Adjoint Bernoulli’s Kantorovich–Schurer-Type Operators: Univariate Approximations in Functional Spaces

Abstract

In this work, we first establish a new connection between adjoint Bernoulli’s polynomials and gamma function as a new sequence of linear positive operators denoted by ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$. Further, convergence results for these sequences of operators, i.e., ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$ are derived in various functional spaces with the aid of the Korovkin theorem, the Voronovskaja-type theorem, the first order of the modulus of continuity, the second order of the modulus of continuity, Peetre’s K-functional, the Lipschitz condition, etc. In the last section, we focus our research on the bivariate extension of these sequences of operators; their uniform rate of approximation and order of approximation are investigated in different functional spaces.

1. Introduction

Weierstrass (1885) [1] introduced an elegant result named the Weierstrass approximation theorem, i.e., all continuous functions over a closed interval $[a,b]$ can be uniformly approximated by a polynomial function over $[a,b]$ with the desired accuracy. Several branches of approximation theory such as numerical analysis, operator theory, and wavelet analysis are generated in light of this theorem. The proof of this theorem, however, is difficult to understand. It is noteworthy that this theorem is crucial in the field of approximation theory. Therefore, several mathematicians have worked on the proof of this theorem to make it understandable. One of the mathematicians, Bernstein (1912) [2], presented the simplest and shortest proof of this theorem via binomial probability distribution, that is,

$$\begin{array}{c}\hfill B_r(⌀;\varkappa )=\sum _{\varrho =0}^{r+\varsigma }{p}_{r,\varsigma ,\varrho }\left(\varkappa \right)⌀\left({\displaystyle \frac{\varrho }{r+\varsigma }}\right),r\in \mathbb{N},\end{array}$$

(1)

where $p_{r,\varsigma ,\varrho }\left(\varkappa \right)=\left({\displaystyle \genfrac{}{}{0pt}{}{r+\varsigma }{\varrho }}\right){\varkappa }^{\varrho }{(1-\varkappa )}^{r+\varsigma -\varrho }$. He found that $B_r(⌀;.)\rightrightarrows ⌀$ for every bounded function ⌀ defined on $[0,1]$, where ⇉ denotes the uniform convergence. In the past decade, numerous researchers have introduced several modifications to the operators defined in Equation (1), aiming to enhance their flexibility and effectiveness in approximating functions across both unbounded and bounded intervals within diverse functional spaces, e.g., Aslan et al. [3,4], Cai et al. [5], Bhat et al. [6,7], Mohiuddine et al. [8,9], Ayman-Mursaleen [10], Savaş and Mursaleen [11] Savaş and Patterson [12], Braha et al. [13], Rao et al. [14,15], Çetin et al. [16], etc. (see also [17,18,19,20,21] and the references therein).

In light of the growing interest in classes of polynomials, particularly within the field of special functions, we recall a notable class introduced by Appell [22], known as the Appell polynomials ${\left\{p_{\varrho }\left(\varkappa \right)\right\}}_{\varrho =0}^{\infty }$, which are associated with the following generating function:

$$A\left(t\right)e^{\varkappa t}=\sum _{\varrho =0}^{\infty }{p}_{\varrho }\left(\varkappa \right){\displaystyle \frac{t^{\varrho }}{\varrho !}},$$

(2)

where $A\left(t\right)={\displaystyle \sum _{\varrho =0}^{\infty }}{a}_{\varrho }{\displaystyle \frac{t^{\varrho }}{\varrho !}},A\left(0\right)\ne 0,$ which is an analytic function at $t=0$, satisfying $a_{\varrho }=p_{\varrho }\left(0\right).$

Recently, Natalini et al. [23] introduced Appell–Bernoulli polynomials by selecting $A\left(t\right)={\displaystyle \frac{t}{e^t-1}}$ in Equation (2). The corresponding adjoint Bernoulli polynomials obtained by opting for $A\left(t\right)$ as ${\displaystyle \frac{1}{A\left(t\right)}}$ are denoted by ${\left\{{\beta }_{\varrho }\left(\varkappa \right)\right\}}_{\varrho =0}^{\infty }$ and defined via an exponential-type generating function:

$${\displaystyle \frac{e^t-1}{t}}{e}^{\varkappa t}=\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\beta }_{\varrho }\left(\varkappa \right)t^{\varrho }}{\varrho !}}.$$

(3)

These adjoint Bernoulli’s polynomials are positive on $[0,\infty )$ and are part of a broader framework of polynomial sequences studied in modern umbral calculus [24,25].

