Improved Approximation Via Hybrid Shepard–Lagrange Operators: Linear and Nonlinear Perspectives

Abstract

This paper introduces a hybrid operator that combines Shepard operators with Lagrange polynomials, proving that the new operator exhibits superior approximation properties compared to the classical Shepard operator. In the linear case, our approach advances known results in the literature, providing a more effective framework for approximation. Building on this foundation, the method is also extended to nonlinear scenarios by employing max-product operations, demonstrating that the nonlinear operator achieves even better approximation characteristics than its linear counterpart. The theoretical findings are validated through numerical computations and graphical representations, strongly supporting the effectiveness of the hybrid operator in both linear and nonlinear contexts. © 2025