Approximation Results: Szász–Kantorovich Operators Enhanced by Frobenius–Euler–Type Polynomials

Abstract

This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and the rate of approximation are analyzed using the classical Korovkin theorem and the modulus of continuity for Lebesgue measurable and continuous functions. A Voronovskaja-type theorem is also established to approximate functions with first- and second-order continuous derivatives. Numerical and graphical analyses are presented to support these findings. Furthermore, a bivariate sequence of these operators is introduced to approximate a bivariate class of Lebesgue measurable and continuous functions in two variables. Finally, numerical and graphical representations of the error are provided to check the rapidity of convergence.

1. Introduction and Preliminaries

Bernstein (1913) [1] introduced a sequence of polynomials to present the simplest proof for an elegant theorem on the basis of Weierstrass (1885) [2], which is termed the Weierstrass approximation theorem in terms of a binomial probability distribution as follows:

$$\begin{array}{c}\hfill {\mathbf{B}}_n(f;y)={\displaystyle \sum _{k=0}^n}{}_{n,k}\left(y\right)f\left({\displaystyle \frac{k}{n}}\right),\mathrm{n}\in \mathbb{N},\end{array}$$

(1)

where ${}_{n,k}\left(y\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)y^k{(1-y)}^{n-k}$ and $y\in [0,1]$. He found that ${\mathbf{B}}_n(f;.)\rightrightarrows f$ for every bounded function f defined on $[0,1]$, where ⇉ depicts the convergence, is uniform.

Kantorovich [3] extended (1) to approximating Lebesgue measurable functions. For $n\in \mathbb{N}$ and $f\in L^p[0,1]$ where $1\le p<\infty$, the Kantorovich operators $K_n:L^p\left([0,1]\right)\to L^p\left([0,1]\right)$ are defined as

$$K_n(f;y)=(n+1)\sum _{k=0}^n{}_{n,k}\left(y\right){\int }_{{\displaystyle \frac{k}{n+1}}}^{{\displaystyle \frac{k+1}{n+1}}}f\left(t\right)dt,$$

(2)

where ${}_{n,k}\left(y\right)=\left({\displaystyle \genfrac{}{}{0pt}{}{n}{k}}\right)y^k{(1-y)}^{n-k}$ and the Kantorovich operators converge almost everywhere with a function on $[0,1]$. Nowadays, operator theory connects various disciplines of science like engineering and medical sciences such as robotics, CAGD, HIV research, etc. [4,5,6,7,8,9]. In the past decade, many mathematicians have constructed various modifications of the operators defined by (1) to achieve better flexibility in the approximation properties over bounded and unbounded intervals in various functional spaces, e.g., $\ddot{\mathrm{O}}$zger et al. [10,11], Cai et al. [12,13], Aslan [14,15], Acu et al. [16,17,18], Mohiuddine et al. [19,20], Mursaleen et al. [21,22], Bustamante [23], Ozsarac et al. [24], Khan et al. [25], Nasiruzzaman [26], Braha et al. [27], Rao et al. [28,29], and Çetin et al. [30,31]. In view of polynomial classes, which are an active field of research as a special function field, we recall a class of polynomials by Simsek [32] which are termed Frobenius–Euler–Simsek-type numbers and polynomials ${\left\{l_k(y;v)\right\}}_{k=0}^{\infty }$ associated with the generating function as

$$F_l(y;w,v):={\displaystyle \frac{w^v}{{\displaystyle \prod _{j=0}^{v-1}}(e^w-j)}}{e}^{wy}=\sum _{k=0}^{\infty }{l}_k(y;v){\displaystyle \frac{w^k}{k!}}.$$

(3)

For $y=0$, we yield Frobenius–Euler–Simsek-type numbers as

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

Putting $v=2$ into the above equation, we acquire

$$F_l(y;w,2):={\displaystyle \frac{w^2}{e^w(e^w-1)}}{e}^{wy}=\sum _{k=0}^{\infty }{l}_k(y;2){\displaystyle \frac{w^k}{k!}}.$$

(4)

For detailed information and applications, see [33,34]. Motivated by the above development in the literature, we introduce a new connection to Kantorovich-type operators involving Frobenius–Euler–Simsek-type numbers as follows:

$$K_n(f;y)={\displaystyle \frac{n}{c_n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}\sum _{k=0}^{\infty }{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}{\int }_{{\displaystyle \frac{k}{n}}{c}_n}^{{\displaystyle \frac{(k+1)}{n}}{c}_n}f\left(t\right)dt,f\in L^p[0,\infty ),$$

(5)

where $1\le p<\infty$ and ${\left\{c_n\right\}}_1^{\infty }$ is a positive increasing sequence of real numbers such that $\underset{n\to \infty }{lim}{c}_n=\infty$ and $\underset{n\to \infty }{lim}{\displaystyle \frac{c_n}{n}}=0$. Now, we discuss some preliminary results to investigate the approximation properties of $K_n(.;.)$ in (5) as follows:

Lemma 1

([35]). In view of (4), one has

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

Lemma 2.

