Rate of Convergence of Shepard's Global Interpolation Formula

Given any data points xx,...,x„ in R5 and values f(xx),... ,/(x„) of a function /, Shepard's global interpolation formula reads as follows: s°pf(x) = !/(*,>,(*). "*(*) -I* *,|"7D* xj\", ' j where | ■ | denotes the Euclidean norm in R*. This interpolation scheme is stable, but if p > 1, the gradient of the interpolating function vanishes in all data points. The interpolation operator Sjj is defined by replacing the values f(x¡) in S°f by Taylor polynomials of / of degree q s N. In this paper, we investigate the approximating power of 5' for all values of p, q and s.