On the Approximations and Symmetric Properties of Frobenius–Euler–Şimşek Polynomials Connecting Szász Operators

Abstract

This study focuses on approximating continuous functions using Frobenius–Euler–Simsek polynomial analogues of Szász operators. Test functions and central moments are computed to study convergence uniformly, approximation order by these operators. Next, we investigate approximation order uniform convergence via Korovkin result and the modulus of smoothness for functions in continuous functional spaces. A Voronovskaja theorem is also explored approximating functions which belongs to the class of function having first and second order continuous derivative. Further, we discuss numerical error and graphical analysis. In the last, two dimensional operators are constructed to discuss approximation for the class of two variable continuous functions.

1. Introduction and Preliminaries

Bernstein (1912) [1] introduced a sequence of polynomials to present the simplest proof of an elegant theorem due to Weierstrass (1885) [2] which is termed as the Weierstrass approximation theorem in terms of the binomial probability distribution, as follows:

$$\begin{array}{c}\hfill {\mathbf{B}}_m(\hslash ;\overline{y})=\sum _{\mu =0}^m{}_{m,\mu }\left(\overline{y}\right)\hslash \left({\displaystyle \frac{\mu }{m}}\right),\mathit{m}\in \mathbb{N},\end{array}$$

(1)

Define ${}_{m,\mu }\left(\overline{y}\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{m}{\mu }}\right){\overline{y}}^{\mu }{(1-\overline{y})}^{m-\mu }$ for $\overline{y}\in [0,1]$. It is well established that the sequence of operators ${\mathbf{B}}_m(\hslash ;·)$ converges uniformly to ℏ on $[0,1]$ for every bounded function ℏ, i.e., ${\mathbf{B}}_m(\hslash ;·)\rightrightarrows \hslash$. In recent developments, operator theory has emerged as a powerful tool, establishing strong interdisciplinary connections across various fields of science, including engineering and medical sciences. Notable applications can be found in areas such as robotics, computer-aided geometric design (CAGD), and the modeling of diseases like HIV, among others [3,4,5,6,7,8]. Over the past decade, numerous mathematicians have proposed various modifications of the operators defined in (1) in order to enhance their flexibility in approximation properties on both bounded and unbounded intervals within different functional spaces, for example, Özger et al. [9,10], Aslan et al. [11,12], Ayman et al. [13,14], Mohiuddine et al. [15,16], Mursaleen et al. [17,18], Khan et al. [19], Nasiruzzaman [20], Braha et al. [21], Rao et al. [22,23,24], Çetin et al. [25,26]. Numerous researchers have developed significant literature in the field of approximation theory, such as [27,28,29,30,31,32,33,34,35,36,37,38,39,40,41]. In the context of polynomial classes, which constitute an active area of research within the broader field of special functions, we recall a class of polynomials introduced by Simsek [42], known as the Frobenius Simsek Euler type polynomials andnumbers denoted by ${\left\{l_{\mu }(y;v)\right\}}_{\mu =0}^{\infty }$. Frobenius–Euler polynomials possess rich symmetric properties [43], often expressed through binomial expansions and translation identities. These symmetries simplify computations and reveal deep structural relationships among the polynomials. Such properties have important applications in combinatorics, number theory, and special functions connected with the generating function, such as:

$$F_l(\overline{y};u,v):={\displaystyle \frac{u^v}{{\displaystyle \prod _{j=0}^{v-1}}(e^u-j)}}{e}^{uy}={\displaystyle \sum _{\mu =0}^{\infty }}{l}_{\mu }(y;v){\displaystyle \frac{u^{\mu }}{\mu !}}.$$

(2)

As $\overline{y}=0$, one yields Frobenious–Simsek–Euler–type numbers as follows:

$${\displaystyle \frac{u^v}{{\displaystyle \prod _{j=0}^{v-1}}(e^u-j)}}=F_l(0;u,v)=\sum _{\mu =0}^{\infty }{\displaystyle \frac{u^{\mu }}{\mu !}}{l}_{\mu }(0;v).$$

Setting $v=2$ in above relation, one obtains the following:

$$F_l(\overline{y};u,2):={\displaystyle \frac{u^2}{e^u(e^u-1)}}{e}^{u\overline{y}}=\sum _{\mu =0}^{\infty }{\displaystyle \frac{u^{\mu }}{\mu !}}{l}_{\mu }(\overline{y};2).$$

(3)

For detailed information and application regarding Frobenius–Euler–Simsek-type numbers, see [44,45]. Motivated by the aforementioned developments, We propose a new formulation of Szász–Chlodowsky operators incorporating Frobenius–Şimşek–Euler–type numbers, defined as:

$${Ł}_s(\hslash ;\overline{y})=e^{-{\displaystyle \frac{s}{c_s}}\overline{y}}(e^2-e)\sum _{\mu =0}^{\infty }{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}y;2\right)}{\mu !}}\hslash \left(\mu {\displaystyle \frac{c_s}{s}}\right), \hslash \in C[0,\infty ),$$

(4)

Lemma 1

([3]). Using (3), we have the followings:

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{du}}{F}_l(\overline{y};u,2)& =& {\displaystyle \frac{u{e}^{u(\overline{y}-1)}(-\overline{y}u+e^u(u\overline{y}-2u+2)+u-2)}{{(e^u-1)}^2}},\hfill \\ \hfill {\displaystyle \frac{d^2}{d{u}^2}}{F}_l(\overline{y};u,2)& =& {\displaystyle \frac{e^{u(\overline{y}-1)}}{{(e^u-1)}^3}}[u^2{(\overline{y}-1)}^2+u{e}^2({(\overline{y}-2)}^2{u}^2+4u(\overline{y}-2)+2)\hfill \\ & +& 2+4u(\overline{y}-1)-e^u(u^2(3+2{\overline{y}}^2-6\overline{y})+4-4u(3-2\overline{y}))].\hfill \end{array}$$

Lemma 2.

