Approximation by Bicomplex Favard–Szász–Mirakjan Operators

Abstract

The aim of this paper is to consider bicomplex Favard–Szász–Mirakjan operators and study some approximation properties on a compact ${\mathbb{C}}_2$ disk. We provide quantitative estimates of the convergence. Moreover, the Voronovskaja-type results for analytic functions and the simultaneous approximation by bicomplex Favard–Szász–Mirakjan operators are investigated.

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, Weierstrass [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 Weierstrass theorem was later proven in 1912 by Bernstein [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, Szász [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. Mirakjan [4] extended the Bernstein operators from a finite interval to an infinite interval, defining a sequence of linear positive operators as follows:

$$S_n(h;x)=e^{-nx}\sum _{m=0}^{\infty }{\displaystyle \frac{{\left(nx\right)}^m}{m!}}h\left({\displaystyle \frac{m}{n}}\right),x\in [0,\infty ),n\in \mathbb{N},$$

(1)

This converges to $h\left(x\right)$ as $n\to \infty$ when the function is continuous. Many researchers have studied various generalizations of the Szász 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 Lorenz [11]. Recently, Gal (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.

Segre [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 [20,24,25,26,27,28,29]). Rönn 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], Cattani et al. introduced the bicomplex Bernstein operators and explored their approximation properties.

In this paper, our primary contribution is to present novel results regarding the bicomplex Favard–Szász–Mirakjan operators within the frame of approximation theory. We have established fundamental inequalities satisfied by these operators, along with a Voronovskaja-type theorem for analytic functions derived from them. Our work bridges the gap between classical approximation theory and bicomplex analysis. More interestingly, it provides a motivational foundation for further investigation of linear positive operators in hypercomplex domains.

This paper includes four sections. In Section 1, we provide an introduction and literature survey regarding the brief history of the development of approximation theory, as well as Bernstein and Bernstein-type polynomials such as Favard–Szász–Mirakjan operators. An elegant motivation regarding bicomplex numbers and their relationships with other similar types of numbers in the corpus of numerical data is also presented. In Section 2, an extensive use of the principles of the concept of a bicomplex number and numerical operations acting on the space of the bicomplex numbers, as well as their important properties, is made. Section 3 is a cornerstone of the current work. In this context, we first define a bicomplex extended version of the Favard–Szász–Mirakjan operators and calculate an upper estimate for them, as well as their derivatives. Also, a Voronovskaja-type approximation result is investigated. In Section 4, we finalize the paper, summarizing our main contributions. Additionally, we mention that our findings have potential to open up new avenues in the intersection of approximation theory and bicomplex analysis.

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:

$$x_1+i_1{x}_2+i_2{x}_3+i_1{i}_2{x}_4;x_1,\dots ,x_4\in \mathbb{R},$$

(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

$$0,1,e_1={\displaystyle \frac{1+i_1{i}_2}{2}},e_2={\displaystyle \frac{1-i_1{i}_2}{2}},$$

(3)

and they have the following properties:

(i) $e_1{e}_2=0$;

(ii) $\left|\right|e_1\left|\right|=\left|\right|e_2\left|\right|={\displaystyle \frac{\sqrt{2}}{2}}$;

(iii) ${\left(e_1\right)}^2=e_1,$${\left(e_2\right)}^2=e_2$.

Moreover, each element $(t_1+i_2{t}_2)$ in ${\mathbb{C}}_2$ is defined as the idempotent representation

$$t_1+i_2{t}_2=(t_1-i_1{t}_2)e_1+(t_1+i_1{t}_2)e_2,$$

(4)

(please, see, [20]) and let $\left|\right|.\left|\right|$ represent the norm of each elements in ${\mathbb{C}}_2$; then,

$$\left|\right|t_1+i_2{t}_2\left|\right|={\left({\displaystyle \frac{|t_1-i_1{t}_2{|}^2+{|t_1+i_1{t}_2|}^2}{2}}\right)}^{{\displaystyle \frac{1}{2}}}.$$

(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]). A Cartesian set $\mathbb{E}$ in ${\mathbb{C}}_2$ is constructed from the sets ${\mathbb{E}}_1$ in $S_1=\left\{t_1-i_1{t}_2:t_1,t_2\in \mathbb{C}\right\}$ and ${\mathbb{E}}_2$ in $S_2=\left\{t_1+i_1{t}_2:t_1,t_2\in \mathbb{C}\right\}$ as follows:

$$\mathbb{E}=\left\{(t_1+i_2{t}_2)\in {\mathbb{C}}_2:t_1+i_2{t}_2=(t_1-i_1{t}_2)e_1+(t_1+i_1{t}_2)e_2\right\}$$

(6)

for $(t_1-i_1{t}_2,t_1+i_1{t}_2)\in {\mathbb{E}}_1\times {\mathbb{E}}_2.$ Suppose that $a=(\alpha +i_2\beta )$ for $(\alpha ,\beta )\in \mathbb{C}$. If

$${\mathbb{E}}_1=\{(t_1-i_1{t}_2)\in S_1:|(t_1-i_1{t}_2)-(\alpha -i_1\beta )|<r_1\}$$

and

$${\mathbb{E}}_2=\{(t_1+i_1{t}_2)\in S_2:|(t_1+i_1{t}_2)-(\alpha +i_1\beta )|<r_2\}$$

then the set $\mathbb{E}$ is referred to as an open discus with center a and radii $r_1$ and $r_2$, denoted by $\mathbf{D}(a;r_1,r_2).$ If

$${\mathbb{E}}_1=\{(t_1-i_1{t}_2)\in S_1:|(t_1-i_1{t}_2)-(\alpha -i_1\beta )|\le r_1\}$$

and

$${\mathbb{E}}_2=\{(t_1+i_1{t}_2)\in S_2:|(t_1+i_1{t}_2)-(\alpha +i_1\beta )|\le r_2\}$$

then the set $\mathbb{E}$ is referred to as a closed discus with center a and radii $r_1$ and $r_2$, denoted by $\overline{\mathbf{D}}(a;r_1,r_2)$. Moreover,