Motivated by the aforementioned literature, we introduce a novel connection between the adjoint Bernoulli polynomials and the Gamma function, which leads to the formulation of a new sequence of linear positive operators as follows:

$${\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )=\sum _{\varrho =0}^{\infty }{g}_{\varrho }\left((r+\varsigma )\varkappa \right){\int }_0^{\infty }{b}_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right)⌀\left(\theta \right)d\theta ,$$

(4)

where $\varsigma \in [0,\infty ),⌀\in {\mathcal{L}}_{\beta }[0,\infty )$ (a space of bounded and Lebesgue measurable functions) and $g_{\varrho }\left((r+\varsigma )\varkappa \right)={\displaystyle \frac{e^{-(r+\varsigma )\varkappa }}{e-1}}{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}$, $b_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right)={\displaystyle \frac{{(r+\varsigma )}^{\varrho +\lambda +1}}{\Gamma \left(\varrho +\lambda \right)}}{\theta }^{\varrho +\lambda }{e}^{-(r+\varsigma )\theta }$.

1.1. Motivation and Theoretical Need

While classical polynomial approximation operators (e.g., Bernstein, Szász) provide foundational results, they often lack flexibility in weighted spaces, exhibit slow convergence for oscillatory functions, and have limited adaptability to functions with singularities or rapid variations. Recent advancements in umbral calculus [24] have revealed that carefully constructed polynomial sequences, particularly those of the Appell type can yield operators with superior approximation properties.

The primary theoretical motivation for this work is to bridge the gap between classical approximation theory and modern special function analysis by constructing a new family of linear positive operators that combine the following:

• The structural advantages of adjoint Bernoulli polynomials, which are positive on $[0,\infty )$ and possess favorable recurrence and orthogonality properties;
• The gamma function in the kernel, which provides an additional parameter $\lambda >0$ to control weight distribution and moment behavior;
• A Szász-type exponential factor for approximation on unbounded intervals.

This combination addresses several limitations of existing operators:

• Bernstein-type operators are restricted to bounded intervals;
• Classical Szász operators lack polynomial precision beyond linear functions;
• Most existing modifications do not simultaneously handle both bounded and unbounded domains with optimal convergence rates.

The numerical motivation stems from the need for operators that fulfil the following conditions:

• Exhibit faster convergence for oscillatory and nonsmooth functions;
• Provide tunable parameters for adaptive approximation;
• Maintain positivity and stability in floating-point computations.

Our proposed operator ${\mathfrak{S}}_{r,\varsigma ,\lambda }(·;·)$ addresses these needs by integrating concepts from umbral calculus, special functions, and approximation theory into a unified framework with provable convergence guarantees.

We now outline various preliminary results required for analyzing the approximation properties of the operator sequence ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$ defined in Equation (4).

Lemma 1 ([26]).

Through the generating function for $\varkappa \in [0,\infty )$ given in (3), we have

$$\begin{array}{cc}\hfill \sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}& =e^{(r+\varsigma )\varkappa }(e-1),\hfill \\ \hfill \sum _{\varrho =0}^{\infty }\varrho {\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}& =e^{(r+\varsigma )\varkappa }\left[(r+\varsigma )\varkappa (e-1)+1\right],\hfill \\ \hfill \sum _{\varrho =0}^{\infty }{\varrho }^2{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}& =e^{(r+\varsigma )\varkappa }\left[{(r+\varsigma )}^2{\varkappa }^2(e-1)+(r+\varsigma )\varkappa (e+1)+(e-1)\right],\hfill \\ \hfill \sum _{\varrho =0}^{\infty }{\varrho }^3{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}& =e^{(r+\varsigma )\varkappa }[{(r+\varsigma )}^3{\varkappa }^3(e-1)+3e{(r+\varsigma )}^2{\varkappa }^2\hfill \\ \hfill & +(r+\varsigma )\varkappa (4e-1)+(e+1)],\hfill \\ \hfill \sum _{\varrho =0}^{\infty }{\varrho }^4{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}& =e^{(r+\varsigma )\varkappa }[{(r+\varsigma )}^4{\varkappa }^4(e-1)+{(r+\varsigma )}^3{\varkappa }^3(6e-2)\hfill \\ \hfill & +{(r+\varsigma )}^2{\varkappa }^2(13e-1)+(r+\varsigma )\varkappa (11e+1)+(4e-1)].\hfill \end{array}$$

Lemma 2.

