# Approximation by Bicomplex Favard-Szasz-Mirakjan Operators
## 2 Theoretical Background on Bicomplex Numbers and Polynomials  
In this section, the necessary background on operations with bicomplex numbers and their associated polynomials are provided. Additionally, the fundamental definitions and theorems related to bicomplex numbers and bicomplex polynomials shall be discussed. These foundational details shall serve as a basis for bicomplex analysis theory and its approximation results in the subsequent sections of this study. Primarily, several fundamental notations \(\mathbb{R}\), \(\mathbb{C}\), and \(\mathbb{C}_{2}\) are used for the spaces of the real, complex, and bicomplex numbers, respectively. As well-known, an element from the space \(\mathbb{C}\) is denoted by the expression \(x_{1}+i_{1}x_{2}\) with \(x_{1},x_{2}\in\mathbb{R}\) such that \(i_{1}^{2}=-1.\) The basis \(1,\ i_{1},\ i_{2}\), and \(i_{1}i_{2}\) are used to write an element from the space \(\mathbb{C}_{2}\) as follows:  
\[x_{1}+i_{1}x_{2}+i_{2}x_{3}+i_{1}i_{2}x_{4};\ \ x_{1},\ldots,x_{4}\in\mathbb{R}, \tag{2}\]  
where \(i_{1},i_{2}\), and \(i_{1}i_{2}\) satisfy the properties \(i_{1}^{2}=-1,i_{2}^{2}=-1\) and \(i_{1}i_{2}=i_{2}i_{1}\). The space of the bicomplex numbers, denoted by \(\mathbb{C}_{2}\), is an algebraic function over the field of real numbers \(\mathbb{R}\) with a dimension of four. This can be observed from Equation (2), which shows that since \(\mathbb{C}\) has a dimension of two over \(\mathbb{R}\), it follows that \(\mathbb{C}_{2}\), consisting of bicomplex numbers, forms an algebraic structure over \(\mathbb{R}\) with a dimension of four. Building on this, if Equation (2) is rearranged in terms of complex numbers as \(t_{1}=x_{1}+i_{1}x_{2}\) and \(t_{2}=x_{3}+i_{1}x_{4}\), we can then formulate \(t_{1}+i_{2}t_{2}\) with \(t_{1},t_{2}\in\mathbb{C}\), \(i_{2}^{2}=-1\). In addition, the concept of an idempotent element refers to an element that remains unchanged when operated upon by a particular operation. There are four idempotent elements in \(\mathbb{C}_{2}\), denoted by  
\[0,1,e_{1}=\frac{1+i_{1}i_{2}}{2},e_{2}=\frac{1-i_{1}i_{2}}{2}, \tag{3}\]  
and they have the following properties:  
(i) \(e_{1}e_{2}=0\) ;  
(ii) \(||e_{1}||=||e_{2}||=\frac{\sqrt{2}}{2}\);  
(iii) \((e_{1})^{2}=e_{1}\), \((e_{2})^{2}=e_{2}\). Moreover, each element \((t_{1}+i_{2}t_{2})\) in \(\mathbb{C}_{2}\) is defined as the idempotent representation  
\[t_{1}+i_{2}t_{2}=(t_{1}-i_{1}t_{2})e_{1}+(t_{1}+i_{1}t_{2})e_{2}, \tag{4}\]  
(please, see, [20]) and let \(||.||\) represent the norm of each elements in \(\mathbb{C}_{2}\); then,  
\[||t_{1}+i_{2}t_{2}||=\bigg{(}\frac{|t_{1}-i_{1}t_{2}|^{2}+|t_{1}+i_{1}t_{2}|^{ 2}}{2}\bigg{)}^{\frac{1}{2}}. \tag{5}\]  
In addition, we aim to guide the reader toward foundational works on bicomplex numbers, bicomplex analysis, and related topics. So far, we have provided an introductory understanding of bicomplex numbers and offered a concise overview of the existing literature on bicomplex studies [14; 18; 24]. Nevertheless, it is essential to emphasize the key difference \(\mathbb{C}\) and \(\mathbb{C}_{2}\); whereas complex numbers constitute a field, bicomplex numbers do not due to the presence of zero divisors. **Definition 1** (see [20]).