$$\begin{array}{ccc}\hfill \mathbf{D}(a,r_1,r_2)& =& \left\{(t_1+i_2{t}_2)\in {\mathbb{C}}_2:\right.\hfill \\ & & t_1+i_2{t}_2=(t_1-i_1{t}_2)e_1+(t_1+i_1{t}_2)e_2,\hfill \\ & & \left.|(t_1-i_1{t}_2)-(\alpha -i_1\beta )|<r_1,|(t_1+i_1{t}_2)-(\alpha +i_1\beta )|<r_2\right\}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \overline{\mathbf{D}}(a;r_1,r_2)& =& \left\{(t_1+i_2{t}_2)\in {\mathbb{C}}_2:\right.\hfill \\ & & t_1+i_2{t}_2=(t_1-i_1{t}_2)e_1+(t_1+i_1{t}_2)e_2,\hfill \\ & & \left.|(t_1-i_1{t}_2)-(\alpha -i_1\beta )|\le r_1,|(t_1+i_1{t}_2)-(\alpha +i_1\beta )|\le r_2\right\}.\hfill \end{array}$$

If $\mathbf{D}$ is an open discus in ${\mathbb{C}}_2$ and we take the Cartesian product of two disks with centers at $t_1+i_2{t}_2$ and radii $r_1$ and $r_2$ in $\mathbb{C}$, then it is defined by the following:

$$\mathbf{D}(a,r_1,r_2)=\mathbf{B}(\alpha -i_1\beta ;r_1)\times \mathbf{B}(\alpha +i_1\beta ;r_2)$$

and

$$\overline{\mathbf{D}}(a;r_1,r_2)=\overline{\mathbf{B}(\alpha -i_1\beta ;r_1)}\times \overline{\mathbf{B}(\alpha +i_1\beta ;r_2)}\subset \overline{\mathbf{D}(a;r_1,r_2)}.$$

Let $a=0$; thus, we can write

$$\begin{array}{ccc}\hfill \mathbf{D}(0;r_1,r_2)& =& \mathbf{B}(0,r_1)\times \mathbf{B}(0,r_2)\hfill \\ & =& \left\{t_1+i_2{t}_2\in {\mathbb{C}}_2:\right.\hfill \\ & & t_1+i_2{t}_2=(t_1-i_1{t}_2)e_1+(t_1+i_1{t}_2)e_2,\hfill \\ & & \left.|t_1-i_1{t}_2|\le r_1,|t_1+i_1{t}_2|\le r_2\right\}.\hfill \end{array}$$

We also point the reader to [20] for more detail.

Theorem 1

(see [20]). The complex power series is obtained as the idempotent component series of the bicomplex power series for all $k\in \mathbb{N}$, where ${\sum }_{k=0}^{\infty }(s_k+i_2{v}_k){(t_1+i_2{t}_2)}^k$ with $s_k,v_k,t_1,t_2\in \mathbb{C},$ defined as

$$\begin{array}{ccc}\hfill \sum _{k=0}^{\infty }(s_k+i_2{v}_k){(t_1+i_2{t}_2)}^k& =& \left[\sum _{k=0}^{\infty }(s_k-i_1{v}_k){(t_1-i_1{t}_2)}^k\right]e_1\hfill \\ & & +\left[\sum _{k=0}^{\infty }(s_k+i_1{v}_k){(t_1+i_1{t}_2)}^k\right]e_2.\hfill \end{array}$$

(7)

Moreover, suppose a function g is defined by a power series as follows:

$$g(t_1+i_2{t}_2)=\sum _{k=0}^{\infty }(s_k+i_2{v}_k){(t_1+i_2{t}_2)}^k,$$

(8)

for $(t_1+i_2{t}_2)\in \mathbf{B}(0,r_1)\times \mathbf{B}(0,r_2)$ and $r_1>0,r_2>0$.

Define two complex-valued functions, $g_1(t_1-i_1{t}_2)$ and $g_2(t_1+i_1{t}_2)\in \mathbb{C}$, where $t_1$ and $t_2$ are complex variables. The functions are defined on the circles of convergence of the idempotent component power series. It is assumed that $r_1>0$ and $r_2>0$ represent the radii of convergence for the power series. The expression $(t_1+i_2{t}_2)$ is used as the argument for the functions. The idempotent power series is as follows:

$$\begin{array}{ccc}\hfill g_1(t_1-i_1{t}_2)& =& \sum _{k=0}^{\infty }(s_k-i_1{v}_k){(t_1-i_1{t}_2)}^k,|t_1-i_1{t}_2|<r_1,\hfill \\ \hfill g_2(t_1+i_1{t}_2)& =& \sum _{k=0}^{\infty }(s_k+i_1{v}_k){(t_1+i_1{t}_2)}^k,|t_1+i_1{t}_2|<r_2.\hfill \end{array}$$

For this case,

$$g(t_1+i_2{t}_2)=g_1(t_1-i_1{t}_2)e_1+g_2(t_1+i_1{t}_2)e_2.$$

(9)

Also, for $r_1>0,$ the norm $\left|\right|g_1{\left|\right|}_{r_1}$ is defined by

$$\left|\right|g_1{\left|\right|}_{r_1}=sup\left\{|g_1(t_1-i_1{t}_2)|;|t_1-i_1{t}_2|<r_1\right\}.$$

A similar norm definition applies to the function $g_2.$

3. Construction of Bicomplex Favard–Szász–Mirakjan Operators

In this section, we aim to introduce the approximation properties of Favard–Szász–Mirakjan operators for bicomplex numbers. We begin by defining bicomplex Szász–Mirakjan operators and obtaining a quantitative upper estimate for these operators and their derivatives on a compact ${\mathbb{C}}_2$ disk. Later, we shall prove a qualitative Voronovskaja-type result for these operators. In other words, our main goal is to extend the concept of Szász–Mirakjan operators to a bicomplex version of themselves, allowing for the approximation of functions defined on a compact interval.