In light of the generating function introduced in (4), we yield

$$\begin{array}{cc}\hfill {\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}& ={\displaystyle \frac{e^{\frac{n}{c_n}y}}{e^2-e}},\hfill \\ \hfill {\displaystyle \sum _{k=0}^{\infty }}k{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}& ={\displaystyle \frac{ny{e}^{\frac{n}{c_n}y}}{c_n(e^2-e)}}-{\displaystyle \frac{e^{\frac{n}{c_n}y}}{e{(e-1)}^2}},\hfill \\ \hfill {\displaystyle \sum _{k=0}^{\infty }}{k}^2{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}& ={\left({\displaystyle \frac{n}{c_n}}\right)}^2{\displaystyle \frac{e^{\frac{n}{c_n}y}{y}^2}{e(e-1)}}-{\displaystyle \frac{2n}{c_n}}{\displaystyle \frac{e^{\frac{n}{c_n}y}y}{(e-1)}}-{\displaystyle \frac{2e^2-5e+1}{e{(e-1)}^3}}{e}^{{\displaystyle \frac{n}{c_n}}y}.\hfill \end{array}$$

Proof. 

On account of Lemma 1, we can prove Lemma 2 considering $w=1$ and y as ${\displaystyle \frac{n}{c_n}}y$. □

Lemma 3.

Let $f_i\left(t\right)=t^i$, $i\in \{0,1,2\}$. Then, we obtain

$$\begin{array}{cc}\hfill K_n(f_0;y)& =1,\hfill \\ \hfill K_n(f_1,y)& =y-{\displaystyle \frac{c_n}{n}}{\displaystyle \frac{1}{(e-1)}}+{\displaystyle \frac{c_n}{2n}},\hfill \\ \hfill K_n(f_2,y)& =y^2+{\displaystyle \frac{c_n}{n}}(1-2e)y+{\left({\displaystyle \frac{c_n}{n}}\right)}^2\left[{\displaystyle \frac{13-2e-5e^2}{3{(e-1)}^2}}\right].\hfill \end{array}$$

Proof. 

On account of proving Lemma 3 and from the operators $K_n(.;.)$ in (5), we have

$$\begin{array}{c}\hfill K_n(f_i;y)={\displaystyle \frac{n}{c_n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}{\int }_{{\displaystyle \frac{k}{n}}{c}_n}^{{\displaystyle \frac{(k+1)}{n}}{c}_n}{t}^{i}dt.\end{array}$$

For $i=0$,

$$\begin{array}{ccc}\hfill K_n(f_0;y)& =& {\displaystyle \frac{n}{c_n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}{\int }_{{\displaystyle \frac{k}{n}}{c}_n}^{{\displaystyle \frac{(k+1)}{n}}{c}_n}1dt\hfill \\ & =& {\displaystyle \frac{n}{c_n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}\left[{\displaystyle \frac{(k+1)}{n}}{c}_n-{\displaystyle \frac{k}{n}}{c}_n\right]\hfill \\ & =& (e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}=1.\hfill \end{array}$$

For $i=1$, we yield

$$\begin{array}{ccc}\hfill K_n(f_1;y)& =& {\displaystyle \frac{n}{c_n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}{\int }_{{\displaystyle \frac{k}{n}}{c}_n}^{{\displaystyle \frac{(k+1)}{n}}{c}_n}tdt\hfill \\ & =& {\displaystyle \frac{n}{c_n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}{\left[{\displaystyle \frac{t^2}{2}}\right]}_{{\displaystyle \frac{k}{n}}{c}_n}^{{\displaystyle \frac{k+1}{n}}{c}_n}\hfill \\ & =& {\displaystyle \frac{n}{c_n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}{\displaystyle \frac{1}{2}}\left[{\left({\displaystyle \frac{k+1}{n}}\right)}^2{c}_n^2-{\displaystyle \frac{k^2}{n^2}}{c}_n^2\right]\hfill \\ & =& {\displaystyle \frac{c_n}{n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}k{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}\hfill \\ & +& {\displaystyle \frac{c_n}{2n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}.\hfill \end{array}$$

In light of Lemma 2, we obtain

$$\begin{array}{ccc}\hfill K_n(f_1;y)& =& y+{\displaystyle \frac{c_n}{n}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{e-1}}\right).\hfill \end{array}$$

For $i=2$,

$$\begin{array}{ccc}\hfill K_n(f_2;y)& =& {\displaystyle \frac{n}{c_n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}{\int }_{{\displaystyle \frac{k}{n}}{c}_n}^{{\displaystyle \frac{(k+1)}{n}}{c}_n}{t}^{2}dt\hfill \\ & =& {\displaystyle \frac{n}{c_n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}{\left[{\displaystyle \frac{t^3}{3}}\right]}_{{\displaystyle \frac{k}{n}}{c}_n}^{{\displaystyle \frac{k+1}{n}}{c}_n}\hfill \\ & =& {\displaystyle \frac{n}{c_n}}(e^2-e)e^{-{\displaystyle \frac{n}{c_n}}y}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_k\left(\frac{n}{c_n}y;2\right)}{k!}}{\displaystyle \frac{1}{3}}\left[{\left({\displaystyle \frac{k+1}{n}}\right)}^3{c}_n^3-{\displaystyle \frac{k^3}{n^3}}{c}_n^3\right].\hfill \end{array}$$

Similarly, in light of Lemma 2, we find

$$\begin{array}{cc}\hfill K_n(f_2;y)& =y^2+{\displaystyle \frac{c_n}{n}}(1-2e)y+{\left({\displaystyle \frac{c_n}{n}}\right)}^2\left[{\displaystyle \frac{13-2e-5e^2}{3{(e-1)}^2}}\right].\end{array}$$

□

Remark 1.

