Optimal order Jackson type inequality for scaled Shepard approximation

We study a variation of the Shepard (1968) approximation scheme by introducing a dilation factor into the base function, which synchronizes with the Hausdorff distance between the data set and the domain. The novelty enables us to establish an optimal order Jackson (1911) type error estimate (with an explicit constant) for bounded continuous functions on any given convex domain. We also improve en route an upper bound estimate due to Narcowich and Ward (1991) for the numbers of well-separated points in thin annuli, which is of independent interest.