Let $g:\mathbf{B}(0,r_1)\times \mathbf{B}(0,r_2)⟶{\mathbb{C}}_2$ be a bicomplex-valued function; then, the bicomplex Szász–Mirakjan operator $K_n(.)$ (n is a positive integer) of g is defined as

$$\left(K_{n}g\right)(t_1+i_2{t}_2)=e^{-n(t_1+i_2{t}_2)}\sum _{k=0}^{\infty }{\displaystyle \frac{{\left(n(t_1+i_2{t}_2)\right)}^k}{k!}}g\left({\displaystyle \frac{k}{n}}\right),$$

(10)

where $k=0,1,2\dots$ and for $t_1+i_2{t}_2\in {\mathbb{C}}_2$.

Analysis of Approximation Properties

Lemma 1.

Let $e_k(t_1+i_2{t}_2)={(t_1+i_2{t}_2)}^k$ and $T_{n,k}(t_1+i_2{t}_2)=(K_n,e_k)(t_1+i_2{t}_2)$ be for $k=0,1,2\dots$; then, the recurrence formula is defined as

$$\begin{array}{ccc}\hfill T_{n,k+1}(t_1+i_2{t}_2)& =& {\displaystyle \frac{(t_1+i_2{t}_2)}{n}}{D}_{t_1+i_2{t}_2}\left(T_{n,k}\right)(t_1+i_2{t}_2)\hfill \\ & & +(t_1+i_2{t}_2)T_{n,k}(t_1+i_2{t}_2),\hfill \end{array}$$

(11)

for all $n\in \mathbb{N}$, $(t_1+i_2{t}_2)\in$${\mathbb{C}}_2$.

Proof of Lemma 1.

Let us write that $T_{n,0}(t_1+i_2{t}_2)=1,$$T_{n,1}(t_1+i_2{t}_2)=t_1+i_2{t}_2$ for $k=0,1$.

Furthermore, for any $k,$ we have

$$\begin{array}{cc}\hfill T_{n,k}\left(t_1+i_2{t}_2\right)& =K_n(e_k;t_1+i_2{t}_2)\hfill \\ \hfill & =\left(e^{-n(t_1-i_1{t}_2)}\sum _{j=0}^{\infty }{{\displaystyle \frac{\left(n\left(t_1-i_1{t}_2\right)\right)}{j!}}}^j{\left({\displaystyle \frac{j}{n}}\right)}^k\right)e_1\hfill \\ \hfill & +\left(e^{-n(t_1+i_1{t}_2)}\sum _{j=0}^{\infty }{{\displaystyle \frac{\left(n\left(t_1+i_1{t}_2\right)\right)}{j!}}}^j{\left({\displaystyle \frac{j}{n}}\right)}^k\right)e_2.\hfill \end{array}$$

Accordingly, differentiating $T_{n,k}\left(t_1+i_2{t}_2\right)$ with respect to $t_1+i_2{t}_2\ne 0,$ we have

$$\begin{array}{cc}\hfill D_{t_1+i_2{t}_2}\left(T_{n,k}\right)\left(t_1+i_2{t}_2\right)& =D_{t_1-i_1{t}_2}\left[\left(e^{-n(t_1-i_1{t}_2)}\sum _{j=0}^{\infty }{{\displaystyle \frac{\left(n\left(t_1-i_1{t}_2\right)\right)}{j!}}}^j{\left({\displaystyle \frac{j}{n}}\right)}^k\right)e_1\right]\hfill \\ \hfill & +D_{t_1+i_1{t}_2}\left[\left(e^{-n(t_1+i_1{t}_2)}\sum _{j=0}^{\infty }{{\displaystyle \frac{\left(n\left(t_1+i_1{t}_2\right)\right)}{j!}}}^j{\left({\displaystyle \frac{j}{n}}\right)}^k\right)e_2\right]\hfill \end{array}$$

for $k=0,1,\dots .$ Employing the basic calculations, we obtain the following result:

$$\begin{array}{cc}\hfill D_{t_1+i_2{t}_2}\left(T_{n,k}\right)\left(t_1+i_2{t}_2\right)& =\left(-n{e}^{-n(t_1-i_1{t}_2)}\sum _{j=0}^{\infty }{{\displaystyle \frac{\left(n\left(t_1-i_1{t}_2\right)\right)}{j!}}}^j{\left({\displaystyle \frac{j}{n}}\right)}^k\right.\hfill \\ \hfill & \left.+e^{-n(t_1-i_1{t}_2)}\sum _{j=0}^{\infty }{{\displaystyle \frac{jn\left(n\left(t_1-i_1{t}_2\right)\right)}{j!}}}^{j-1}{\left({\displaystyle \frac{j}{n}}\right)}^k\right)e_1\hfill \\ \hfill & +\left(-n{e}^{-n(t_1+i_1{t}_2)}\sum _{j=0}^{\infty }{{\displaystyle \frac{\left(n\left(t_1+i_1{t}_2\right)\right)}{j!}}}^j{\left({\displaystyle \frac{j}{n}}\right)}^k\right.\hfill \\ \hfill & \left.+e^{-n(t_1+i_1{t}_2)}\sum _{j=0}^{\infty }{{\displaystyle \frac{jn\left(n\left(t_1+i_1{t}_2\right)\right)}{j!}}}^{j-1}{\left({\displaystyle \frac{j}{n}}\right)}^k\right)e_2,\hfill \end{array}$$

for $n\in \mathbb{N}$. As a conclusion, we can write that

$$\begin{array}{cc}\hfill D_{t_1+i_2{t}_2}\left(T_{n,k}\right)\left(t_1+i_2{t}_2\right)& =\left(-n{T}_{n,k}(t_1-i_1{t}_2)+\left({\displaystyle \frac{n}{t_1-i_1{t}_2}}\right)T_{n,k+1}(t_1-i_1{t}_2)\right)e_1\hfill \\ \hfill & \left(-n{T}_{n,k}(t_1+i_1{t}_2)+\left({\displaystyle \frac{n}{t_1+i_1{t}_2}}\right)T_{n,k+1}(t_1+i_1{t}_2)\right)e_2,\hfill \end{array}$$

which implies

$$\begin{array}{ccc}\hfill T_{n,k+1}(t_1+i_2{t}_2)& =& \left({\displaystyle \frac{t_1+i_2{t}_2}{n}}\right)D_{t_1+i_2{t}_2}\left(T_{n,k}\right)\left(t_1+i_2{t}_2\right)\hfill \\ & & +\left(t_1+i_2{t}_2\right)T_{n,k}\left(t_1+i_2{t}_2\right),\hfill \end{array}$$

for all $(t_1+i_2{t}_2)\in \mathbf{B}(0,R_1)\times \mathbf{B}(0,R_2)$ with $1<R_1,R_2<\infty$, $k=0,1,\dots$ and $n\in \mathbb{N}$. □

Lemma 2.

