Rate of convergence of Shepard’s global interpolation formula

Abstract

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