For ${⌀}_j\left(\theta \right)={\theta }^j$, $j\in \{0,1,2,3,4\}$, we have

$$\begin{array}{cc}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }(1;\varkappa )& =1,\hfill \\ \hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1,\varkappa )& =\varkappa +{\displaystyle \frac{1}{r+\varsigma }}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2,\varkappa )& ={\varkappa }^2+{\displaystyle \frac{\varkappa }{r+\varsigma }}(2e(\lambda +1)-2\lambda )+{\displaystyle \frac{1}{{(r+\varsigma )}^2}}\left(1+{\displaystyle \frac{2\lambda +1}{e-1}}+{\lambda }^2+\lambda \right),\hfill \\ \hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_3,\varkappa )& ={\varkappa }^3+{\displaystyle \frac{{\varkappa }^2}{r+\varsigma }}\left[3e(\lambda +2)-3(\lambda +1)\right]+{\displaystyle \frac{\varkappa }{{(r+\varsigma )}^2}}[3{\lambda }^2(e-1)+3\lambda (3e-1)\hfill \\ \hfill & +3(3e-1)]+{\displaystyle \frac{1}{{(r+\varsigma )}^3}}\left[{\lambda }^3+6{\lambda }^2+8\lambda +3+{\displaystyle \frac{e+1}{e-1}}\right],\hfill \\ \hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_4,\varkappa )& ={\varkappa }^4+o\left({\displaystyle \frac{1}{r+\varsigma }}\right).\hfill \end{array}$$

Proof. 

We present the proof for the operators ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.,.)$ in Equation (4) as follows:

$$\begin{array}{c}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_j;\varkappa )={\displaystyle \frac{e^{-(r+\varsigma )\varkappa }}{e-1}}\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}{\int }_0^{\infty }{b}_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right){\theta }^{j}d\theta .\end{array}$$

(5)

For $j=0$, ${\int }_0^{\infty }{b}_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right)d\theta =1$, which implies that

$$\begin{array}{c}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_0;\varkappa )={\displaystyle \frac{e^{-(r+\varsigma )\varkappa }}{e-1}}\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}=1.\end{array}$$

For $j=1$,

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{b}_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right)\theta d\theta & ={\displaystyle \frac{{(r+\varsigma )}^{\varrho +\lambda +1}}{\Gamma \left(\varrho +\lambda \right)}}{\int }_0^{\infty }{\theta }^{\varrho +\lambda +1}{e}^{-(r+\varsigma )\theta }d\theta \\ & ={\displaystyle \frac{{(r+\varsigma )}^{\varrho +\lambda +1}}{\Gamma \left(\varrho +\lambda \right)}}{\displaystyle \frac{\Gamma \left(\varrho +\lambda +1\right)}{{(r+\varsigma )}^{\varrho +\lambda +2}}}={\displaystyle \frac{\varrho +\lambda }{r+\varsigma }}.\end{array}$$

(6)

Clubbing Equations (5) and (6), we yield

$$\begin{array}{c}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1;\varkappa )={\displaystyle \frac{1}{r+\varsigma }}\left[{\displaystyle \frac{e^{-(r+\varsigma )\varkappa }}{e-1}}\sum _{\varrho =0}^{\infty }{\displaystyle \frac{\varrho {\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}\right]+{\displaystyle \frac{\lambda }{r+\varsigma }}.\end{array}$$

Via Lemma 1, we acquire

$$\begin{array}{cc}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1,\varkappa )& =\varkappa +{\displaystyle \frac{1}{r+\varsigma }}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right).\end{array}$$

For $j=2$,

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{b}_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right){\theta }^{2}d\theta & ={\displaystyle \frac{{(r+\varsigma )}^{\varrho +\lambda +1}}{\Gamma \left(\varrho +\lambda \right)}}{\int }_0^{\infty }{\theta }^{\varrho +\lambda +2}{e}^{-(r+\varsigma )\theta }d\theta \\ & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}={\displaystyle \frac{{(r+\varsigma )}^{\varrho +\lambda +1}}{\Gamma \left(\varrho +\lambda \right)}}{\displaystyle \frac{\Gamma \left(\varrho +\lambda +2\right)}{{(r+\varsigma )}^{\varrho +\lambda +3}}}={\displaystyle \frac{(\varrho +\lambda +1)(\varrho +\lambda )}{{(r+\varsigma )}^2}}.\end{array}$$

(7)

Clubbing Equations (5) and (7), we yield

$$\begin{array}{c}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2;\varkappa )=\left[{\displaystyle \frac{e^{-(r+\varsigma )\varkappa }}{e-1}}\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}\right]{\displaystyle \frac{(\varrho +\lambda +1)(\varrho +\lambda )}{{(r+\varsigma )}^2}}.\end{array}$$

In view of Lemma 1, we obtain

$$\begin{array}{cc}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2,\varkappa )& ={\varkappa }^2+{\displaystyle \frac{\varkappa }{r+\varsigma }}(2e(\lambda +1)-2\lambda )+{\displaystyle \frac{1}{{(r+\varsigma )}^2}}\left(1+{\displaystyle \frac{2\lambda +1}{e-1}}+{\lambda }^2+\lambda \right).\end{array}$$

The remaining part of this lemma can be established through a similar procedure. □

Lemma 3.