Let us define $e_k$ and $T_{n,k}$ as stated in Lemma 1. Also, let

$$E_{n,k}(t_1+i_2{t}_2)=T_{n,k}(t_1+i_2{t}_2)-e_k(t_1+i_2{t}_2)-{\displaystyle \frac{{(t_1+i_2{t}_2)}^{k}k(k-1)}{n}}$$

Then, it yields the following argument:

$$\left|E_{n,k}\left(t_1+i_2{t}_2\right)\right|\le {\displaystyle \frac{3(k+1)!}{n^2}}(r_1^{k-2}+r_2^{k-2}).$$

Proof of Lemma 2.

For all $n\in \mathbb{N}$ and $(t_1+i_2{t}_2)\in {\mathbb{C}}_2$, by the recurrence relationship in Lemma 1 satisfied by $T_{n,k}(t_1+i_2{t}_2),$ for all $k\ge 2,$ we obtain the new recurrence

$$\begin{array}{ccc}\hfill E_{n,k}\left(t_1+i_2{t}_2\right)& =& {\displaystyle \frac{t_1+i_2{t}_2}{n}}{D}_{t_1+i_2{t}_2}{E}_{n,k-1}\left(t_1+i_2{t}_2\right)\hfill \\ & & +\left(t_1+i_2{t}_2\right)E_{n,k-1}\left(t_1+i_2{t}_2\right)\hfill \\ & & +{\left(t_1+i_2{t}_2\right)}^{k-2}{\displaystyle \frac{(k-1){(k-2)}^2}{2n^2}},\hfill \end{array}$$

and $n\in \mathbb{N}.$ Also, the idempotent representation of $E_{n,k}\left(t_1+i_2{t}_2\right)$ is

$$\begin{array}{c}\hfill E_{n,k}\left(t_1+i_2{t}_2\right)=E_{n,k}\left(t_1-i_1{t}_2\right)e_1+E_{n,k}\left(t_1+i_1{t}_2\right)e_2\end{array}$$

for all $\left|t_1-i_1{t}_2\right|<r_1$, $\left|t_1+i_1{t}_2\right|<r_2$ and $n\in \mathbb{N}$. Thus, we obtain

$$\begin{array}{ccc}\hfill \left|E_{n,k}\left(t_1-i_1{t}_2\right)\right|& \le & {\displaystyle \frac{r_1}{2n}}{∥2D_{t_1-i_1{t}_2}{E}_{n,k-1}∥}_{r_1}+r_1\left|E_{n,k-1}\left(t_1-i_1{t}_2\right)\right|\hfill \\ & & +r_1^{k-2}{\displaystyle \frac{(k-1){(k-2)}^2}{2n^2}},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \left|E_{n,k}\left(t_1+i_1{t}_2\right)\right|& \le & {\displaystyle \frac{r_2}{2n}}{∥2D_{t_1+i_1{t}_2}{E}_{n,k-1}∥}_{r_2}+r_2\left|E_{n,k-1}\left(t_1+i_1{t}_2\right)\right|\hfill \\ & & +r_2^{k-2}{\displaystyle \frac{(k-1){(k-2)}^2}{2n^2}}.\hfill \end{array}$$

Now, we first investigate the inequality $\left|E_{n,k}\left(t_1-i_1{t}_2\right)\right|$. According to Bernstein’s inequality ${∥D_{t_1-i_1{t}_2}{E}_{n,k-1}∥}_{r_1}\le {\displaystyle \frac{\left(k-1\right)}{r_1}}{∥E_{n,k-1}∥}_{r_1}$ (see p. 3, Theorem 1.0.7 in [12]), we evidently have

$$\begin{array}{ccc}\hfill \left|E_{n,k}\left(t_1-i_1{t}_2\right)\right|& \le & r_1\left|E_{n,k-1}\left(t_1-i_1{t}_2\right)\right|+{\displaystyle \frac{r_1}{2n}}\left\{{\displaystyle \frac{2(k-1)}{r_1}}{∥T_{n,k-1}-e_{k-1}∥}_{r_1}\right.\hfill \\ & & \left.+r_1^{k-3}{\displaystyle \frac{{(k-1)}^2(k-2)}{n}}+r_1^{k-3}{\displaystyle \frac{(k-1){(k-2)}^2}{n}}\right\}\hfill \end{array}$$

and for $k\ge 2$ and $n\in \mathbb{N}$. Thus, we can write

$$\left|E_{n,k}\left(t_1-i_1{t}_2\right)\right|\le r_1\left|E_{n,k-1}\left(t_1-i_1{t}_2\right)\right|+{\displaystyle \frac{3(k+1)!}{2n^2}}{r}_1^{k-2}.$$

Taking $k=2,3,\dots$ in last inequality, step by step, we obtain

$$\left|E_{n,k}\left(t_1-i_1{t}_2\right)\right|\le {\displaystyle \frac{3r_1^{k-2}}{2n^2}}\sum _{j=3}^{k+1}j!.$$

In a similar way,

$$\begin{array}{ccc}\hfill \left|E_{n,k}\left(t_1+i_1{t}_2\right)\right|& \le & {\displaystyle \frac{3r_2^{k-2}}{2n^2}}\sum _{j=3}^{k+1}j!\hfill \end{array}$$

is found. Clearly, we earn

$$\begin{array}{ccc}\hfill \left|E_{n,k}\left(t_1+i_2{t}_2\right)\right|& \le & {\displaystyle \frac{3r_1^{k-2}}{2n^2}}\sum _{j=3}^{k+1}j!+{\displaystyle \frac{3r_2^{k-2}}{2n^2}}\sum _{j=3}^{k+1}j!\hfill \\ & \le & {\displaystyle \frac{3(k+1)!}{n^2}}(r_1^{k-2}+r_2^{k-2}),\hfill \end{array}$$

for $(t_1+i_2{t}_2)\in \mathbf{B}(0,R_1)\times \mathbf{B}(0,R_2)$. □

Our first main result is the following theorem for the upper bound:

Theorem 2.

