Effective Localization and Forward Stability of Multinode Shepard Operators

We study the effective localization and forward stability of multinode Shepard operators for scattered data approximation. Although these operators are globally supported, the product structure of their inverse-distance weights yields a quantitatively controlled decay of the normalized weights, resulting in an effectively local numerical action. Using a geometric mean distance associated with each multinode subset, we derive explicit decay estimates for the normalized weights and algebraic bounds for the contribution of distant subsets. These results provide a rigorous basis for truncated implementations with controlled error. We also derive weighted approximation and stability estimates in terms of local Lebesgue functions, and develop a finite-precision analysis based on logarithmic weight evaluation, log-sum-exp normalization, and backward stable local Vandermonde solves. Under the stated assumptions, the fully computed operator is shown to be first-order forward stable. Numerical experiments confirm the effective localization mechanism, the practical sharpness of the stability bounds, and the accuracy of the truncated approximations.