Let ${⌀}_j^{\varkappa }\left(\theta \right)={(\theta -\varkappa )}^j$ be the central moments for $j\in \{0,1,2,3,4\}$. Then, for the sequence of operators given by (4), one has the following equalities:

$$\begin{array}{ccc}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_o^{\varkappa };\varkappa )& =& 1,\hfill \\ \hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1^{\varkappa };\varkappa )& =& {\displaystyle \frac{1}{r+\varsigma }}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2^{\varkappa };\varkappa )& =& {\displaystyle \frac{\varkappa }{r+\varsigma }}\left(2e(\lambda +1)-2\lambda -2\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)\right)\hfill \\ & +& {\displaystyle \frac{1}{{(r+\varsigma )}^2}}\left(1+{\displaystyle \frac{2\lambda +1}{e-1}}+{\lambda }^2+\lambda \right),\hfill \\ \hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_4^{\varkappa };\varkappa )& =& o\left({\displaystyle \frac{1}{{(r+\varsigma )}^2}}\right){\varkappa }^2.\hfill \end{array}$$

Proof. 

Utilizing the operators defined in Equation (4) with the linearity property, we derive the following:

$$\begin{array}{ccc}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_o^{\varkappa };\varkappa )& =& {\mathfrak{S}}_{r,\varsigma ,\lambda }(1;\varkappa )=1,\hfill \\ \hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1^{\varkappa };\varkappa )& =& {\mathfrak{S}}_{r,\varsigma ,\lambda }(\theta -\varkappa ;\varkappa )={\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1;\varkappa )-\varkappa {\mathfrak{S}}_{r,\varsigma ,\lambda }(1;\varkappa ),\hfill \\ \hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2^{\varkappa };\varkappa )& =& {\mathfrak{S}}_{r,\varsigma ,\lambda }({(\theta -\varkappa )}^2;\varkappa )={\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2;\varkappa )-2\varkappa {\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1;\varkappa )\hfill \\ & +& {\varkappa }^2{\mathfrak{S}}_{r,\varsigma ,\lambda }(1;\varkappa ).\hfill \end{array}$$

Following this approach, the desired result is obtained. □

Remark 1.

The sequences of operators introduced in (4) are linear, i.e., for all $k_1,k_2\in \mathbb{R}$ and functions ${\phi }_1,{\phi }_2\in {\mathcal{L}}_{\beta }[0,\infty )$, we have

$$\begin{array}{c}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }(k_1{\theta }_1+k_2{\theta }_2;\varkappa )=k_1{\mathfrak{S}}_{r,\varsigma ,\lambda }({\theta }_1;\varkappa )+k_2{\mathfrak{S}}_{r,\varsigma ,\lambda }({\theta }_2;\varkappa ).\end{array}$$

Remark 2.

The sequences of operators introduced in (4) are positive, i.e., ${\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )\ge 0$ for $⌀\ge 0$.

To examine the approximation characteristics of the sequence of operators defined in Equation (4), the manuscript is organized into several subsequent sections. These include an analysis of the uniform rate of convergence, direct approximation results, and properties of weighted approximation. Furthermore, a bivariate extension of operators is explored, along with their respective rates of convergence and orders of approximation in several functional spaces. This structured approach is aimed at establishing enhanced approximation behavior for the proposed sequence of operators.

1.2. Theoretical Advantages over Existing Operators

The proposed operator ${\mathfrak{S}}_{r,\varsigma ,\lambda }(·;·)$ in (4) offers several theoretical advantages over classical and recent operators [27]:

• Flexibility via Parameterization:

The inclusion of the gamma function with parameter $\lambda >0$ in the kernel

$$b_{r,\varsigma ,k}^{\lambda }\left(\theta \right)=\frac{(r+\varsigma )k+\lambda +1}{\Gamma (k+\lambda )}{\theta }^{k+\lambda }{e}^{-(r+\varsigma )\theta }$$

allows adaptive weight adjustment, unlike fixed-kernel operators such as classical Bernstein or Szász operators. This enables better approximation of functions with varying smoothness across the domain.

2

Optimal Moment Decay:

As shown in Lemmas 2 and 3, the central moments satisfy

$${\mathfrak{S}}_{r,\varsigma ,\lambda }({(t-x)}^j;x)=O\left({\displaystyle \frac{1}{{(r+\varsigma )}^{\lceil j/2\rceil }}}\right),$$

which is optimal for positive linear operators and leads to faster convergence rates compared to operators with slower moment decay.

3

Unbounded Domain Approximation:

Unlike Bernstein-type operators restricted to $[0,1]$, our operator naturally extends to $[0,\infty )$ through the Szász-type exponential factor combined with adjoint Bernoulli polynomials, which remain positive on the entire half-line.

4

Connection to Umbral Calculus:

The use of adjoint Bernoulli polynomials places our operator within the rigorous framework of modern umbral calculus [24,25], ensuring well-defined algebraic and analytic properties. Other conjugate polynomial sequences (e.g., Appell–Euler) could be used; see Theorem 8.4 in [24].