Assume that $1<R_1,R_2<\infty$ and $g:\mathbf{B}(0,R_1)\times \mathbf{B}(0,R_2)\to {\mathbb{C}}_2$ is analytic in $\mathbf{B}(0,R_1)\times \mathbf{B}(0,R_2)$. Then, for all $(t_1+i_2{t}_2)\in \mathbf{B}(0,R_1)\times \mathbf{B}(0,R_2)$,

$$g(t_1+i_2{t}_2)=\sum _{k=0}^{\infty }\left(c_k+i_2{d}_k\right){(t_1+i_2{t}_2)}^k$$

is satisfied. Let $M>0$ and $S_1\in ({\displaystyle \frac{1}{R_1}},1),$$S_2\in ({\displaystyle \frac{1}{R_2}},1)$ be given numbers and suppose that

$$\left|g(t_1+i_2{t}_2)\right|\le M(e^{S_1\left|t_1-i_1{t}_2\right|}+e^{S_2\left|t_1+i_1{t}_2\right|}).$$

Then, the following statements hold:

$\left(i\right)$ Let $r_1,$$r_2$ be arbitrary constants such that $1\le r_1<{\displaystyle \frac{1}{S_1}}$ and $1\le r_2<{\displaystyle \frac{1}{S_2}},$ with $\left|c_k-i_1{d}_k\right|\le M{\displaystyle \frac{S_1^k}{k!}}$ and $\left|c_k+i_1{d}_k\right|\le M{\displaystyle \frac{S_2^k}{k!}}$. Then, we have

$$∥K_n\left(g\right)-g∥\le {\displaystyle \frac{A_{r_1,S_1}\left(g\right)}{n}}+{\displaystyle \frac{A_{r_2,S_2}\left(g\right)}{n}}$$

for all $\left|t_1-i_1{t}_2\right|<r_1$, $\left|t_1+i_1{t}_2\right|<r_2$ and $n\in \mathbb{N}$.

$\left(ii\right)$ Let $r_1^{*}$, $r_2^{*}$ be any constants satisfying $1\le r_1<r_1^{*}<{\displaystyle \frac{1}{S_1}}$ and $1\le r_2<r_2^{*}<{\displaystyle \frac{1}{S_2}}$ for all $\left|t_1-i_1{t}_2\right|<r_1$, $\left|t_1+i_1{t}_2\right|<r_2$ and $n,p\in \mathbb{N}$; then, we have

$$∥D_{t_1+i_2{t}_2}^p{K}_n\left(g\right)-D_{t_1+i_2{t}_2}^{p}g∥\le {\displaystyle \frac{p!}{n}}{\displaystyle \frac{A_{r_1,S_1}\left(g\right)r_1^{*}}{{(r_1^{*}-r_1)}^{p+1}}}+{\displaystyle \frac{A_{r_2,S_2}\left(g\right)r_2^{*}}{{(r_2^{*}-r_2)}^{p+1}}},$$

for the simultaneous approximation by bicomplex Favard–Szász–Mirakjan operators, where

$$A_{r_1,S_1}\left(g\right)={\displaystyle \frac{M}{2r_1}}\sum _{k=2}^{\infty }(k+1){\left(r_1{S}_1\right)}^k<\infty$$

and

$$A_{r_2,S_2}\left(g\right)={\displaystyle \frac{M}{2r_2}}\sum _{k=2}^{\infty }(k+1){\left(r_2{S}_2\right)}^k<\infty .$$

Proof of Theorem 2.

$\left(i\right)$ Denoting $e_k\left(t_1+i_2{t}_2\right)={\left(t_1+i_2{t}_2\right)}^k,$$T_{n,k}\left(t_1+i_2{t}_2\right)=K_n(e_k;t_1+i_2{t}_2)$ for $k=0,1\dots$ and $\mathbf{B}(0,R_1)\times \mathbf{B}(0,R_2)=\left\{t_1+i_2{t}_2\in {\mathbb{C}}_2:t_1+i_2{t}_2=\left(t_1-i_1{t}_2\right)e_1\right.$ $\left.+\left(t_1+i_1{t}_2\right)e_2,\left|t_1-i_1{t}_2\right|<R_1,\left|t_1+i_1{t}_2\right|<R_2\right\}$.

We obtain that $T_{n,k}$ ($k=0,1\dots$) is a polynomial of degree $\le k$ and for all $(t_1+i_2{t}_2)\in {\mathbb{C}}_2.$

Now, as in (6) by using the idempotent representation of $g(t_1+i_2{t}_2)$, we obtain the inequality below. If we use the “iteration relation” (please, see [12]), then

$$K_n(g,t_1+i_2{t}_2)=\sum _{k=0}^{\infty }\left(c_k+i_2{d}_k\right)K_n(e_k;t_1+i_2{t}_2).$$

For $r_1\ge 1,$$r_2\ge 1$, let $\overline{\mathbf{D}}\left(0;r_1,r_2\right)=\left\{t_1+i_2{t}_2\in {\mathbb{C}}_2:t_1+i_2{t}_2=\left(t_1-i_1{t}_2\right)e_1\right.\left.+\left(t_1+i_1{t}_2\right)e_2,\left|t_1-i_1{t}_2\right|\le r_1,\left|t_1+i_1{t}_2\right|\le r_2\right\}$. Given that $K_n(g;t_1+i_2{t}_2)$ is analytic in $\mathbf{B}(0,R_1)\times \mathbf{B}(0,R_2),$ we obtain

$$\begin{array}{ccc}\hfill ∥K_n\left(g\right)-g∥& \le & \sqrt{2}\left(\sum _{k=2}^{\infty }\left|c_k-i_1{d}_k\right|{∥T_{n,k}-e_k∥}_{r_1}\right.\hfill \\ & & \left.+\sum _{k=2}^{\infty }\left|c_k+i_1{d}_k\right|{∥T_{n,k}-e_k∥}_{r_2}\right).\hfill \end{array}$$