The sequence of operators introduced in (5) is linear, i.e., for all $k_1,k_2\in \mathbb{R}$ and $f,g\in L^p[0,\infty )$, we yield

$$\begin{array}{c}\hfill K_n(k_{1}f+k_{2}g;y)=k_1{K}_n(f;y)+k_2{K}_n(g;y).\end{array}$$

Remark 2.

The sequence of operators introduced in (5) is positive, i.e., $K_n(f;y)\ge 0$ for $f\ge 0$.

Lemma 4.

For the sequence of operators given in (5) and $f_i^y\left(t\right)={(t-y)}^i,i\in \{0,1,2\}$, the following equalities hold:

$$\begin{array}{ccc}\hfill K_n(f_o^y;y)& =& 1,\hfill \\ \hfill K_n(f_1^y;y)& =& {\displaystyle \frac{c_n}{n}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill K_n(f_2^y;y)& =& {\displaystyle \frac{c_n}{n}}{\displaystyle \frac{2(1+e-e^2)}{e-1}}y+{\left({\displaystyle \frac{c_n}{n}}\right)}^2\left({\displaystyle \frac{13-2e-5e^2}{3{(e-1)}^2}}\right).\hfill \end{array}$$

Proof. 

Using (5) and the linearity property, we obtain

$$\begin{array}{ccc}\hfill K_n(f_o^y;y)& =& K_n(f_0;y)=1,\hfill \\ \hfill K_n(f_1^y;y)& =& K_n(t-y;y)=K_n(f_1;y)-y{K}_n(f_0;y),\hfill \\ \hfill K_n(f_2^y;y)& =& K_n({(t-y)}^2;y)=K_n(f_2;y)-2y{K}_n(f_1;y)+y^2{K}_n(f_0;y).\hfill \end{array}$$

In light of Lemma 3, we arrive at the desired result. □

This manuscript presents the approximation results for the operators defined by (5). It covers several key aspects: the uniform convergence and the pointwise and Voronovskaja-type theorems which deal with functions that grow at different rates. It also extends the operators to a bivariate case, analyzing their uniform rate of approximation and approximation order. Finally, it examines how these operators perform in different functional spaces to demonstrate their improved approximation behavior.

2. Approximation Properties—Uniform Convergence and the Approximation Order

Definition 1

([36]). Let $f\in C_B[0,\infty )$ (the space of a bounded and continuous function). Then, the modulus of continuity is introduced as

$$\begin{array}{c}\hfill \omega (f;\tilde{\delta })={\displaystyle \underset{|y_1-y_2|\le \tilde{\delta }}{sup}}|f\left(y_1\right)-f\left(y_2\right)|,y_1,y_2\in [0,\infty ),\end{array}$$

(6)

and

$$\begin{array}{c}\hfill |f\left(y_1\right)-f\left(y_2\right)|\le \left(1+{\displaystyle \frac{|y_1-y_2|}{\tilde{\delta }}}\right)\omega (f;\tilde{\delta }).\end{array}$$

(7)

Theorem 1.

Let $K_n(.;.)$ be given in (5) and for all $f\in L^p[0,\infty )$. Then, $K_n(f;.)$ uniformly converges to f on a closed and bounded subset of $[0,\infty ).$

Proof. 

Based on the classical Korovkin theorem [37], it is enough to show that

$$\begin{array}{c}\hfill K_n(f_i;y)=y^i,i\in \{0,1,2\},\end{array}$$

uniformly on every closed and bounded subset of $[0,\infty )$. Operating Lemma 3, we quickly reach the required result. □

The next result is the study of the order of approximation of (5) in terms of the modulus of continuity in Equation (6) as

Theorem 2.

For $f\in C_B[0,\infty )$ and the sequence of operators $K_n(.;.)$ in Equation (5), we have

$$\begin{array}{c}\hfill |K_n(f;y)-f\left(y\right)|\le 2\omega (f;\tilde{\delta }),\end{array}$$

where $\tilde{\delta }=\sqrt{K_n(f_2^y;y)}$.

Proof. 