5

Weighted Approximation Capability:

The operator preserves positivity and demonstrates excellent behavior in weighted spaces (Section 3), making it suitable for approximating functions with polynomial growth at infinity—a challenging scenario for many classical operators.

2. On the Uniform Convergence and the Order of Approximation Properties

Definition 1 ([28]).

Let $⌀\in C_B[0,\infty )$. Then, the modulus of continuity is

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

(8)

and

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

(9)

Theorem 1.

Let ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$ be given in (4), a sequence of operators for all $⌀\in C_B[0,\infty )$. Then, ${\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;.)\rightrightarrows ⌀$ on a subset (closed and bounded) of $[0,\infty )$, where ⇉ depicts that convergence is uniform.

Proof. 

By virtue of the classical Korovkin theorem [29], it suffices to verify

$$\begin{array}{c}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }({\theta }^j;\varkappa )={\varkappa }^j,and j=0,1,2.\end{array}$$

uniformly on every bounded and closed subset of $[0,\infty ).$ Invoking Lemma 2, the conclusion follows directly. □

The next result concerns the order of approximation of the operator given in Equation (4), expressed in the form of the modulus of continuity given in Equation (8).

Theorem 2.

For $⌀\in C_B[0,\infty )$ and operators ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$ in Equation (4), we have

$$\begin{array}{c}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)|\le 2\omega (⌀;\tilde{\delta }),\end{array}$$

where $\tilde{\delta }=\sqrt{{\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2^{\varkappa };\varkappa )}$.

Proof. 

With the definition of Equation (4), we have

$$\begin{array}{ccc}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)|& =& \left|{\displaystyle \frac{e^{-(r+\varsigma )\varkappa }}{e-1}}\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}{\int }_0^{\infty }{b}_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right)\{⌀\left(\theta \right)-⌀\left(\varkappa \right)\}d\theta \right|,\hfill \\ & \le & {\displaystyle \frac{e^{-(r+\varsigma )\varkappa }}{e-1}}\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}{\int }_0^{\infty }{b}_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right)|⌀\left(\theta \right)-⌀\left(\varkappa \right)|d\theta \hfill \\ & \le & {\displaystyle \frac{e^{-(r+\varsigma )\varkappa }}{e-1}}\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}\hfill \\ & \times & {\int }_0^{\infty }{b}_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right)\left(1+{\displaystyle \frac{|\theta -\varkappa |}{\tilde{\delta }}}\right)\omega (⌀;\tilde{\delta })d\theta \hfill \\ & \le & [1+{\displaystyle \frac{1}{\tilde{\delta }}}\left({\displaystyle \frac{e^{-(r+\varsigma )\varkappa }}{e-1}}\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}\right)\hfill \\ & \times & \left({\int }_0^{\infty }{b}_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right)|\theta -\varkappa |d\theta \right)]\omega (⌀;\tilde{\delta }).\hfill \end{array}$$

Via Cauchy–Schwarz inequality, we acquire

$$\begin{array}{ccc}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)|& \le & \{1+{\displaystyle \frac{1}{\tilde{\delta }}}{\left({\displaystyle \frac{e^{-(r+\varsigma )\varkappa }}{e-1}}\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}{\int }_0^{\infty }{b}_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right)d\theta \right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ & \times & {\left({\displaystyle \frac{e^{-(r+\varsigma )\varkappa }}{e-1}}\sum _{\varrho =0}^{\infty }{\displaystyle \frac{{\beta }_{\varrho }\left((r+\varsigma )\varkappa \right)}{\varrho !}}{\int }_0^{\infty }{b}_{r,\varsigma ,\varrho }^{\lambda }\left(\theta \right){(\theta -\varkappa )}^{2}d\theta \right)}^{{\displaystyle \frac{1}{2}}}\}\hfill \\ & \times & \omega (⌀;\tilde{\delta })\hfill \\ & \le & \left\{1+{\displaystyle \frac{\sqrt{{\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2^{\varkappa };\varkappa )}}{\tilde{\delta }}}\right\}\omega (⌀;\tilde{\delta }).\hfill \end{array}$$

On choosing $\tilde{\delta }={\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2^{\varkappa };\varkappa )$, we yield

$$\begin{array}{c}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)|\le 2\omega (⌀;\tilde{\delta }).\end{array}$$

Hence, we conclude the proof of the above theorem. □

We now proceed to establish a Voronovskaja-type theorem to approximate a class of continuous first- and second-order differential functions, utilizing the operators defined in Equation (4) as follows:

Theorem 3.

