Approximation by bivariate Bernstein–Kantorovich–Stancu operators that reproduce exponential functions

In this study, we construct a Stancu-type generalization of bivariate Bernstein–Kantorovich operators that reproduce exponential functions. Then, we investigate some approximation results for these operators. We use test functions to prove a Korovkin-type convergence theorem. Then, we show the rate of convergence by the modulus of continuity and give a Voronovskaya-type theorem. We give a covergence comparison about bivariate Bernstein–Kantorovich–Stancu operators and their exponential form.


Introduction
The goal of approximation theory is to approximate a target function using straightforward, computable, and more useful functions.In 1912, Bernstein [1] defined the Bernstein operators for every function on the interval [0, 1].Later, the various generalizations of Bernstein polynomials were investigated in [2][3][4].
In 2019, Aral et al. [30] gave the Bernstein-Kantorovich operators that reproduce exponential functions for n ∈ N and α, β, μ > 0 and x ∈ [0, 1].They considered the operator This paper consists of 6 sections.In Sect.2, we give the definition of generalized bivariate Bernstein-Kantorovich-Stancu operators and we obtain some auxiliary results.In Sect.3, we mention the rate of convergence with the help of the modulus of continuity.In Sect.4, we present Voronovskaya-type results.In Sect.5, we illustrate numerical examples with graphics.In Sect.6, we give the conclusions.
Other results can be obtained in a similar way.

Voronovskaya-type theorem
In this section, we mention a Voronovskaya-type theorem for the K α,β,μ,ν m,n (f ; x, y).Let the inverse of the exponential function for the first variable t be denoted by log ν μ and the inverse of the exponential function for the second variable s be shown as log μ ν .

Graphical and numerical analysis
In this section, we give a graphical and numerical analysis of K α,β,μ,ν m,n (f ; x, y) operators that illustrate the modeling of the approximation for the function f .by choosing x = y ∈ [0.1, 0.9] and step size h = 0.1 in Table 1 for m = n ∈ {70, 80, 90}.

Conclusion
In this work, we construct the exponential bivariate Bernstein-Kantorovich-Stancu operators.Then, we calculate the rate of convergence with the modulus of continuity of the functions defined on C(S μ,ν ).Also, we give the Voronovskaya-type theorem.Finally, the error tables of the exponential bivariate Berntein-Kantorovich operators are given for different values of m, n, μ, ν, α, and β.

Table 3
Comparison of K