With the definition of Equation (5), we yield

$$\begin{array}{ccc}\hfill \left|K_n(f;y)-f\left(y\right)\right|& =& K_n\left(\left|f\right(t)-f(y\left)\right|;y\right).\hfill \end{array}$$

In view of Equation (7), we have

$$\begin{array}{c}\hfill \left|K_n(f;y)-f\left(y\right)\right|\le K_n\left(\left(1+{\displaystyle \frac{|t-y|}{\tilde{\delta }}}\right)\omega (f;\tilde{\delta });y\right)\\ \hfill =\left\{1+{\displaystyle \frac{1}{\tilde{\delta }}}{K}_n\left(\right|t-y\left|\right)\right\}\omega (f;\tilde{\delta }).\end{array}$$

Using the Cauchy–Schwarz inequality, we yield

$$\begin{array}{ccc}\hfill |K_n(f;y)-f\left(y\right)|& \le & \left\{1+{\displaystyle \frac{1}{\tilde{\delta }}}\sqrt{\left(K_n{(t-y)}^2;y\right)}\right\}\omega (f;\tilde{\delta })\hfill \\ & \le & \left\{1+{\displaystyle \frac{\sqrt{K_n(f_2^y;y)}}{\tilde{\delta }}}\right\}\omega (f;\tilde{\delta }).\hfill \end{array}$$

On choosing $\tilde{\delta }=\sqrt{K_n(f_2^y;y)}$, we yield

$$\begin{array}{c}\hfill |K_n(f;y)-f\left(y\right)|\le 2\omega (f;\tilde{\delta }).\end{array}$$

Thus, we prove the required result. □

Now, we introduce the Voronovskaja-type theorem to approximate functions with continuous first- and second-order derivatives, using the operators from (5), as follows:

Theorem 3.

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

$$\begin{array}{ccc}\hfill {\displaystyle \underset{n\to \infty }{lim}}{\displaystyle \frac{n}{c_n}}(K_n(f;y)-f\left(y\right))& =& f^{\prime }\left(y\right)\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{e-1}}\right)+{\displaystyle \frac{f^{″}\left(y\right)}{2!}}\left({\displaystyle \frac{2(1+e-e^2)}{e-1}}\right)y.\hfill \end{array}$$

Proof. 

To approximate the functions, we first recall the Taylor series expansion:

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

(8)

where $\xi (t,y)$ is the Peano remainder such that $\xi (t,y)\in C[0,\infty )\cap E$ and ${lim}_{t\to y}\xi (t,y)=0$. Applying the operators $K_n(.;.)$ defined in Equation (5) to Equation (8), we have

$$\begin{array}{c}\hfill K_n(f;y)=f\left(y\right)+f^{\prime }\left(y\right)K_n(f_1^y;y)+{\displaystyle \frac{f^{″}}{2!}}{K}_n(f_2^y;y)+K_n(\xi (t,y){(t-y)}^2;y).\end{array}$$

(9)

By applying the limit to both sides of the expression in (9), we obtain

$$\begin{array}{ccc}\hfill {\displaystyle \underset{n\to \infty }{lim}}{\displaystyle \frac{n}{c_n}}(K_n(f;y)-f\left(y\right))& =& f^{\prime }\left(y\right){\displaystyle \underset{n\to \infty }{lim}}{\displaystyle \frac{n}{c_n}}{K}_n(f_1^y;y)+{\displaystyle \frac{f^{″}}{2!}}{\displaystyle \underset{n\to \infty }{lim}}{\displaystyle \frac{n}{c_n}}{K}_n(f_2^y;y)\hfill \\ \hfill & +& {\displaystyle \underset{n\to \infty }{lim}}{\displaystyle \frac{n}{c_n}}{K}_n(\xi (t,y){(t-y)}^2;y)\hfill \\ \hfill & =& f^{\prime }\left(y\right)\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{e-1}}\right)+{\displaystyle \frac{f^{″}\left(y\right)}{2!}}\left({\displaystyle \frac{2(1+e-e^2)}{e-1}}\right)y\hfill \\ & +& {\displaystyle \underset{n\to \infty }{lim}}{\displaystyle \frac{n}{c_n}}{K}_n(\xi (t,y){(t-y)}^2;y).\hfill \end{array}$$

(10)

The last term of the equation is obtained using the Cauchy–Schwarz inequality:

$$\begin{array}{c}\hfill {\displaystyle \frac{n}{c_n}}{K}_n(\xi (t,y){(t-y)}^2;y)\le \sqrt{{\left({\displaystyle \frac{n}{c_n}}\right)}^2{K}_n({(t-y)}^4;y)K_n({\xi }^2(t,y);y)}.\end{array}$$

(11)

From Equations (10) and (11), Lemma 4, and ${lim}_{n\to \infty }{K}_n({\xi }^2(t,y);y)=0$, we yield

$$\begin{array}{ccc}\hfill {\displaystyle \underset{c_n\to \infty }{lim}}{\displaystyle \frac{n}{c_n}}(K_n(f;y)-f\left(y\right))& =& f^{\prime }\left(y\right)\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{e-1}}\right)\hfill \\ & +& {\displaystyle \frac{f^{″}\left(y\right)}{2!}}\left({\displaystyle \frac{2(1+e-e^2)}{e-1}}\right)y.\hfill \end{array}$$

