Approximation by Bivariate Half-Bernstein Operators on the Rectangular Domain

Abstract

The problem of approximating a function in two dimensions is a central topic in approximation theory. This subject has played a major role in approximation. Many modifications of the classical Bernstein operator in two dimensions are introduced and studied to obtain an operator with a higher approximation order or a faster approximation rate. This paper defined the bivariate Bernstein operators by taking the half terms (even terms) to approximate a function in the space . Convergence was verified using Korovkin's theorem, while the asymptotic behavior was described using a Voronovskaja-type formula. Quantitative estimates for error are also derived using continuity measures suitable for bivariate functions. Also, the theoretical study is supported by a numerical example for a specified function in the space . A table and some graphs of the spacefied function and its approximations are plotted. The numerical example compares the performance of the proposed operator with that of the classical Bernstein operator in two dimensions. The numerical results showed that the operator has lower numerical approximation errors than the classical Bernstein operator in two dimensions. This work showed that the operator under study has a lower order of approximation than that of the classical Bernstein operator in two dimensions