# Approximation by Bicomplex Favard-Szasz-Mirakjan Operators
## 1 Introduction  
As is well known, the basic logic in approximation theory is to express an arbitrary function in terms of an alternative representation obtained by simpler elementary functions with certain properties. In 1885, [PERSON] [1] made a groundbreaking discovery by demonstrating the density of algebraic polynomials within the class of continuous real-valued functions on a closed interval. This theorem has made significant contributions to the development of functional analysis, function theory, and different fields of mathematics. The [PERSON] theorem was later proven in 1912 by [PERSON] [2], who employed a sequence of polynomials and principles from probability theory to establish his result. Bernstein polynomials are widely used for plenty of reasons, including their simplicity in differentiation and integration, their value in solving a range of problems, their clarity and simplicity in representation, and their numerous shape-preserving properties. The study of the approximation properties of linear positive operators is a key area in approximation theory. Following the emergence of the uniform convergence of operator sequences to a continuous function on a finite closed interval, the concept of linear positive operator gained considerable significance. In 1950, [PERSON] [3] proposed operators for a function \(f\) defined on positive semi-axis and subjected to an optimal growth condition. Research into the approximation properties of operators defined on infinite intervals, alongside those on compact intervals, grew as the study of linear positive operators in approximation theory advanced. [PERSON] [4] extended the Bernstein operators from a finite interval to an infinite interval, defining a sequence of linear positive operators as follows:\[S_{n}(h;x)=e^{-nx}\sum_{m=0}^{\infty}\frac{(nx)^{m}}{m!}h\Big{(}\frac{m}{n}\Big{)},\ \ x\in[0,\infty),n\in\mathbb{N}, \tag{1}\]  
This converges to \(h(x)\) as \(n\to\infty\) when the function is continuous. Many researchers have studied various generalizations of the Szasz operators and identified their approximation characteristics. One can refer to publications such as [5; 6; 7; 8; 9; 10]. The initial study of approximation properties of complex Bernstein polynomials was conducted by [PERSON] [11]. Recently, [PERSON] (see [12; 13]) provided calculations regarding the convergence result of these polynomials to an analytic function in closed disks. Although the growing interest in complex variable operators exists (see, e.g., [14; 15; 16; 17]), bicomplex Bernstein-type operators have not yet been deeply investigated. [PERSON] [18] was the first to introduce the concept of a bicomplex number in 1892, with the aim of representing physical problems in a four-dimensional natural world. An in-depth examination of the set of bicomplex numbers, which serves as a suitable generalization of complex numbers and hyperbolic numbers, can be found in [19; 20; 21; 22; 23]. On the other hand, bicomplex algebra--two-dimensional Clifford algebra--satisfying the commutative rule of multiplication on \(\mathbb{C}\), plays a significant role in a broad range of disciplines, including theoretical physics, digital image processing, and geometry (for more, please, see [24; 20; 25; 26; 27; 28; 29]). [PERSON] has contributed a cult work in [25], revealing the connections between bicomplex algebra and function theory. Moreover, bicomplex functions are holomorphic, that is, \"differentiable\" functions. Since the operations of bicomplex algebra are isomorphic to those of complex algebra, bicomplex functions are also isomorphic to complex holomorphic functions. However, it is important to note that there are also singular numbers other than zero in bicomplex algebra that have no inverse. In fact, the fields of bicomplex theory and bicomplex algebra are growing rapidly thanks to interesting results regarding approximation theory. For instance, in [30], [PERSON] et al. introduced the bicomplex Bernstein operators and explored their approximation properties.