Based on the generating function given in (3), we derive the following:

$$\begin{array}{cc}\hfill \sum _{\mu =0}^{\infty }{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}& ={\displaystyle \frac{e^{\frac{s}{c_s}\overline{y}}}{e^2-e}},\hfill \\ \hfill \sum _{\mu =0}^{\infty }\mu {\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}& ={\displaystyle \frac{s\overline{y}{e}^{\frac{s}{c_s}\overline{y}}\overline{y}}{c_s(e^2-e)}}-{\displaystyle \frac{e^{\frac{s}{c_s}\overline{y}}}{e{(e-1)}^2}},\hfill \\ \hfill \sum _{\mu =0}^{\infty }{\mu }^2{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}& ={\left({\displaystyle \frac{s}{c_s}}\right)}^2{\displaystyle \frac{e^{\frac{s}{c_s}\overline{y}}{\overline{y}}^2}{e(1-e)}}+{\displaystyle \frac{2s}{c_s}}{\displaystyle \frac{e^{\frac{s}{c_s}\overline{y}}\overline{y}}{(1-e)}}+{\displaystyle \frac{2e^2-5e+1}{e{(1-e)}^3}}{e}^{{\displaystyle \frac{s}{c_s}}\overline{y}}.\hfill \end{array}$$

Proof. 

Applying Lemma (1), Lemma (2) can be established by taking $w=1$ and replacing y with ${\displaystyle \frac{s}{c_s}}y$. □

Lemma 3.

Let ${\hslash }_j\left(\theta \right)={\theta }^j$, $j=0,1,2$. Then, one yields:

$$\begin{array}{cc}\hfill {Ł}_s({\hslash }_0;\overline{y})& =1,\hfill \\ \hfill {Ł}_s({\hslash }_1,\overline{y})& =\overline{y}-{\displaystyle \frac{c_s}{s}}{\displaystyle \frac{1}{(e-1)}},\hfill \\ \hfill {Ł}_s({\hslash }_2,\overline{y})& ={\overline{y}}^2-2e{\displaystyle \frac{c_s}{s}}\overline{y}+{\left({\displaystyle \frac{c_s}{s}}\right)}^2{\displaystyle \frac{2e^2-5e+1}{{(1-e)}^2}}.\hfill \end{array}$$

Proof. 

To prove Lemma (3) in view of ${Ł}_s(.;.)$ given in (4), we have the following:

$$\begin{array}{c}\hfill {Ł}_s({\hslash }_j;\overline{y})=e(e-1)e^{-{\displaystyle \frac{s}{c_s}}\overline{y}}\sum _{\mu =0}^{\infty }{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}{\left(\mu {\displaystyle \frac{c_s}{s}}\right)}^j.\end{array}$$

(5)

For $j=0$,

$$\begin{array}{c}\hfill {Ł}_s({\hslash }_0;\overline{y})=e(e-1)e^{-{\displaystyle \frac{s}{c_s}}y}\sum _{\mu =0}^{\infty }{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}=1.\end{array}$$

For $j=1$, we yield the following:

$$\begin{array}{c}\hfill {Ł}_s({\hslash }_1;\overline{y})=(e^2-e)e^{-{\displaystyle \frac{s}{c_s}}\overline{y}}\sum _{\mu =0}^{\infty }{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}{\displaystyle \frac{c_s}{s}}.\end{array}$$

In the light of Equation (2), we yield:

$$\begin{array}{cc}\hfill {Ł}_s({\hslash }_1,\overline{y})& =\overline{y}-{\displaystyle \frac{c_s}{s}}{\displaystyle \frac{1}{(e-1)}}.\end{array}$$

For $j=2$,

$$\begin{array}{c}\hfill {Ł}_s({\hslash }_1,\overline{y})=(e^2-e)e^{-{\displaystyle \frac{s}{c_s}}\overline{y}}\sum _{\mu =0}^{\infty }{\displaystyle \frac{{\mu }^2{l}_{\mu }\left(\frac{s}{c_s}y;2\right)}{\mu !}}{\left({\displaystyle \frac{c_s}{s}}\right)}^2.\end{array}$$

Based on Lemma (2), we yield the following:

$$\begin{array}{cc}\hfill {Ł}_s({\hslash }_2,\overline{y})& ={\overline{y}}^2-2e{\displaystyle \frac{c_s}{s}}\overline{y}-{\displaystyle \frac{2e^2-5e+1}{{(e-1)}^2}}{\left({\displaystyle \frac{c_s}{s}}\right)}^2.\end{array}$$

□

Lemma 4.

Let ${\hslash }_j^{\overline{y}}\left(\theta \right)={(\theta -\overline{y})}^j$. Then, for the sequence of operators given by (4), the following properties hold:

$$\begin{array}{ccc}\hfill {Ł}_s({\hslash }_o^{\overline{y}};\overline{y})& =& 1,\hfill \\ \hfill {Ł}_s({\hslash }_1^{\overline{y}};\overline{y})& =& -{\displaystyle \frac{c_s}{s}}{\displaystyle \frac{1}{(e-1)}},\hfill \\ \hfill {Ł}_s({\hslash }_2^{\overline{y}};\overline{y})& =& 2{\displaystyle \frac{c_s}{s}}\left({\displaystyle \frac{1-e^2+e}{(e-1)}}\right)\overline{y}-{\displaystyle \frac{2e^2-5e+1}{{(e-1)}^2}}{\left({\displaystyle \frac{c_s}{s}}\right)}^2.\hfill \end{array}$$

Proof. 

Using (4) and its linearity property, we yield:

$$\begin{array}{ccc}\hfill {Ł}_s({\hslash }_o^{\overline{y}};\overline{y})& =& {Ł}_s(1;\overline{y})=1,\hfill \\ \hfill {Ł}_s({\hslash }_1^{\overline{y}};\overline{y})& =& {Ł}_s(\theta -\overline{y};\overline{y})={Ł}_s({\hslash }_1;\overline{y})-\overline{y}{Ł}_s(1;\overline{y}),\hfill \\ \hfill {Ł}_s({\hslash }_2^{\overline{y}};\overline{y})& =& {Ł}_s({(\theta -\overline{y})}^2;\overline{y})={Ł}_s({\hslash }_2;\overline{y})-2\overline{y}{Ł}_s({\hslash }_1;\overline{y})+{\overline{y}}^2{Ł}_s(1;\overline{y}).\hfill \end{array}$$

In the light of Lemma 3, we arrived required Lemma. □

Remark 1.