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

$$\begin{array}{ccc}\hfill \underset{r\to \infty }{lim}(r+\varsigma )({\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right))& =& {⌀}^{\prime }\left(\varkappa \right)\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)+{\displaystyle \frac{{⌀}^{″}\left(\varkappa \right)}{2!}}\left[\right(2e(\lambda +1)\hfill \\ & -& 2\lambda -2\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)]\varkappa .\hfill \end{array}$$

Proof. 

To examine the approximation of functions, we first revisit Taylor series expansion

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

(10)

where $\xi (\theta ,\varkappa )$ is the Peano remainder with $\xi (\theta ,\varkappa )\in C[0,\infty )\bigcap E$ and ${lim}_{\theta \to \varkappa }\xi (\theta ,\varkappa )=0$. Employing operators ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$ as defined in Equations (4)–(10),

$$\begin{array}{cc}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )& =⌀\left(\varkappa \right)+{⌀}^{\prime }\left(\varkappa \right){\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1^{\varkappa };\varkappa )+{\displaystyle \frac{{⌀}^{″}}{2!}}{\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2^{\varkappa };\varkappa )\hfill \\ \hfill & +{\mathfrak{S}}_{r,\varsigma ,\lambda }(\xi (\theta ,\varkappa ){(\theta -\varkappa )}^2;\varkappa ).\hfill \end{array}$$

By taking the limit for the above expression, we acquire

$$\begin{array}{ccc}\hfill \underset{r\to \infty }{lim}(r+\varsigma )({\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right))& =& {⌀}^{\prime }\left(\varkappa \right)\underset{r\to \infty }{lim}(r+\varsigma ){\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1^{\varkappa };\varkappa )\hfill \\ & +& {\displaystyle \frac{{⌀}^{″}}{2!}}\underset{r\to \infty }{lim}(r+\varsigma ){\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2^{\varkappa };\varkappa )\hfill \\ & +& \underset{r\to \infty }{lim}(r+\varsigma ){\mathfrak{S}}_{r,\varsigma ,\lambda }(\xi (\theta ,\varkappa ){(\theta -\varkappa )}^2;\varkappa )\hfill \\ & =& {⌀}^{\prime }\left(\varkappa \right)\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)+{\displaystyle \frac{{⌀}^{″}\left(\varkappa \right)}{2!}}[2e(\lambda +1)\hfill \\ & -& 2\lambda -2\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)]\varkappa \hfill \\ & +& \underset{r\to \infty }{lim}(r+\varsigma ){\mathfrak{S}}_{r,\varsigma ,\lambda }(\xi (\theta ,\varkappa ){(\theta -\varkappa )}^2;\varkappa ).\hfill \end{array}$$

Utilizing Cauchy–Schwarz inequality, the last term of expression can be written as follows:

$$\begin{array}{c}\hfill (r+\varsigma ){\mathfrak{S}}_{r,\varsigma ,\lambda }(\xi (\theta ,\varkappa ){(\theta -\varkappa )}^2;\varkappa )\le \sqrt{{(r+\varsigma )}^2{\mathfrak{S}}_{r,\varsigma ,\lambda }({(\theta -\varkappa )}^4;\varkappa ){\mathfrak{S}}_{r,\varsigma ,\lambda }({\xi }^2(\theta ,\varkappa );\varkappa )}.\end{array}$$

From Lemma 3 and ${lim}_{r\to \infty }{\mathfrak{S}}_{r,\varsigma ,\lambda }({\xi }^2(\theta ,\varkappa );\varkappa )=0$, we yield

$$\begin{array}{ccc}\hfill \underset{r\to \infty }{lim}(r+\varsigma )({\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right))& =& {⌀}^{\prime }\left(\varkappa \right)\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)\hfill \\ & +& {\displaystyle \frac{{⌀}^{″}\left(\varkappa \right)}{2!}}\left(2e(\lambda +1)-2\lambda -2\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)\right)\varkappa .\hfill \end{array}$$

Thus, the desired result is established. □

3. Analysis of Local Approximation

Let ${\tilde{C}}_{\tilde{B}}[0,\infty )$ denote the space of bounded and continuous functions. Then, Peetre’s K-functional is

$$\begin{array}{c}\hfill \widetilde{K_2}(⌀,\delta )=\underset{g\in {\tilde{C}}_{\tilde{B}}^2[0,\infty )}{inf}\left\{{\parallel ⌀-g\parallel }_{{\tilde{C}}_{\tilde{B}}[0,\infty )}+\tilde{\delta }{\parallel g^{″}\parallel }_{{\tilde{C}}_{\tilde{B}}^2[0,\infty )}\right\},\end{array}$$