In a nutshell, it is necessary to have an estimate for ${∥T_{n,k}-e_k∥}_{r_1}$ and ${∥T_{n,k}-e_k∥}_{r_2}.$

Now, in Equation (11), replacing k by $k-1$, we can write

$$\begin{array}{ccc}\hfill T_{n,k}(t_1+i_2{t}_2)& =& \left({\displaystyle \frac{t_1+i_2{t}_2}{n}}\right)D_{t_1+i_2{t}_2}\left(T_{n,k-1}\right)\left(t_1+i_2{t}_2\right)\hfill \\ & & +\left(t_1+i_2{t}_2\right)T_{n,k-1}\left(t_1+i_2{t}_2\right).\hfill \end{array}$$

So, we obtain

$$\begin{array}{cc}\hfill T_{n,k}\left(t_1+i_2{t}_2\right)-{\left(t_1+i_2{t}_2\right)}^k& =\left({\displaystyle \frac{t_1+i_2{t}_2}{n}}\right)\hfill \\ \hfill & \times D_{t_1+i_2{t}_2}\left(T_{n,k-1}\left(t_1+i_2{t}_2\right)-{\left(t_1+i_2{t}_2\right)}^{k-1}\right)\hfill \\ \hfill & +{\displaystyle \frac{k-1}{n}}{\left(t_1+i_2{t}_2\right)}^{k-1}\hfill \\ \hfill & +\left(t_1+i_2{t}_2\right)\left(T_{n,k-1}\left(t_1+i_2{t}_2\right)-{\left(t_1+i_2{t}_2\right)}^{k-1}\right).\hfill \end{array}$$

Based on the findings of Bernstein’s inequality (see [12,13,14,15,16]) within the closed unit disk and through an appropriate linear transformation, if $P_k$ denotes an algebraic polynomial of a degree at most k while also taking into account the Formula (7), we can write that ${∥D_{t_1-i_1{t}_2}{P}_k∥}_{r_1}\le {\displaystyle \frac{k}{r_1}}{∥P_k∥}_{r_1}$, ${∥D_{t_1+i_1{t}_2}{P}_k∥}_{r_2}\le {\displaystyle \frac{k}{r_2}}{∥P_k∥}_{r_2}$, so we achieve

$$\begin{array}{ccc}\hfill {∥T_{n,k}-e_k∥}_{r_1}& \le & {\displaystyle \frac{r_1}{n}}{\displaystyle \frac{k-1}{r_1}}{∥T_{n,k-1}-e_{k-1}∥}_{r_1}\hfill \\ & & +{\displaystyle \frac{k-1}{n}}{r}_1^{k-1}\hfill \\ & & +r_1{∥T_{n,k-1}-e_{k-1}∥}_{r_1},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {∥T_{n,k}-e_k∥}_{r_2}& \le & {\displaystyle \frac{r_2}{n}}{\displaystyle \frac{k-1}{r_2}}{∥T_{n,k-1}-e_{k-1}∥}_{r_2}\hfill \\ & & +{\displaystyle \frac{k-1}{n}}{r}_2^{k-1}\hfill \\ & & +r_2{∥T_{n,k-1}-e_{k-1}∥}_{r_2}\hfill \end{array}$$

for all $\left|t_1-i_1{t}_2\right|<r_1$ and $\left|t_1+i_1{t}_2\right|<r_2$. Therefore, we have

$$\begin{array}{ccc}\hfill {∥T_{n,k}-e_k∥}_{r_1}& \le & \left(r_1+{\displaystyle \frac{k-1}{n}}\right){∥T_{n,k-1}-e_{k-1}∥}_{r_1}\hfill \\ & & +{\displaystyle \frac{k-1}{n}}{r}_1^{k-1},\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill {∥T_{n,k}-e_k∥}_{r_2}& \le & \left(r_2+{\displaystyle \frac{k-1}{n}}\right){∥T_{n,k-1}-e_{k-1}∥}_{r_2}\hfill \\ & & +{\displaystyle \frac{k-1}{n}}{r}_2^{k-1}.\hfill \end{array}$$

Now, let $n\ge 1$ be an arbitrary constant. Employing the mathematical induction on k, we prove that this recurrence leads to the following inequalities:

$${∥T_{n,k}-e_k∥}_{r_1}\le {\displaystyle \frac{(k+1)!}{2n}}{r}_1^{k-1},$$

(12)

and

$${∥T_{n,k}-e_k∥}_{r_2}\le {\displaystyle \frac{(k+1)!}{2n}}{r}_2^{k-1}$$

(13)

for all $k\ge 2,$$n\ge 1.$ Indeed, given the first inequality above for $k=2$ and $n\ge 1$, the left-hand sides emerge as ${\displaystyle \frac{r_1}{n}}$ and ${\displaystyle \frac{r_2}{n}}$, while the right-hand sides are ${\displaystyle \frac{3r_1}{n}}$ and ${\displaystyle \frac{3r_2}{n}}.$ Assuming that the first inequality holds for k, we proceed to establish the result for $k+1$. Thus, we can express

$${∥T_{n,k+1}-e_{k+1}∥}_{r_1}\le \left(r_1+{\displaystyle \frac{k}{n}}\right){\displaystyle \frac{(k+1)!}{2n}}{r}_1^{k-1}+{\displaystyle \frac{k}{n}}{r}_1.$$

It remains to prove that

$$\left(r_1+{\displaystyle \frac{k}{n}}\right){\displaystyle \frac{(k+1)!}{2n}}{r}_1^{k-1}+{\displaystyle \frac{k}{n}}{r}_1\le {\displaystyle \frac{(k+2)!}{2n}}{r}_1^k$$

or we can write

$$\left(r_1+{\displaystyle \frac{k}{n}}\right)(k+1)!+2k{r}_1\le (k+2)!r_1.$$

It is easy to see that this last inequality holds true for all $k\ge 2,$$n\ge 1.$ We can use a similar method for ${∥T_{n,k}-e_k∥}_{r_2}$.

