Approximation Results for Urysohn Type Two Dimensional Nonlinear Bernstein Operators

In the present work, our aim of this study is generalization and extension of the theory of interpolation of two dimensional functions to functionals or operators by means of Urysohn type nonlinear operators. In accordance with this purpose, we introduce and study a new type of Urysohn type nonlinear operators. In particular, we investigate the convergence problem for nonlinear operators that approximate the Urysohn type operator in two dimensional case. The starting point of this study is motivated by the important applications that approximation properties of certain families of nonlinear operators have in signal-image reconstruction and in other related fields. We construct our nonlinear operators by using a nonlinear form of the kernels together with the Urysohn type operator values instead of the sampling values of the function.