Nonlinear approximation of vector-valued functions by Shepard operators based on max-product and max-min operations

Abstract

In this paper, to approximate vector-valued and continuous functions on the unit hypercube, we modify the linear Shepard operators by using max-product and max-min operations. We also investigate the effects of some regular summability methods in the approximation, such as Cesàro summability and Abel summability. Furthermore, we give some interesting applications and graphical simulations verifying our theoretical results. For example, we approximate a torus surface, a helix curve, a fuzzy point and the LogSumExp function by means of these modified operators. Our applications show that the results obtained here are connected with not only the classical approximation theory but also the theory of fuzzy logic and machine learning algorithms.