Here, ${\tilde{C}}_{\tilde{B}}^2[0,\infty )=\{⌀\in {\tilde{C}}_{\tilde{B}}[0,\infty ):{⌀}^{\prime },{⌀}^{″}\in {\tilde{C}}_{\tilde{B}}[0,\infty )\}$, having the norm $\parallel ⌀\parallel =\underset{0\le \varkappa <\infty }{sup}|⌀\left(\varkappa \right)|$. The second-order Ditzian–Totik modulus of smoothness is

$$\begin{array}{c}\hfill \widetilde{{\omega }_2}(⌀;\sqrt{\tilde{\delta }})=\underset{0<k\le \sqrt{\tilde{\delta }}}{sup}\underset{\varkappa \in [0,\infty )}{sup}|⌀(\varkappa +2k)-2⌀(\varkappa +k)+⌀\left(\varkappa \right)|.\end{array}$$

In accordance with the result of DeVore and Lorentz ([28] p. 177, Theorem 2.4 ), we acquire

$$\begin{array}{c}\hfill \widetilde{K_2}(⌀;\tilde{\delta })\le \tilde{C}\widetilde{{\omega }_2}(⌀;\sqrt{\tilde{\delta }}),\end{array}$$

(11)

Here, $\tilde{C}$ represents an absolute constant. To establish the results of local approximation, we introduce the following auxiliary operators:

$$\begin{array}{c}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}(⌀;\varkappa )={\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )+⌀\left(\varkappa \right)-⌀\left(\varkappa +{\displaystyle \frac{1}{r+\varsigma }}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)\right),\end{array}$$

(12)

Here, $⌀\in {\tilde{C}}_{\tilde{B}}[0,\infty ),$$\varkappa \ge 0$. Via Equation (12), one has

$$\begin{array}{c}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}(1;\varkappa )=1,{\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}({⌀}_1^{\varkappa };\varkappa )=0\mathrm{and}|{\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}(⌀;\varkappa )|\le 3\parallel ⌀\parallel .\end{array}$$

(13)

Lemma 4.

For operators introduced in Equation (4) with $\varkappa \ge 0$,

$$\begin{array}{c}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}(⌀;\varkappa )-⌀\left(\varkappa \right)|\le \Theta \left(\varkappa \right)\parallel {⌀}^{″}\parallel ,\end{array}$$

where $⌀\in {\tilde{C}}_{\tilde{B}}^2[0,\infty )$ and $\Theta \left(\varkappa \right)={\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_2^{\varkappa };\varkappa )+{\left({\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1^{\varkappa };\varkappa )\right)}^2$.

Proof. 

By Taylor’s series expansion for $⌀\in {\tilde{C}}_{\tilde{B}}^2[0,\infty )$, we yield

$$\begin{array}{c}\hfill ⌀\left(\theta \right)=⌀\left(\varkappa \right)+(\theta -\varkappa ){⌀}^{\prime }\left(\varkappa \right)+\underset{\varkappa }{\overset{\theta }{\int }}(\theta -v){⌀}^{″}\left(v\right)dv.\end{array}$$

(14)

Employing the operator ${\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}(.;.)$ introduced in Equations (12)–(14), one has

$$\begin{array}{ccc}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}(⌀;\varkappa )-⌀\left(\varkappa \right)& =& {⌀}^{\prime }\left(\varkappa \right){\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1^{\varkappa };\varkappa )+{\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}\left(\underset{\varkappa }{\overset{\theta }{\int }}(\theta -v){⌀}^{″}\left(v\right)dv;\varkappa \right).\hfill \end{array}$$

Combining the Equations (13) and (14), we acquire

$$\begin{array}{cc}\hfill {\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}(⌀;\varkappa )-⌀\left(\varkappa \right)& ={\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}\left(\underset{\varkappa }{\overset{\theta }{\int }}(\theta -v){⌀}^{″}\left(v\right)dv;\varkappa \right)\hfill \\ \hfill & ={\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}\left(\underset{\varkappa }{\overset{\theta }{\int }}(\theta -v){⌀}^{″}\left(v\right)dv;\varkappa \right)\hfill \\ \hfill & -\underset{\varkappa }{\overset{\varkappa +{\displaystyle \frac{1}{r+\varsigma }}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)}{\int }}\left(\varkappa +{\displaystyle \frac{1}{r+\varsigma }}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)-v\right){⌀}^{″}\left(v\right)dv,\hfill \end{array}$$

$$\begin{array}{cc}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}(⌀;\varkappa )-⌀\left(\varkappa \right)|& \le \left|{\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}\left(\underset{\varkappa }{\overset{\theta }{\int }}(\theta -v){⌀}^{″}\left(v\right)dv;\varkappa \right)\right|\hfill \\ \hfill & +\left|\underset{\varkappa }{\overset{\varkappa +{\displaystyle \frac{1}{r+\varsigma }}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)}{\int }}\left(\varkappa +{\displaystyle \frac{1}{r+\varsigma }}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)-v\right){⌀}^{\prime \prime }\left(v\right)dv\right|.\hfill \end{array}$$