This immediately implies the estimates for both ${∥T_{n,k}-e_k∥}_{r_1}$ and ${∥T_{n,k}-e_k∥}_{r_2}$ for all $k=0,1,\dots$ and $n\in \mathbb{N}.$

Note that from this hypothesis, $g(t_1+i_2{t}_2)={\sum }_{k=0}^{\infty }\left(c_k+i_2{d}_k\right){(t_1+i_2{t}_2)}^k$ is absolutely and uniformly convergent in $|t_1-i_1{t}_2|<r_1$ and $|t_1+i_1{t}_2|<r_2$, respectively. Then, for all $1\le r_1<{\displaystyle \frac{1}{S_1}}$, $1\le r_2<{\displaystyle \frac{1}{S_2}}$ and $S_1\in ({\displaystyle \frac{1}{R_1}},1),$$S_2\in ({\displaystyle \frac{1}{R_2}},1)$, the power series on the right hand side can be differentiated twice, i.e.,

$$\begin{array}{ccc}\hfill ∥K_n\left(g\right)-g∥& \le & {\displaystyle \frac{M}{2n{r}_1}}\sum _{k=2}^{\infty }(k+1){\left(r_1{S}_1\right)}^k+{\displaystyle \frac{M}{2n{r}_2}}\sum _{k=2}^{\infty }(k+1){\left(r_2{S}_2\right)}^k\hfill \\ & =& {\displaystyle \frac{A_{r_1,S_1}\left(g\right)}{n}}+{\displaystyle \frac{A_{r_2,S_2}\left(g\right)}{n}},\hfill \end{array}$$

where $\left|c_k-i_1{d}_k\right|\le M{\displaystyle \frac{S_1^k}{k!}},$$\left|c_k+i_1{d}_k\right|\le M{\displaystyle \frac{S_2^k}{k!}}$, and

$$A_{r_1,S_1}\left(g\right)={\displaystyle \frac{M}{2r_1}}\sum _{k=2}^{\infty }(k+1){\left(r_1{S}_1\right)}^k<\infty ,A_{r_2,S_2}\left(g\right)={\displaystyle \frac{M}{2r_2}}\sum _{k=2}^{\infty }(k+1){\left(r_2{S}_2\right)}^k<\infty ,$$

by taking into account the series ${\displaystyle \sum _{k=2}^{\infty }}{\left(r_1{S}_1\right)}^{k+1},$${\displaystyle \sum _{k=2}^{\infty }}{\left(r_2{S}_2\right)}^{k+1}$, meaning their derivatives ${\displaystyle \sum _{k=2}^{\infty }}(k+1){\left(r_1{S}_1\right)}^k,$${\displaystyle \sum _{k=2}^{\infty }}(k+1){\left(r_2{S}_2\right)}^k$ are uniformly and absolutely convergent.

$\left(ii\right)$ By using the Cauchy derivation formula (see [12,13,16]), we prove the required results as follows:

For the radii $r_1^{*}>r_1$, $r_2^{*}>r_2$ and $t_1+i_2{t}_2=\left(t_1-i_1{t}_2\right)e_1+\left(t_1+i_1{t}_2\right)e_2$, if $\eta =\mathbf{B}(0,r_1^{*})\times \mathbf{B}(0,r_2^{*})$ is an open ball with zero center, where $\left|t_1-i_1{t}_2\right|<r_1$ and $\left|t_1+i_1{t}_2\right|<r_2$, then

$$\begin{array}{cc}\hfill ∥\left({\eta }_1+i_2{\eta }_2\right)-\left(t_1+i_2{t}_2\right)∥& \ge ∥{\eta }_1-i_1{\eta }_2-\left(t_1-i_1{t}_2\right)∥-∥{\eta }_1+i_1{\eta }_2-\left(t_1+i_1{t}_2\right)∥\hfill \\ \hfill & \ge \left(r_1^{*}-r_1\right)-\left(r_2^{*}-r_2\right)\hfill \end{array}$$

is obtained for $\left({\eta }_1+i_2{\eta }_2\right)\in \eta$

$$\begin{array}{cc}\hfill & D_{t_1+i_2{t}_2}^p\left(K_{n}g\right)\left(t_1+i_2{t}_2\right)-D_{t_1+i_2{t}_2}^{p}g\left(t_1+i_2{t}_2\right)\hfill \\ \hfill & ={\displaystyle \frac{p!}{2\pi i_1}}{\int }_{{\eta }_1}{\displaystyle \frac{\left[\left(K_n{g}_1\right)\left({\eta }_1-i_1{\eta }_2\right)-g_1\left({\eta }_1-i_1{\eta }_2\right)\right]}{{\left[\left({\eta }_1-i_1{\eta }_2\right)-\left(t_1-i_1{t}_2\right)\right]}^{p+1}}}d\left({\eta }_1-i_1{\eta }_2\right)e_1\hfill \\ \hfill & +{\displaystyle \frac{p!}{2\pi i_1}}{\int }_{{\eta }_2}{\displaystyle \frac{\left[\left(K_n{g}_2\right)\left({\eta }_1+i_1{\eta }_2\right)-g_2\left({\eta }_1+i_1{\eta }_2\right)\right]}{{\left[\left({\eta }_1+i_1{\eta }_2\right)-\left(t_1+i_1{t}_2\right)\right]}^{p+1}}}d\left({\eta }_1+i_1{\eta }_2\right)e_2,\hfill \end{array}$$

This implies that

$$\begin{array}{cc}\hfill & ∥D_{t_1+i_2{t}_2}^p\left(K_{n}g\right)-D_{t_1+i_2{t}_2}^p\left(g\right)∥\hfill \\ \hfill & \le {\displaystyle \frac{p!}{n}}\left\{{\displaystyle \frac{A_{r_1,S_1}\left(g\right)r_1^{*}}{{\left(r_1^{*}-r_1\right)}^{p+1}}}+{\displaystyle \frac{A_{r_2,S_2}\left(g\right)r_2^{*}}{{\left(r_2^{*}-r_2\right)}^{p+1}}}\right\}.\hfill \end{array}$$

Now, the proof is completed. □

Finally, we present the next theorem, which provides the Voronovskaja-type approximation result by making use of Theorem 2.

Theorem 3.