The operators introduced in (4) are found as positive, i.e., for $\hslash \ge 0$, ${Ł}_s(\hslash ;\overline{y})\ge 0$.

Remark 2.

The operators introduced in (4) are found as linear, i.e., for every $k_1,k_2\in \mathbb{R}$ and $\hslash ,g\in C[0,\infty )$, we yield the following:

$$\begin{array}{c}\hfill {Ł}_s(k_1\hslash +k_{2}g;\overline{y})=k_1{Ł}_s(\hslash ;\overline{y})+k_2{Ł}_s(g;\overline{y}).\end{array}$$

This manuscript investigates the approximation results for the sequence of operators defined in (4). It addresses several fundamental aspects, including uniform convergence, pointwise convergence, and theorem due Voronovskaja. Furthermore, a bivariate sequence of operators is constructed, and their uniform rate of approximation as well as the order of approximation are analyzed. Finally, the study explores direct approximation results using these operators in various functional spaces.

2. Voronovskaja-Type Theorem and Order of Approximation

Definition 1

(See [46]). Let $\hslash \in C_B[0,\infty )$ (a space of continuous and bounded functions). Then, modulus of smoothness is given as follows:

$$\begin{array}{c}\hfill \omega (\hslash ;\tilde{\delta })=\underset{|{\overline{y}}_1-{\overline{y}}_2|\le \tilde{\delta }}{\sup }|\hslash \left({\overline{y}}_1\right)-\hslash \left({\overline{y}}_2\right)|,{\overline{y}}_1,{\overline{y}}_2\in [0,\infty ),\end{array}$$

(6)

and

$$\begin{array}{c}\hfill |\hslash \left({\overline{y}}_1\right)-\hslash \left({\overline{y}}_2\right)|\le \left(1+{\displaystyle \frac{|{\overline{y}}_1-{\overline{y}}_2|}{\tilde{\delta }}}\right)\omega (\hslash ;\tilde{\delta }).\end{array}$$

(7)

Theorem 1.

Let ${Ł}_s(.;.)$ be introduced in (4) and for all $\hslash \in C_B[0,\infty )$. Then, ${Ł}_s(\hslash ;.)\rightrightarrows \hslash$ on a bounded subset of $[0,\infty )$ where ⇉ depicts that the convergence is uniform.

Proof. 

According to the theorem due to Korovkin [47], it is only required to demonstrate that:

$$\begin{array}{c}\hfill {Ł}_s({\hslash }_j;\overline{y})={\overline{y}}^j, j=0,1,2,\end{array}$$

On each compact subset of $[0,\infty )$ uniformly, and applying Lemma (3), one obtain the required proof of this theorem.

□

The next approximation result involves the investigation of the order of approximation (4) using modulus of smoothness given in (6), as follows:

Theorem 2.

Let $\hslash \in C_B[0,\infty )$ and operators ${Ł}_s(.;.)$ in (4), one has the following:

$$\begin{array}{c}\hfill |{Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right)|\le 2\omega (\hslash ;\tilde{\delta }),\end{array}$$

choosing $\tilde{\delta }=\sqrt{{Ł}_s({\hslash }_2^{\overline{y}};\overline{y})}$.

Proof. 

Using (4), we get

$$\begin{array}{ccc}\hfill |{Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right)|& =& \left|(e^2-e)e^{-{\displaystyle \frac{s}{c_s}}\overline{y}}\sum _{\mu =0}^{\infty }{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}\left\{\hslash \left(\mu {\displaystyle \frac{c_s}{s}}\right)-\hslash \left(\overline{y}\right)\right\}\right|,\hfill \\ & \le & (e^2-e)e^{-{\displaystyle \frac{s}{c_s}}\overline{y}}\sum _{\mu =0}^{\infty }{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}\left|\hslash \left(\mu {\displaystyle \frac{c_s}{s}}\right)-\hslash \left(\overline{y}\right)\right|d\theta \hfill \\ & \le & (e^2-e)e^{-{\displaystyle \frac{s}{c_s}}\overline{y}}\sum _{\overline{y}=0}^{\infty }{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}\left(1+{\displaystyle \frac{|\mu \frac{c_s}{s}-\overline{y}|}{\tilde{\delta }}}\right)\omega (\hslash ;\tilde{\delta })\hfill \\ & \le & \left\{1+{\displaystyle \frac{1}{\tilde{\delta }}}\left(e^{-{\displaystyle \frac{s}{c_s}}\overline{y}}(e^2-e)\sum _{\mu =0}^{\infty }{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}\left|\mu {\displaystyle \frac{c_s}{s}}-\overline{y}\right|\right)\right\}\omega (\hslash ;\tilde{\delta }).\hfill \end{array}$$

By applying the Cauchy–Schwarz inequality, we have

$$\begin{array}{ccc}\hfill |{Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right)|& \le & \{1+{\tilde{\delta }}^{-1}{\left((e^2-e)e^{-{\displaystyle \frac{s}{c_s}}\overline{y}}\sum _{\overline{y}=0}^{\infty }{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \times & {\left(e^{-{\displaystyle \frac{s}{c_s}}\overline{y}}(e^2-e)\sum _{\overline{y}=0}^{\infty }{\displaystyle \frac{l_{\mu }\left(\frac{s}{c_s}\overline{y};2\right)}{\mu !}}{(\mu {\displaystyle \frac{c_s}{s}}-\overline{y})}^2\right)}^{{\displaystyle \frac{1}{2}}}\}\omega (\hslash ;\tilde{\delta })\hfill \\ & \le & \left\{1+{\displaystyle \frac{\sqrt{{Ł}_{s_1,s_2}({\hslash }_2^y;\overline{y})}}{\tilde{\delta }}}\right\}\omega (\hslash ;\tilde{\delta }).\hfill \end{array}$$

On choosing $\tilde{\delta }={Ł}_s({\hslash }_2^y;\overline{y})$, we yield the following:

$$\begin{array}{c}\hfill |{Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right)|\le 2\omega (\hslash ;\tilde{\delta }).\end{array}$$

Thus, we conclude the proof of above Theorem. □

We now state the Voronovskaja result for functions havingfirst and second ordered continuous derivatives, employing the sequences in (4), as:

Theorem 3.