which proves the required result. □

3. Graphical and Numerical Approaches to a Convergence Analysis

In this section, we analyze the convergence properties of the operators defined in (5) for different values of n, specifically $n=50$, $n=60$, and $n=70$. Figure 1 visualizes the convergence behavior, demonstrating the impact of increasing n on the operator’s performance, while Figure 2 illustrates the approximation error trends. The numerical errors, computed using the formula $E_n(f;y)=\left|K_n(f;y)-f\left(y\right)\right|$ for the function $f\left(y\right)=y^2+2$, are presented in Table 1. These errors provide a quantitative measure of the approximation accuracy for different values of n and highlight the rate of convergence as n increases. From Figure 1, we observe that larger values of n improve the operators’ convergence to $f\left(y\right)$, while Figure 2 shows a consistent reduction in the approximation error, demonstrating the effectiveness of the operators for higher values of n. Table 1 further confirms that the error decreases significantly as n increases, validating the theoretical convergence properties. This numerical analysis not only supports the theoretical findings but also offers a practical understanding of the operator’s behavior. By evaluating the errors across different y values, we gain insights into the robustness and efficiency of the operators in approximating $f\left(y\right)$, which is essential for applications requiring high accuracy.

Figure 1. Convergence of operator $K_n(f;y)$ for $n=50,60,70$.

Figure 2. Error approximation $E_n(f;y)=\left|K_n(f;y)-f\left(y\right)\right|$.

Table 1. The error approximation of the operators $K_n(f;y)$ to $f\left(y\right)$.

y	$|{\mathit{K}}_{50}(\mathit{f};\mathit{y})-\mathit{f}\left(\mathit{y}\right)|$	$|{\mathit{K}}_{60}(\mathit{f};\mathit{y})-\mathit{f}\left(\mathit{y}\right)\left|\right)$	$|{\mathit{K}}_{70}(\mathit{f};\mathit{y})-\mathit{f}\left(\mathit{y}\right)|$
0.2	4.82078	4.66625	4.54366
0.4	4.88478	4.73025	4.60766
0.6	4.93278	4.77825	4.65566
0.8	4.91678	4.77825	4.63966
1.0	4.86978	4.76228	4.59266
1.2	4.78878	4.71525	4.51166
1.4	4.50078	4.63425	4.22366
1.6	4.00478	4.34625	3.72766
1.8	3.25278	3.85025	2.97566
2.0	2.19678	3.09825	1.91966

4. Local Approximation Results

Now, we recall a Lipschitz-type space [38], which is defined as

$$\begin{array}{c}\hfill Li{p}_{\tilde{M}}^{{\phi }_1,{\phi }_2}\left(\tau \right):=\{f\in {\tilde{C}}_{\tilde{B}}[0,\infty ):|f\left(t\right)-f\left(y\right)|\le \tilde{M}{\displaystyle \frac{{|t-y|}^{\tau }}{{(t+{\phi }_{1}y+{\phi }_2{y}^2)}^{\frac{\tau }{2}}}}:y,t\in (0,\infty )\},\end{array}$$

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

Theorem 4.

Let $K_n(.;.)$ be the operator given by (4). Then, for $f\in Li{p}_M^{{\phi }_1,{\phi }_2}\left(\tau \right)$, we have

$$\begin{array}{c}\hfill |K_n(f;y)-f\left(y\right)|\le \tilde{M}{\left({\displaystyle \frac{\lambda \left(y\right)}{{\phi }_{1}y+{\phi }_2{y}^2}}\right)}^{{\displaystyle \frac{\tau }{2}}},\end{array}$$

(12)

where $0<\tau \le 1$, ${\phi }_1,{\phi }_2\in (0,\infty )$ and $\lambda \left(y\right)=K_n(f_2^y;y)$.

Proof. 

For $\tau =1$ and $y>0$, we yield

$$\begin{array}{cc}\hfill |K_n(f;y)-f\left(y\right)|& \le K_n\left(\right|f\left(t\right)-f\left(y\right)|;y)\hfill \\ \hfill & \le \tilde{M}{K}_n\left({\displaystyle \frac{|t-y|}{{(t+{\phi }_{1}y+{\phi }_2{y}^2)}^{\frac{1}{2}}}};y\right).\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill |K_n(f;y)-f\left(y\right)|& \le {\displaystyle \frac{\tilde{M}}{{({\phi }_{1}y+{\phi }_2{y}^2)}^{\frac{1}{2}}}}{\left(K_n(f_2;y)\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le \tilde{M}{\left({\displaystyle \frac{\lambda \left(y\right)}{{\phi }_{1}y+{\phi }_2{y}^2}}\right)}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

This implies that Theorem 4 holds for $\tau =1$. Next, we consider a case where $\tau \in (0,1)$, and in view of Hölder’s inequality using $p={\displaystyle \frac{2}{\tau }}$ and $q={\displaystyle \frac{2}{2-\tau }}$, obtain

$$\begin{array}{cc}\hfill |K_n(f;y)-f\left(y\right)|& \le {(K_n{\left(\right|f\left(t\right)-f\left(y\right)|}^{{\displaystyle \frac{2}{\tau }}};y))}^{{\displaystyle \frac{\tau }{2}}}\hfill \\ \hfill & \le \tilde{M}{\left(K_n\left({\displaystyle \frac{{|t-y|}^2}{(t+{\phi }_{1}y+{\phi }_2{y}^2)}};y\right)\right)}^{{\displaystyle \frac{\tau }{2}}}.\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill |K_n(f;y)-f\left(y\right)|& \le \tilde{M}{\left({\displaystyle \frac{K_n{\left(\right|t-y|}^2;y)}{{\phi }_{1}y+{\phi }_2{y}^2}}\right)}^{{\displaystyle \frac{\tau }{2}}}\le \tilde{M}{\left({\displaystyle \frac{\lambda \left(y\right)}{{\phi }_{1}y+{\phi }_2{y}^2}}\right)}^{{\displaystyle \frac{\tau }{2}}}.\hfill \end{array}$$