Let $r_1,$$r_2$ be arbitrary constants satisfying the inequalities $1\le r_1<{\displaystyle \frac{1}{S_1}}$ and $1\le r_2<{\displaystyle \frac{1}{S_2}}$. Under the hypotheses of Theorem 2, for all $\left|t_1-i_1{t}_2\right|<r_1$, $\left|t_1+i_1{t}_2\right|<r_2$ and $n\in \mathbb{N}$, the following Voronovskaja-type result is valid:

$$\begin{array}{ccc}& & \left|K_n(g;t_1+i_2{t}_2)-g(t_1+i_2{t}_2)-{\displaystyle \frac{t_1+i_2{t}_2}{2n}}{D}_{t_1+i_2{t}_2}^2\left(g\right)\right|\hfill \\ & \le & {\displaystyle \frac{3M{S}_1}{n^2{r}_1}}\sum _{k=2}^{\infty }(k+1){\left(r_1{S}_1\right)}^{k-1}+{\displaystyle \frac{3M{S}_2}{n^2{r}_2}}\sum _{k=2}^{\infty }(k+1){\left(r_2{S}_2\right)}^{k-1}\hfill \end{array}$$

Proof of Theorem 3.

Denoting $e_k\left(t_1+i_2{t}_2\right)={\left(t_1+i_2{t}_2\right)}^k$ and $T_{n,k}(t_1+i_2{t}_2)=K_n(e_k;t_1+i_2{t}_2)$ for $k=0,1,\dots ,$ by the proof of Theorem 2 (i), we can write $K_n(g;t_1+i_2{t}_2)={\sum }_{k=0}^{\infty }(c_k+i_2{d}_k)T_{n,k}(t_1+i_2{t}_2)$, which immediately implies that

$$\begin{array}{ccc}& & \left|K_n(g;t_1+i_2{t}_2)-g(t_1+i_2{t}_2)-{\displaystyle \frac{t_1+i_2{t}_2}{2n}}{D}_{t_1+i_2{t}_2}^2\left(g\right)\right|\hfill \\ & \le & \sum _{k=2}^{\infty }\left|c_k+i_2{d}_k\right|\left|T_{n,k}(t_1+i_2{t}_2)-e_k\left(t_1+i_2{t}_2\right)-{\displaystyle \frac{k(k-1)}{2n}}{(t_1+i_2{t}_2)}^{k-1}\right|\hfill \\ & =& \sum _{k=2}^{\infty }\left|c_k+i_2{d}_k\right|\left|E_{n,k}(t_1+i_2{t}_2)\right|\hfill \end{array}$$

for all $(t_1+i_2{t}_2)\in \mathbf{B}(0,R_1)\times \mathbf{B}(0,R_2)$ and $n\in \mathbb{N}.$ As a consequence, from Lemma 1, we can write

$$\begin{array}{ccc}\hfill \left|K_n(f;t_1+i_2{t}_2)-g(t_1+i_2{t}_2)-{\displaystyle \frac{t_1+i_2{t}_2}{2n}}{D}_{t_1+i_2{t}_2}^2\left(g\right)\right|& \le & {\displaystyle \frac{3M{S}_1}{n^2{r}_1}}\sum _{k=2}^{\infty }(k+1){\left(r_1{S}_1\right)}^{k-1}\hfill \\ & & +{\displaystyle \frac{3M{S}_2}{n^2{r}_2}}\sum _{k=2}^{\infty }(k+1){\left(r_2{S}_2\right)}^{k-1}\hfill \end{array}$$

for all $\left|t_1-i_1{t}_2\right|<r_1$, $\left|t_1+i_1{t}_2\right|<r_2$ and $n\in \mathbb{N},$ where for $r_1{S}_1<1$ and $r_2{S}_2<1.$ Obviously, we have ${\displaystyle \sum _{k=2}^{\infty }}(k+1){\left(r_1{S}_1\right)}^{k-1}<\infty ,$${\displaystyle \sum _{k=2}^{\infty }}(k+1){\left(r_2{S}_2\right)}^{k-1}<\infty .$ □

The precise order of approximation can be derived as follows:

Corollary 1.

Suppose that $r_1,r_2>1$, $f:\mathbf{B}(0,r_1)\times \mathbf{B}(0,r_2)\to {\mathbb{C}}_2$ is analytic in $\mathbf{B}(0,r_1)\times \mathbf{B}(0,r_2).$ If f is not a polynomial of a degree at most $\le 1$, then for each $r_1^{*}\in [1,r_1)$ and $r_2^{*}\in [1,r_2)$, the following relations hold:

$${∥S_n\left(f\right)-f∥}_{r_1^{*}}\sim {\displaystyle \frac{1}{n}}\mathit{and}{∥S_n\left(f\right)-f∥}_{r_2^{*}}\sim {\displaystyle \frac{1}{n}}$$

such that $n\in \mathbb{N}$.

4. Conclusions and Future Directions

Recently, bicomplex numbers and bicomplex analysis have attracted increasing attention, particularly in the fields of mathematics and physics. Motivated by the theoretical advantages offered by this area, the present work extends the related framework and introduces certain generalizations to approximation theory. To this end, we have employed bicomplex Favard–Szász–Mirakjan operators, defined analogously to the classical complex Favard–Szász–Mirakjan operators. In other words, this article can be considered a study that allows algebraic structures to interact with the analytical behavior of mathematical arguments. As part of future research, similar approaches may be incorporated within the scope of other suitable polynomials or operators to obtain improved approximation results.

The definitions and results presented in this study are considered within the framework of bicomplex spaces. Naturally, this raises the question of whether these structures and findings can be extended to higher-dimensional multicomplex spaces. However, transitioning to higher dimensions may involve considerable structural and analytical challenges. Thus, exploring such generalizations in a rigorous and comprehensive manner remains an important open problem for future research.

In addition to pure mathematics, the theoretical perspective developed in our work may provide new insights into areas where hypercomplex and complex numbers find application, such as signal processing, Hopfield neural networks, electromagnetic theory, quantum mechanics, etc.

There is undoubtedly still a lot to be done with convenient linear and positive operators, especially with respect to their approximation results and bicomplex modifications. Moreover, by extending the current theoretical background, interesting properties of bicomplex operators such as univalence, starlikeness, convexity, and spirallikeness may also be investigated.