Let $\hslash ,{\hslash }^{\prime },{\hslash }^{″}\in C[0,\infty )\bigcap E=\{\hslash :{\displaystyle \frac{\hslash \left(\overline{y}\right)}{1+y^2}}$ converge as $\overline{y}\to \infty \}$ and $\overline{y}\in [0,\infty ).$ Then, we receive the following:

$$\begin{array}{ccc}\hfill \underset{s\to \infty }{\lim }{\displaystyle \frac{s}{c_s}}({Ł}_s(\hslash \left(\overline{y}\right)-\hslash ;\overline{y})& =& {\hslash }^{\prime }\left(\overline{y}\right)\left({\displaystyle \frac{1}{1-e}}\right)+\left({\displaystyle \frac{\overline{y}(e+1)-2}{1-e}}\right){\displaystyle \frac{{\hslash }^{″}\left(\overline{y}\right)}{2!}}\overline{y}.\hfill \end{array}$$

Proof. 

For function approximation, first, we recall the expansion of the Taylor series:

$$\begin{array}{c}\hfill \hslash \left(\theta \right)=\hslash \left(\overline{y}\right)+{\hslash }^{\prime }\left(\overline{y}\right)(\theta -\overline{y})+{\hslash }^{″}\left(\overline{y}\right){\displaystyle \frac{{(\theta -\overline{y})}^2}{2!}}+{(\theta -\overline{y})}^2\xi (\theta ,\overline{y}),\end{array}$$

(8)

where $\xi (\theta ,\overline{y})$ is a remainder term and $\xi (\theta ,\overline{y})\in C[0,\infty )\cap E$ with ${\lim }_{\theta \to \overline{y}}\xi (\theta ,\overline{y})=0$. We apply ${Ł}_s(.;.)$ given in Equations (4)

$$\begin{array}{c}\hfill {Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right)={\hslash }^{\prime }\left(\overline{y}\right){Ł}_s({\hslash }_1^{\overline{y}};\overline{y})+{\displaystyle \frac{{\hslash }^{″}}{2!}}{Ł}_s({\hslash }_2^{\overline{y}};\overline{y})+{Ł}_s({(\theta -\overline{y})}^2\xi (\theta ,\overline{y});\overline{y}).\end{array}$$

(9)

Taking limits both the sides (9), we get following result:

$$\begin{array}{ccc}\hfill \underset{r\to \infty }{\lim }{\displaystyle \frac{s}{c_s}}({Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right))& =& {\hslash }^{\prime }\left(\overline{y}\right)\underset{s\to \infty }{\lim }{\displaystyle \frac{s}{c_s}}{Ł}_s({\hslash }_1^{\overline{y}};\overline{y})+{\displaystyle \frac{{\hslash }^{″}}{2!}}\underset{r\to \infty }{\lim }{\displaystyle \frac{s}{c_s}}{Ł}_s({\hslash }_2^{\overline{y}};\overline{y})\hfill \\ & +& \underset{s\to \infty }{\lim }{\displaystyle \frac{s}{c_s}}{Ł}_s(\xi (\theta ,\overline{y}){(\theta -\overline{y})}^2;\overline{y})\hfill \\ & =& {\hslash }^{\prime }\left(\overline{y}\right)\left({\displaystyle \frac{1}{e-1}}\right)\hfill \\ & +& {\displaystyle \frac{{\hslash }^{″}\left(\overline{y}\right)}{2!}}\left({\displaystyle \frac{\overline{y}(e+1)-2}{e-1}}\right)\overline{y}\hfill \\ & +& \underset{r\to \infty }{\lim }{\displaystyle \frac{s}{c_s}}{Ł}_s(\xi (\theta ,\overline{y}){(\theta -\overline{y})}^2;\overline{y}).\hfill \end{array}$$

(10)

Using the inequality due to Cauchy–Schwarz inequality:

$$\begin{array}{c}\hfill {\displaystyle \frac{s}{c_s}}{Ł}_s({(\theta -\overline{y})}^2\xi (\theta ,\overline{y});\overline{y})\le \sqrt{{\left({\displaystyle \frac{s}{c_s}}\right)}^2{Ł}_s({(\theta -\overline{y})}^4;\overline{y}){Ł}_s({\xi }^2(\theta ,\overline{y});\overline{y})}.\end{array}$$

(11)

Using Equations (10) and (11), Lemma (4), and ${\lim }_{s\to \infty }{Ł}_s({\xi }^2(\theta ,\overline{y});\overline{y})=0$, we derive the following:

$$\begin{array}{ccc}\hfill \underset{c_s\to \infty }{\lim }{\displaystyle \frac{s}{c_s}}({Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right))& =& \left({\displaystyle \frac{1}{e-1}}\right){\hslash }^{\prime }\left(\overline{y}\right)\hfill \\ & +& \left({\displaystyle \frac{\overline{y}(e+1)-2}{e-1}}\right){\displaystyle \frac{{\hslash }^{″}\left(\overline{y}\right)}{2!}}\overline{y}.\hfill \end{array}$$

Hence we arrived the desired result. □

3. Convergence Analysis via Numerical and Graphical Methods

This section, we focus on examination of the convergence behavior of the operators defined in (4). The analysis is conducted for different values of m, specifically $s=100$, $s=120$, and $s=140$. The convergence behavior is visualized in Figure 1, which highlights the impact of increasing m on the operator’s performance. Furthermore, the approximation error for these cases is illustrated in Figure 2, providing a clear depiction of the error trends.