Hence, Theorem 4 is proved. □

5. The Bivariate of a Frobenius–Euler–Simsek Polynomial Analog of Szász Operators

In this section, we extend the operators described in (5) to their bivariate form. Let

$$T^2=\left\{(x,y):0\le x<\infty ,0\le y<\infty \right\},$$

and $C\left(T^2\right)$ denotes the class of two variable functions that are continuous on $T^2$, equipped with the supremum norm

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

For all $f\in C\left(T^2\right)$ and $n,m\in \mathbb{N}$, we define a bivariate version of $K_n(.;.)$ as follows:

$$\begin{array}{ccc}\hfill R_{n,m}(f;x,y)& =& {\displaystyle \frac{n}{c_n}}{\displaystyle \frac{m}{c_m}}(e_1^2-e_1)(e_2^2-e_2)e^{-({\displaystyle \frac{n}{c_n}}x+{\displaystyle \frac{m}{c_m}}y)}\hfill \\ \hfill & \times & {\displaystyle \sum _{j=0}^{\infty }}{\displaystyle \sum _{k=0}^{\infty }}{\displaystyle \frac{l_j\left(\frac{n}{c_n}x;2\right)}{j!}}{\displaystyle \frac{l_k\left(\frac{m}{c_m}y;2\right)}{k!}}\hfill \\ & \times & {\int }_{{\displaystyle \frac{j}{n}}{c}_n}^{{\displaystyle \frac{j+1}{n}}{c}_n}{\int }_{{\displaystyle \frac{k}{m}}{c}_m}^{{\displaystyle \frac{k+1}{m}}{c}_m}f\left(t,s\right)dtds,\hfill \end{array}$$

(13)

where ${\left\{c_n\right\}}_1^{\infty },{\left\{c_m\right\}}_1^{\infty }$ are positive increasing sequences of real numbers such that ${lim}_{n\to \infty }{c}_n=\infty ,{lim}_{m\to \infty }{c}_m=\infty$ and ${lim}_{n\to \infty }{\displaystyle \frac{c_n}{n}}=0,{lim}_{m\to \infty }{\displaystyle \frac{c_m}{m}}=0$. We consider the two-dimensional test functions $e_{i,j}=x^i{y}^j$ and the central moments $p_{i,j}^{x,y}(t_1,t_2)={\eta }_{i,j}(t_1,t_2)={(t_1-x)}^i{(t_2-y)}^j$ for $i,j\in \{0,1,2\}$ such that $0\le i+j\le 2$.

Lemma 5.

For $f\in C\left(T^2\right)$ and the operator $R_{n,m}(.;.)$ given by Equation (13), along with the test functions $e_{i,j}(.;.)$, we have

$$\begin{array}{ccc}\hfill R_{n,m}(e_{0,0};x,y)& =& 1,\hfill \\ \hfill R_{n,m}(e_{1,0};x,y)& =& x-{\displaystyle \frac{c_n}{n}}{\displaystyle \frac{1}{(e-1)}}+{\displaystyle \frac{c_n}{2n}},\hfill \\ \hfill R_{n,m}(e_{0,1};x,y)& =& y-{\displaystyle \frac{c_m}{m}}{\displaystyle \frac{1}{(e-1)}}+{\displaystyle \frac{c_m}{2m}},\hfill \\ \hfill R_{n,m}(e_{2,0};x,y)& =& x^2+{\displaystyle \frac{c_n}{n}}(1-2e)x+{\left({\displaystyle \frac{c_n}{n}}\right)}^2\left[{\displaystyle \frac{13-2e-5e^2}{3{(e-1)}^2}}\right],\hfill \\ \hfill R_{n,m}(e_{0,2};x,y)& =& y^2+{\displaystyle \frac{c_m}{m}}(1-2e)y+{\left({\displaystyle \frac{c_m}{m}}\right)}^2\left[{\displaystyle \frac{13-2e-5e^2}{3{(e-1)}^2}}\right].\hfill \end{array}$$

Proof. 

We prove the above lemma using the concept of positive linear operators and by applying Lemma 3, as shown below:

$$\begin{array}{ccc}\hfill R_{n,m}(e_{0,0};x,y)& =& R_{n,m}(e_0;x)R_{n,m}(e_0;y),\hfill \\ \hfill R_{n,m}(e_{1,0};x,y)& =& R_{n,m}(e_1;x)R_{n,m}(e_0;y),\hfill \\ \hfill R_{n,m}(e_{0,1};x,y)& =& R_{n,m}(e_0;x)R_{n,m}(e_1;y),\hfill \\ \hfill R_{n,m}(e_{2,0};x,y)& =& R_{n,m}(e_2;x)R_{n,m}(e_0;y),\hfill \\ \hfill R_{n,m}(e_{0,2};x,y)& =& R_{n,m}(e_0;x)R_{n,m}(e_2;y).\hfill \end{array}$$

From the above equalities and Lemma 3, we can easily prove the lemma. □

Lemma 6.

