# Optimal order Jackson type inequality for scaled Shepard approximation

## TL;DR

This paper introduces a modified Shepard approximation with a dilation factor, achieving optimal error estimates for continuous functions on convex domains and improving bounds on well-separated points in annuli.

## Contribution

It presents a novel variation of Shepard approximation with an explicit optimal Jackson-type error bound and enhances existing bounds on point separation in annuli.

## Key findings

- Established an optimal order Jackson type error estimate with explicit constant.
- Improved bounds on the number of well-separated points in thin annuli.
- Demonstrated applicability to bounded continuous functions on convex domains.

## Abstract

We study a variation of the Shepard 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 \cite{jackson} 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 for the numbers of well-separated points in thin annuli, which is of independent interest.