To quantitatively evaluate the convergence, we compute the numerical errors using the following formula:

$$K_s(\hslash ;\overline{y})=\left|\hslash \left(\overline{y}\right){-}_s(\hslash ;\overline{y})\right|,$$

(12)

where $\hslash \left(\overline{y}\right)$ is defined as follows:

$$\hslash \left(\overline{y}\right)=10{\overline{y}}^3+4{\overline{y}}^2-\overline{y}+1.$$

(13)

The computed numerical errors are presented in

Table 1. The approximation error of the operators $K_s(\mathit{\hslash };\overline{y})$ with respect to $\mathit{\hslash }\left(\overline{y}\right)$.

$\overline{\mathit{y}}$	$|{\mathit{k}}_{\mathit{s}}(\mathit{\hslash };\overline{\mathit{y}})-\mathit{\hslash }\left(\overline{\mathit{y}}\right)|$	$|{\mathit{k}}_{\mathit{s}}(\mathit{\hslash };\overline{\mathit{y}})-\mathit{\hslash }\left(\overline{\mathit{y}}\right)\left|\right)$	$|{\mathit{k}}_{\mathit{s}}(\mathit{\hslash };\overline{\mathit{y}})-\mathit{\hslash }\left(\overline{\mathit{y}}\right)|$
0.1	2.29412	2.11053	1.98317
0.2	2.68187	2.46853	2.30031
0.3	2.68187	2.91136	2.70543
0.4	3.17009	3.43292	3.18315
0.5	4.40275	4.03316	3.73363
0.6	5.16538	4.73213	4.37807
0.7	6.07566	5.57675	5.16583
0.8	7.21082	6.64978	6.18452
0.9	8.69213	8.08191	7.57317

. These errors are crucial for understanding the approximation accuracy of the operators for different values of m and demonstrate the rate of convergence as m increases.

Significance of Numerical Analysis

The numerical analysis confirms the theoretical results and shows how the operators behave in practice. By calculating the errors for various y-values, we can understand how well the operators approximate the target function $\hslash \left(\overline{y}\right)$. This is important for applications where accuracy is key.

4. Results on Local Approximation

Let us recall the definition of the Lipschitz-type space [48], given by:

$$\begin{array}{c}\hfill Li{p}_{\tilde{M}}^{{\phi }_1,{\phi }_2}\left(\tau \right):=\left\{\hslash \in {\tilde{C}}_{\tilde{B}}[0,\infty ):|\hslash \left(t\right)-\hslash \left(\overline{y}\right)|\le {\displaystyle \frac{|t-\overline{y}{|}^{\tau }}{{(t+{\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2)}^{\frac{\tau }{2}}}}\tilde{M}:\overline{y},t\in (0,\infty )\right\},\end{array}$$

where $0<\tilde{M}$, $0<\tau \le 1$ and ${\phi }_1,{\phi }_2>0$.

Theorem 4.

Let ${Ł}_s(.;.)$ be the sequences of the operator given by (4). Then, for $\hslash \in Li{p}_M^{{\phi }_1,{\phi }_2}\left(\tau \right)$, one has the following:

$$\begin{array}{c}\hfill |{Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right)|\le \tilde{M}{\left({\displaystyle \frac{\lambda \left(\overline{y}\right)}{{\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2}}\right)}^{{\displaystyle \frac{\tau }{2}}},\end{array}$$

(14)

choosing $0<\tau \le 1$, ${\phi }_1,{\phi }_2\in (0,\infty )$ and $\lambda \left(\overline{y}\right)={Ł}_s({\eta }_2;\overline{y})$.

Proof. 

Given $\tau =1$ and $\overline{y}\ge 0$, one yields the following:

$$\begin{array}{cc}\hfill |{Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right)|& \le {Ł}_s\left(\right|\hslash \left(t\right)-\hslash \left(\overline{y}\right)|;\overline{y})\hfill \\ \hfill & \le \tilde{M}{Ł}_s\left({\displaystyle \frac{|t-\overline{y}|}{{(t+{\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2)}^{\frac{1}{2}}}};\overline{y}\right).\hfill \end{array}$$

Since ${\displaystyle \frac{1}{t+{\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2}<\frac{1}{{\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2}}$, for each $\overline{y}\in (0,\infty )$, one has

$$\begin{array}{cc}\hfill |{Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right)|& \le {\displaystyle \frac{\tilde{M}}{{({\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2)}^{\frac{1}{2}}}}{\left({Ł}_s({\eta }_2;\overline{y})\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le \tilde{M}{\left({\displaystyle \frac{\lambda \left(\overline{y}\right)}{{\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2}}\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

This means that Theorem 4 holds for $\tau =1$. Next, for $\tau \in (0,1)$, using Hölder’s inequality with $p={\displaystyle \frac{2}{\tau }}$ and $q={\displaystyle \frac{2}{2-\tau }}$, we get the following:

$$\begin{array}{cc}\hfill |{Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right)|& \le {\left({Ł}_s\left(\right|\hslash \left(t\right)-\hslash \left(\overline{y}\right){|}^{{\displaystyle \frac{2}{\tau }}};y)\right)}^{{\displaystyle \frac{\tau }{2}}}\hfill \\ \hfill & \le \tilde{M}{\left({Ł}_s\left({\displaystyle \frac{|t-\overline{y}{|}^2}{(t+{\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2)}};\overline{y}\right)\right)}^{{\displaystyle \frac{\tau }{2}}}.\hfill \end{array}$$

Since ${\displaystyle \frac{1}{t+{\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2}<\frac{1}{{\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2}}$, for all $\overline{y}\in (0,\infty )$, we yields

$$\begin{array}{cc}\hfill |{Ł}_s(\hslash ;\overline{y})-\hslash \left(\overline{y}\right)|& \le \tilde{M}{\left({\displaystyle \frac{{Ł}_s\left(\right|t-\overline{y}{|}^2;\overline{y})}{{\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2}}\right)}^{{\displaystyle \frac{\tau }{2}}}\le \tilde{M}{\left({\displaystyle \frac{\lambda \left(\overline{y}\right)}{{\phi }_1\overline{y}+{\phi }_2{\overline{y}}^2}}\right)}^{{\displaystyle \frac{\tau }{2}}}.\hfill \end{array}$$

