Optimal order Jackson type inequality for scaled Shepard approximation

Abstract

We study a variation of the Shepard [13] 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 [10] 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 [12] for the numbers of well-separated points in thin annuli, which is of independent interest.

Approximation by a class of neural network operators on scattered data

“…For a bounded set on $$$ {\mathbb{R}}^d $$$, from Senger et al, 18 the condition that $$$ \boldsymbol{T} $$$ is $$$ \rho $$$‐uniform implies that there are two positive constants $$$ {C}_1 $$$ and $$$ {C}_2 $$$ such that $$$$ {C}_1{n}^{-1/d}\le {q}_{\boldsymbol{T}}\le {h}_{\boldsymbol{T}}\le {C}_2{n}^{-1/d}. $$$$ …”

“…We omit the details here. For a bounded set on R d , from Senger et al, 18 the condition that T is 𝜌-uniform implies that there are two positive constants C 1 and C 2 such that…”