For $p_{i,j}={(t_1-x)}^i{(t_2-y)}^j$ for $i,j=0,1,2,$ then we have the following equalities:

$$\begin{array}{ccc}\hfill R_{n,m}(t_{0,0};x,y)& =& 1,\hfill \\ \hfill R_{n,m}(t_{1,0};x,y)& =& {\displaystyle \frac{c_n}{n}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill R_{n,m}(t_{0,1};x,y)& =& {\displaystyle \frac{c_m}{m}}\left({\displaystyle \frac{1}{2}}-{\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill R_{n,m}(t_{2,0};x,y)& =& {\displaystyle \frac{c_n}{n}}{\displaystyle \frac{2(1+e-e^2)}{e-1}}x+{\left({\displaystyle \frac{c_n}{n}}\right)}^2\left({\displaystyle \frac{13-2e-5e^2}{3{(e-1)}^2}}\right),\hfill \\ \hfill R_{n,m}(t_{0,2};x,y)& =& {\displaystyle \frac{c_m}{m}}{\displaystyle \frac{2(1+e-e^2)}{e-1}}y+{\left({\displaystyle \frac{c_m}{m}}\right)}^2\left({\displaystyle \frac{13-2e-5e^2}{3{(e-1)}^2}}\right).\hfill \end{array}$$

Proof. 

Using Lemma 5 and the property of linearity, the required result can easily be proven. □

6. Order of Approximation

To analyze the convergence rate of the operators given in (13), we refer to the result established by Volkov [39]. This result provides a framework for addressing the convergence behavior effectively.

Theorem 5.

Let I and J be compact intervals on the real line. Consider the linear positive operators $R_{n,m}:C(I\times J)\to C(I\times J)$, where $(n,m)\in \mathbb{N}\times \mathbb{N}$. If

$$\begin{array}{ccc}\hfill \underset{n,m\to \infty }{lim}{R}_{n,m}\left(e_{ij}\right)& =& e_{i,j}\in \{(0,0),(1,0),(0,1)\}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \underset{n,m\to \infty }{lim}{R}_{n,m}(e_{20}+e_{02})& =& e_{20}+e_{02},\hfill \end{array}$$

and if $\left(R_{n,m}f\right)$ converges uniformly on $I\times J$, then for any $f\in C(I\times J)$, the sequence $\left(R_{n,m}f\right)$ converges uniformly to f on $I\times J$.

Theorem 6.

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

$$\begin{array}{c}\hfill \underset{n,m\to \infty }{lim}{R}_{n,m}(e_{ij};x,y)=e_{ij}(x,y)\end{array}$$

and

$$\begin{array}{c}\hfill \underset{n,m\to \infty }{lim}{R}_{n,m}(e_{20}+e_{02};x,y)=e_{20}(x,y)+e_{02}(x,y),\end{array}$$

uniformly on $T^2$, then

$$\begin{array}{c}\hfill \underset{n,m\to \infty }{lim}{R}_{n,m}(f;x,y)=f(x,y),\end{array}$$

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

Proof. 

In light of Equation (5), it is evident for $i=j=0$

$$\begin{array}{c}\hfill \underset{n,m\to \infty }{lim}{R}_{n,m}(e_{00};x,y)=e_{00}(x,y).\end{array}$$

For $i=1$, $j=0$, we obtain

$$\begin{array}{ccc}\hfill \underset{n,m\to \infty }{lim}{R}_{n,m}(e_{10};x,y)& =& x,\hfill \\ \hfill \underset{n,m\to \infty }{lim}{R}_{n,m}(e_{10};x,y)& =& e_{10}(x,y).\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill \underset{n,m\to \infty }{lim}{R}_{n,m}(e_{01};x,y)& =& y,\hfill \\ \hfill \underset{n,m\to \infty }{lim}{R}_{n,m}(e_{01};x,y)& =& e_{01}(x,y),\hfill \end{array}$$

and in light of Lemma 5, we obtain

$$\begin{array}{ccc}\hfill \underset{n,m\to \infty }{lim}{R}_{n,m}(e_{20}+e_{02};x,y)& =& x^2+y^2,\hfill \\ & =& e_{20}(x,y)+e_{02}(x,y).\hfill \end{array}$$

Using Theorems 5 and 6, we arrive at the desired result. □

In the last result, we deal with the approximation order of the sequence of operators $R_{n,m}(.;.)$ given by (13) as

Theorem 7

([40]). Let $L:C\left(T^2\right)\to B\left(T^2\right)$ be a linear positive operator. For any $f\in C\left(T^2\right)$, any $(x,y)\in T^2$, and any ${\tilde{\delta }}_1,{\tilde{\delta }}_2>0$, the following inequality

$$\begin{array}{ccc}\hfill \left|\right(Lf\left)\right(x,y)-f(x,y\left)\right|& \le & |L{e}_{0,0}(x,y)-1\left|\right|f(x,y)|+[L{e}_{0,0}(x,y)\hfill \\ & +& {\tilde{\delta }}_1^{-1}\sqrt{L{e}_{0,0}(x,y){\left(L(·-x)\right)}^2(x,y)}\hfill \\ & +& {\tilde{\delta }}_2^{-1}\sqrt{L{e}_{0,0}(x,y){\left(L(\ast -y)\right)}^2(x,y)}\hfill \end{array}$$

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

holds.