Thus, we have proved Theorem 4. □

5. A Bivariate Frobenius–Şimşek–Euler Polynomial Representation of Szász Operators

In this part, we construct the bivariate form of the operators described in (4). $T^2=\left\{(w_1,w_2):0\le w_1<\infty ,0\le w_2<\infty \right\},$ And, let $C\left(T^2\right)$ denote the category of functions that are continuous on $T^2,$ equipped with the supremum norm:

$$\begin{array}{c}\hfill {\left|\right|\hslash \parallel }_{C\left(T^2\right)}=\underset{(w_1,w_2)\in T^2}{\sup }|\hslash (w_1,w_2)|.\end{array}$$

For all $\hslash \in C\left(T^2\right)$ and $s_1,s_2\in \mathbb{N}$, we introduce a bivariate version of ${Ł}_m(.;.)$ as follows:

$$\begin{array}{ccc}\hfill {Ł}_{s_1,s_2}(\hslash ;w_1,w_2))& =& (e_1^2-e_1)(e_2^2-e_2)e_1^{-\left({\displaystyle \frac{s_1}{c_{s_1}}}{w}_1+{\displaystyle \frac{s_2}{c_{s_2}}}{w}_2\right)}\hfill \\ & \times & \sum _{{\mu }_1=0}^{\infty }\sum _{{\mu }_2=0}^{\infty }{Q}_{l_1,l_2,{\mu }_1,{\mu }_2}(w_1;w_2)\hslash \left({\mu }_1{\displaystyle \frac{c_{s_1}}{s_1}},{\mu }_2{\displaystyle \frac{c_{s_2}}{s_2}}\right),\hfill \end{array}$$

(15)

where

$$\begin{array}{ccc}\hfill Q_{l_1,l_2,{\mu }_1,{\mu }_2}(w_1;w_2)& =& E_{l_1,{\mu }_1}\left(w_1\right)E_{l_2,{\mu }_2}\left(w_2\right),\hfill \end{array}$$

with

$$\begin{array}{ccc}\hfill E_{l_1,{\mu }_1}\left(w_1\right)& =& {\displaystyle \frac{l_{1{\mu }_1}\left(\frac{s_1}{c_{s_1}}{w}_1;2\right)}{{\mu }_1!}},E_{l_2,{\mu }_2}\left(w_2\right)={\displaystyle \frac{l_{2{\mu }_2}\left(\frac{s_2}{c_{s_2}}{w}_2;2\right)}{{\mu }_2!}},\hfill \end{array}$$

and Let ${\left\{c_{s_i}\right\}}_{i=1}^{\infty }$ be a sequence of positive, increasing real numbers. The sequence satisfies the following conditions:

• $\underset{i\to \infty }{\lim }{c}_{s_i}=\infty$,
• $\underset{i\to \infty }{\lim }{\displaystyle \frac{c_{s_i}}{s_i}}=0$ for $i=1,2,\dots$.

Let $i,j\in \{0,1,2\}$ with $0\le i+j\le 2$. We define the two-dimensional test functions as $e_{i,j}=w_1^i{w}_2^j$, and the central moments as $p_{i,j}^{w_1,w_2}(t_1,t_2)={\eta }_{i,j}(t_1,t_2)={(t_1-w_1)}^i{(t_2-w_2)}^j$.

Lemma 5.

Given $\hslash \in C\left(T^2\right)$, the operator ${Ł}_{s_1,s_2}(.;.)$ from Equation (15), and the test functions $e_{i,j}(.;.)$, we have the following:

$$\begin{array}{ccc}\hfill {Ł}_{s_1,s_2}(e_{0,0};w_1,w_2)& =& 1,\hfill \\ \hfill {Ł}_{s_1,s_2}(e_{1,0};w_1,w_2)& =& w_1+{\displaystyle \frac{c_{s_1}}{s_1}}\left({\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {Ł}_{s_1,s_2}(e_{0,1};w_1,w_2)& =& w_2+{\displaystyle \frac{c_{s_2}}{s_2}}\left({\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {Ł}_{s_1,s_2}(e_{2,0};w_1,w_2)& =& w_1^2+{\displaystyle \frac{c_{s_1}}{s_1}}\left({\displaystyle \frac{e+1}{e-1}}\right)+{\left({\displaystyle \frac{c_{s_1}}{s_1}}\right)}^2\left({\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {Ł}_{s_1,s_2}(e_{0,2};w_1,w_2)& =& w_2^2+{\displaystyle \frac{c_{s_2}}{s_2}}\left({\displaystyle \frac{e+1}{e-1}}\right)+{\left({\displaystyle \frac{c_{s_2}}{s_2}}\right)}^2\left({\displaystyle \frac{1}{e-1}}\right).\hfill \end{array}$$

Proof. 

The above lemma is proved using the linearity property and Lemma (3) as follows:

$$\begin{array}{ccc}\hfill {Ł}_{s_1,s_2}(e_{0,0};w_1,w_2)& =& {Ł}_{s_1,s_2}(e_0;w_1,w_2){Ł}_{s_1,s_2,}(e_0;w_1,w_2),\hfill \\ \hfill {Ł}_{s_1,s_2}(e_{1,0};w_1,w_2)& =& {Ł}_{s_1,s_2}(e_1;w_1,w_2){Ł}_{s_1,s_2}(e_0;w_1,w_2),\hfill \\ \hfill {Ł}_{s_1,s_2}(e_{0,1};w_1,w_2)& =& {Ł}_{s_1,s_2}(e_0;w_1,w_2){Ł}_{s_1,s_2}(e_1;w_1,w_2),\hfill \\ \hfill {Ł}_{s_1,s_2}(e_{2,0};w_1,w_2)& =& {Ł}_{s_1,s_2}(e_2;w_1,w_2){Ł}_{s_1,s_2}(e_0;w_1,w_2),\hfill \\ \hfill {Ł}_{s_1,s_2}(e_{0,2};w_1,w_2)& =& {Ł}_{s_1,s_2}(e_0;w_1,w_2){Ł}_{s_1,s_2}(e_1;w_1,w_2).\hfill \end{array}$$

□

Lemma 6.

