Shepard Operators Revisited: Linear and Nonlinear Approaches, Accelerated Convergence, and Applications

Shepard operators are important in scattered data approximation. They were first introduced as linear positive operators for multivariate interpolation, and later developed into a broader class of approximation tools. In this survey, we review the classical Shepard operator, its multivariate forms, and several modifications aimed at improving convergence. We consider both linear and nonlinear extensions of Shepard operators. We also give some numerical results and graphical illustrations supporting the theoretical results. We aim to offer a unified and current reference for researchers in approximation theory, scattered data methods, and operator-based interpolation.