Theorem 8.

For $f\in C\left(T^2\right)$ and $(x,y)\in T^2$, $(n,m)\in \mathbb{N}\times \mathbb{N}$ and ${\tilde{\delta }}_1,{\tilde{\delta }}_2>0$, one has

$$\begin{array}{c}\hfill |R_{n,m}(f;x,y)-f(x,y)|\le 4{\omega }_{total}(f;{\tilde{\delta }}_1,{\tilde{\delta }}_2)\end{array}$$

where ${\tilde{\delta }}_1=\sqrt{R_{n,m}\left({(t_1-x)}^2;x,y\right)}$ and ${\tilde{\delta }}_2=\sqrt{R_{n,m}\left({(t_2-y)}^2;x,y)\right)}$.

Proof. 

From Theorem 7, we have

$$\begin{array}{ccc}& & |\left(R_{n,m}f\right)(x,y)-f(x,y)|\hfill \\ & \le & [1+{\tilde{\delta }}_1^{-1}\sqrt{R_{n,m}\left({(t_1-x)}^2;x,y\right)}\hfill \\ & +& {\tilde{\delta }}_2^{-1}\sqrt{R_{n,m}\left({(t_2-y)}^2;x,y\right)}\hfill \\ & +& {\tilde{\delta }}_1^{-1}{\tilde{\delta }}_2^{-1}\sqrt{R_{n,m}\left({(t_1-x)}^2;x,y\right)R_{n,m}\left({(t_2-y)}^2;x,y\right)}]\hfill \\ & \times & {\omega }_{total}(f;{\tilde{\delta }}_1,{\tilde{\delta }}_2).\hfill \end{array}$$

Selecting ${\tilde{\delta }}_1=\sqrt{R_{n,m}\left({(t_1-x)}^2;x,y\right)}$ and ${\tilde{\delta }}_2=\sqrt{R_{n,m}\left({(t_2-y)}^2;x,y)\right)}$, we arrive at the required result. □

7. Numerical and Graphical Analyses of Bivariate Operators

We present graphical and numerical analyses of the bivariate operators defined in (13) to demonstrate their convergence. Using the test function $f(x,y)=2{(x+y)}^2+4$, the convergence behavior is illustrated in Figure 3. To examine the error approximation, we employ the formula $S_{n,m}(f;x,y)=\left|R_{n,m}(f;x,y)-f(x,y)\right|$, analyzing the error for different values of n and m, specifically $n,m=50,60,$ and 70. The corresponding error approximations are graphically represented in Figure 4 and numerically tabulated in Table 2. Collectively, the figures and the table provide insights into the behavior and accuracy of the operator as n and m increase, confirming its convergence to the target function $f(x,y)$.

Figure 3. Convergence of the operator $K_{n,m}(f;x,y)$ for $n=m=50,60,70$.

Figure 4. Error approximation $S_{n,m}(f;x,y)=\left|R_{n,m}(f;x,y)-f(x,y)\right|$.

Table 2. Error approximation table: $S_{n,m}(f;x,y)=\left|R_{n,m}(f;x,y)-f(x,y)\right|$.

$\mathit{x},\mathit{y}$	${\mathit{S}}_{50,50}(\mathit{f};\mathit{x},\mathit{y})$	${\mathit{S}}_{60,60}(\mathit{f};\mathit{x},\mathit{y})$	${\mathit{S}}_{70,70}(\mathit{f};\mathit{x},\mathit{y})$
0.2, 0.2	2.59048	1.80191	1.210319
0.4, 0.4	2.27578	1.45012	0.829109
0.6, 0.6	1.81534	0.927824	0.257771
0.8, 0.8	1.30501	0.331003	0.407696
1.0, 1.0	0.87933	0.205938	1.032911
1.2, 1.2	0.71105	0.510199	1.44504
1.4, 1.4	1.01139	0.370581	1.432913
1.6, 1.6	2.02992	0.462517	0.746917
1.8, 1.8	4.05466	2.277118	0.900942

8. Conclusions

In this study, we explored the approximation capabilities of Frobenius–Euler–Simsek polynomials analogous to Szász–Kantorovich operators for Lebesgue measurable functions. Through extensive testing and calculation of the test functions and central moments, we analyzed their uniform convergence and order of approximation. Our investigation included the Korovkin theorem and the modulus of continuity, assessing how well these operators approximated the functions within continuous functional spaces. We also examined Voronovskaja-type theorems for functions with continuous derivatives, supported by numerical and graphical analyses of the errors. Additionally, we developed a bivariate sequence of operators for approximating bivariate continuous functions, discussing the numerical and graphical error deviations.