For $p_{i,j}={(t_1-w_1)}^i{(t_2-w_2)}^j$ for $i,j\in \{0,1,2\}$ such that $0\le i+j\le 2$, we have the following equalities:

$$\begin{array}{ccc}\hfill {Ł}_{s_1,s_2}(t_{0,0};w_1,w_2)& =& 1,\hfill \\ \hfill {Ł}_{s_1,s_2}(t_{1,0};w_1,w_2)& =& {\displaystyle \frac{c_{s_1}}{s_1}}\left({\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {Ł}_{s_1,s_2}(t_{0,1};w_1,w_2)& =& {\displaystyle \frac{c_{s_2}}{s_2}}\left({\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {Ł}_{s_1,s_2}(t_{2,0};w_1,w_2)& =& {\displaystyle \frac{c_{s_1}}{s_1}}\left({\displaystyle \frac{w_1(e+1)-2}{(e-1)}}\right)+{\left({\displaystyle \frac{c_{s_1}}{s_1}}\right)}^2\left({\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {Ł}_{s_1,s_2}(t_{0,2};w_1,w_2)& =& {\displaystyle \frac{c_{s_2}}{s_2}}\left({\displaystyle \frac{w_2(e+1)-2}{(e-1)}}\right)+{\left({\displaystyle \frac{c_{s_2}}{s_2}}\right)}^2\left({\displaystyle \frac{1}{e-1}}\right).\hfill \end{array}$$

Proof. 

By Lemma 5 and the linearity property, the desired result follows easily. □

6. Error Bounds and Approximation Order

To study the rate of convergence of the operators defined in (15), we make use of the result established by Volkov [49], which offers an effective framework for analyzing the convergence behavior, as outlined below:

Theorem 5.

Consider compact intervals I and J on the real line. Let $(s_1,s_2)\in \mathbb{N}\times \mathbb{N}$, and define ${Ł}_{s_1,s_2}:C(I\times J)\to C(I\times J)$ as linear positive operators.

If

$$\begin{array}{ccc}\hfill \underset{s_1,s_2\to \infty }{\lim }{Ł}_{s_1,s_2}\left(e_{i,j}\right)& =& p_{w_1,w_2},(i,j)\in \{(0,0),(1,0),(0,1)\}.\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \underset{s_1,s_2\to \infty }{\lim }{Ł}_{s_1,s_2}(e_{20}+e_{02})& =& e_{20}+e_{02},\hfill \end{array}$$

uniformly on $T^2$. Then, for any $\hslash \in C\left(T^2\right)$, the sequence $({Ł}_{s_1,s_2}\hslash )$ converges uniformly to ℏ.

Theorem 6.

Let $e_{i,j}(w_1,w_2)=w_1^i{w}_2^j(0\le i+j\le 2,i,j\in \mathbb{N})$ be the test functions restricted on $T^2$. If

$$\begin{array}{c}\hfill \underset{s_1,s_2\to \infty }{\lim }{Ł}_{s_1,s_2}(e_{i,j};w_1,w_2)=e_{i,j}(w_1,w_2)(i,j)\in \{(0,0),(1,0),(0,1)\}.\end{array}$$

and

$$\begin{array}{c}\hfill \underset{s_1,s_2\to \infty }{\lim }{Ł}_{s_1,s_2}(e_{20}+e_{02};w_1,w_2)=e_{20}(w_1,w_2)+e_{02}(w_1,w_2),\end{array}$$

uniformly on $T^2$. Then,

$$\begin{array}{c}\hfill \underset{s_1,s_2\to \infty }{\lim }{Ł}_{s_1,s_2}(\hslash ;w_1,w_2)=\hslash (w_1,w_2),\end{array}$$

uniformly for all $\hslash \in C\left(T^2\right)$.

Proof. 

In the light of Lemma 5, it is evident for $i=j=0$

$$\begin{array}{c}\hfill \underset{s_1,s_2\to \infty }{\lim }{Ł}_{s_1,s_2}(e_{00};w_1,w_2)=e_{00}(w_1,w_2).\end{array}$$

For $i=1$, $j=0$, we obtain the following:

$$\begin{array}{ccc}\hfill \underset{s_1,s_2\to \infty }{\lim }{Ł}_{s_1,s_2}(e_{10};w_1,w_2)& =& w_1,\hfill \\ \hfill \underset{s_1,s_2\to \infty }{\lim }{Ł}_{s_1,s_2}(e_{10};w_1,w_2)& =& e_{10}(w_1,w_2).\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill \underset{s_1,s_2\to \infty }{\lim }{Ł}_{s_1,s_2}(e_{01};w_1,w_2)& =& w_2,\hfill \\ \hfill \underset{s_1,s_2\to \infty }{\lim }{Ł}_{s_1,s_2}(e_{01};w_1,w_2)& =& e_{01}(w_1,w_2),\hfill \end{array}$$

Using Lemma (5), we obtain:

$$\begin{array}{ccc}\hfill \underset{s_1,s_2\to \infty }{\lim }{Ł}_{s_1,s_2}(e_{20}+e_{02};w_1,w_2)& =& w_1^2+w_2^2,\hfill \\ & =& e_{20}(w_1,w_2)+e_{02}(w_1,w_2).\hfill \end{array}$$

In the direction of Theorem 5, we establish the desired result. □

In the final result, we address the order of approximation for the sequence of operators ${Ł}_{s_1,s_2}(.;.)$ defined in (15) as follows:

Theorem 7.

