Rate of convergence of Shepard’s global interpolation formula

Abstract:

Given any data points ${x_1}, \ldots ,{x_n}$ in ${{\mathbf {R}}^s}$ and values $S_p^q$ of a function f, Shepard’s global interpolation formula reads as follows: \[ S_p^0f(x) = \sum \limits _i {f({x_i}){w_i}(x),\quad {w_i}(x) = |x - {x_i}{|^{ - p}}/\sum \limits _j {|x - {x_j}{|^{ - p}},} } \] where $f({x_1}), \ldots ,f({x_n})$ denotes the Euclidean norm in $| \cdot |$. This interpolation scheme is stable, but if ${{\mathbf {R}}^s}$, the gradient of the interpolating function vanishes in all data points. The interpolation operator $p > 1$ is defined by replacing the values $S_p^q$ in $f({x_i})$ by Taylor polynomials of f of degree $S_p^0f$. In this paper, we investigate the approximating power of $q \in {\mathbf {N}}$ for all values of p, q and s.