq i}} d^{s-2}(x, x_k)}= \sum^n_{j=1} f(x_j) {1/d^{s- 2}(x, x_j)\over \sum^n_{i=1} (1/d^{s-2}(x, x_i))}, \] where \(\Delta_n:= \{x_1,\dots, x_n\}\). The authors prove some properties of the operator and propose some computational performances. Numerical tests are postponed to a forthcoming paper.