([50]). Let $L:C\left(T^2\right)\to B\left(T^2\right)$ be a linear positive operator. Then, for any $\hslash \in C\left(T^2\right)$, any $(z_1,z_2)\in T^2$, and any ${\tilde{\delta }}_1,{\tilde{\delta }}_2>0$, the following inequality holds:

$$\begin{array}{ccc}\hfill |(L\hslash )(w_1,w_2)-\hslash (w_1,w_2)|& \le & |L{e}_{0,0}(w_1,w_2)-1\left|\right|\hslash (w_1,w_2)|+[L{e}_{0,0}(w_1,w_2)\hfill \\ & +& {\tilde{\delta }}_1^{-1}\sqrt{L{e}_{0,0}(w_1,w_2){\left(L(·-w_1)\right)}^2(w_1,w_2)}\hfill \\ & +& {\tilde{\delta }}_2^{-1}\sqrt{L{e}_{0,0}(w_1,w_2){\left(L(*-w_2)\right)}^2(w_1,w_2)}\hfill \end{array}$$

$$\begin{array}{ccc}& +& {\tilde{\delta }}_1^{-1}{\tilde{\delta }}_2^{-1}\sqrt{{\left(L{e}_{0,0}\right)}^2(w_1,w_2){\left(L(·-w_1)\right)}^2(w_1,w_2){\left(L(*-w_2)\right)}^2(w_1,w_2)}]\hfill \\ & \times & {\omega }_{total}(\hslash ;{\tilde{\delta }}_1,{\tilde{\delta }}_2),\hfill \end{array}$$

Theorem 8.

For $\hslash \in C\left(T^2\right)$ and $(w_1,w_2)\in T^2$, $(s_1,s_2)\in \mathbb{N}\times \mathbb{N}$, and ${\tilde{\delta }}_1,{\tilde{\delta }}_2>0$, one has the following:

$$\begin{array}{c}\hfill |{Ł}_{s_1,s_2}(\hslash ;w_1,w_2)-\hslash (w_1,w_2)|\le 4{\omega }_{total}(\hslash ;{\tilde{\delta }}_1,{\tilde{\delta }}_2),\end{array}$$

where ${\tilde{\delta }}_1=\sqrt{{Ł}_{s_1,s_2}\left({({\theta }_1-w_1)}^2;w_1,w_2\right)}$ and ${\tilde{\delta }}_2=\sqrt{{Ł}_{s_1,s_2}\left({({\theta }_2-w_2)}^2;w_1,w_2)\right)}$.

Proof. 

From Theorem 7, we have the following:

$$\begin{array}{cc}.& |({Ł}_{s_1,s_2}\hslash )(w_1,w_2)-\hslash (w_1,w_2)|\hfill \\ \le & [1++{\tilde{\delta }}_1^{-1}\sqrt{{Ł}_{s_1,s_2}\left({({\theta }_1-w_1)}^2;w_1,w_2\right)}\hfill \\ +& {\tilde{\delta }}_2^{-1}\sqrt{{Ł}_{s_1,s_2}\left({({\theta }_2-w_2)}^2;w_1,w_2\right)}\hfill \\ +& {\tilde{\delta }}_1^{-1}{\tilde{\delta }}_2^{-1}\sqrt{{Ł}_{s_1,s_2}\left({(t_1-w_1)}^2;w_1,w_2\right){Ł}_{s_1,s_2}\left({(t_2-w_2)}^2;w_1,w_2\right)}]\hfill \\ \times & {\omega }_{total}(f;{\tilde{\delta }}_1,{\tilde{\delta }}_2).\hfill \end{array}$$

Selecting ${\tilde{\delta }}_1=\sqrt{{Ł}_{s_1,s_2}\left({(t_1-w_1)}^2;w_1,w_2\right)}$

and ${\tilde{\delta }}_2=\sqrt{{Ł}_{s_1,s_2}\left({(t_2-w_2)}^2;w_1,w_2)\right)}$, we arrive at the required result. □

7. Bivariate Operators: Graphical and Tabular Representations

To verify the convergence of the operator defined in (15), we perform both graphical and numerical analyses. For this purpose, we consider the following test function:

$$\hslash (w_1,w_2)=x^4{y}^{3}siny,$$

The corresponding results are displayed in Figure 3.

Furthermore, to evaluate the error approximation, we utilize the formula:

$$T_{s_1,s_2}(\hslash ;w_1,w_2)=\left|{Ł}_{s_1,s_2}(\hslash ;w_1,w_2)-\hslash (w_1,w_2)\right|.$$

The error is examined for various values of $s_1$ and $s_2$, specifically $s_1,s_2=100, 120,$ and 140. The corresponding error approximations are illustrated graphically in Figure 4 and listed in

Table 2. Error approximation table $T_{s_1,s_2}(\hslash ;w_1,w_2)=\left|{Ł}_{s_1,s_2}(\hslash ;w_1,w_2)-\hslash (w_1,w_2)\right|$.

${\mathit{w}}_1,{\mathit{w}}_2$	${\mathit{T}}_{100,100}(\mathit{\hslash };{\mathit{w}}_1,{\mathit{w}}_2)$	${\mathit{T}}_{120,120}(\mathit{\hslash };{\mathit{w}}_1,{\mathit{w}}_2)$	${\mathit{T}}_{140,140}(\mathit{\hslash };{\mathit{w}}_1,{\mathit{w}}_2)$
0.1, 0.1	0.00291072	0.00203763	0.00173559
0.2, 0.2	0.00547611	0.00394756	0.00340736
0.3, 0.3	0.102474	0.00759292	0.006634474
0.4, 0.4	0.0187716	0.0142067	0.0125241
0.5, 0.5	0.0327879	0.0250319	0.0221139
0.6, 0.6	0.0529577	0.0399807	0.034999
0.7, 0.7	0.0763755	0.055093	0.0467623
0.8, 0.8	0.0931754	0.0592102	0.0456669
0.9, 0.9	0.0838775	0.317595	0.106255

. These representations together offer insights into the accuracy and behavior of the operator as the values of $s_1$ and $s_2$ increase, highlighting the convergence of the operator to the target function $\hslash (w_1,w_2)$.

8. Conclusions

In conclusion, this research explores the approximation of continuous functions using Frobenius–Euler–Simsek polynomial analogues of Szász operators. The study demonstrates uniform convergence, approximation order, and the application of Korovkin and modulus of continuity for continuous functional spaces. A Voronovskaja-type theorem is presented for functions with continuous first and second derivatives. Numerical and graphical analyses, including the construction of a bivariate sequence, further illustrate the accuracy and efficiency of the operators in approximating both univariate and bivariate continuous functions. The findings contribute valuable insights for applications requiring high precision.