(15)

Since

$$\begin{array}{c}\hfill \left|\underset{\varkappa }{\overset{\theta }{\int }}(\theta -v){⌀}^{″}\left(v\right)dv\right|\le {(\theta -\varkappa )}^2\Vert {⌀}^{″}\Vert ,\end{array}$$

(16)

$$\begin{array}{c}\left|\underset{\varkappa }{\overset{\varkappa +{\displaystyle \frac{1}{r+\varsigma }}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)}{\int }}\left(\varkappa +{\displaystyle \frac{1}{r+\varsigma }}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)-v\right){⌀}^{″}\left(v\right)dv\right|\hfill \\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\le {\left({\displaystyle \frac{1}{r+\varsigma }}\left({\displaystyle \frac{1}{e-1}}+\lambda \right)\right)}^2\Vert {⌀}^{″}\Vert .\hfill \end{array}$$

(17)

According to Equations (15)–(17), we yield

$$\begin{array}{cc}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}(⌀;\varkappa )-⌀\left(\varkappa \right)|& \le \left\{{\mathfrak{S}}_{r,\varsigma ,\lambda }^{*}({⌀}_2^{\varkappa };\varkappa )+{\left({\displaystyle \frac{1}{r+\varsigma }}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)\right)}^2\right\}\parallel {⌀}^{″}\parallel \hfill \\ \hfill & =\Theta \left(\varkappa \right)\parallel {⌀}^{″}\parallel .\hfill \end{array}$$

which concludes the proof of the stated result. □

Theorem 4.

Let $⌀\in {\tilde{C}}_{\tilde{B}}^2[0,\infty )$ and the operator given in Equation (4). Then,

$$\begin{array}{c}\hfill \mid {\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)\mid \le \tilde{C}\widetilde{{\omega }_2}\left(⌀;\sqrt{\Theta \left(\varkappa \right)}\right)+\omega (⌀;{\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1^{\varkappa };\varkappa )),\end{array}$$

where $\tilde{C}\ge 0$ and $\Theta \left(\varkappa \right)$ is introduced in Lemma 4.

Proof. 

For $⌀\in {\tilde{C}}_{\tilde{B}}^2[0,\infty )$, $h\in {\tilde{C}}_{\tilde{B}}[0,\infty )$ and operator ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$ given in Equation (4), we obtain

$$\begin{array}{cc}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)|& \le |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀-h;\varkappa )|+|(⌀-h)\left(\varkappa \right)|+|{\mathfrak{S}}_{r,\varsigma ,\lambda }(h;\varkappa )-h\left(\varkappa \right)|\hfill \\ \hfill & +|⌀({\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1,\varkappa ))-⌀\left(\varkappa \right)|.\hfill \end{array}$$

In accordance with Lemma 4 and inequalities in Equation (13), we acquire

$$\begin{array}{cc}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)|& \le 4\parallel ⌀-h\parallel +|{\mathfrak{S}}_{r,\varsigma ,\lambda }(h;\varkappa )-h\left(\varkappa \right)|+|⌀\left({\mathfrak{S}}_{r,\varsigma ,\lambda }({⌀}_1,\varkappa )\right)-⌀\left(\varkappa \right)|\hfill \\ \hfill & \le 4\parallel ⌀-h\parallel +\theta \left(y\right)\parallel h^{″}\parallel +\omega \left(⌀;{\mathfrak{S}}_{r,\varsigma ,\lambda }((\theta -\varkappa );\varkappa )\right).\hfill \end{array}$$

Using Equation (11), we acquire the stated result. □

Following [30], we consider the Lipschitz-type functional space, defined as follows:

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

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

Theorem 5.

For $⌀\in Li{p}_M^{{\phi }_1,{\phi }_2}\left(\tau \right)$, and operator ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$, given in (4), one has

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

(18)

where $0<\tau \le 1$, ${\phi }_1,{\phi }_2\in (0,\infty )$ and $\lambda \left(y\right)={\mathfrak{S}}_{r,\varsigma ,\lambda }({\eta }_2;y)$.

Proof. 

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

$$\begin{array}{cc}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;y)-⌀\left(y\right)|& \le {\mathfrak{S}}_{r,\varsigma ,\lambda }\left(\right|⌀\left(t\right)-⌀\left(y\right)|;y)\hfill \\ \hfill & \le \tilde{M}{\mathfrak{S}}_{r,\varsigma ,\lambda }\left({\displaystyle \frac{|t-y|}{{(t+{\phi }_{1}y+{\phi }_2{y}^2)}^{\frac{1}{2}}}};y\right).\hfill \end{array}$$

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

$$\begin{array}{cc}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;y)-⌀\left(y\right)|& \le {\displaystyle \frac{\tilde{M}}{{({\phi }_{1}y+{\phi }_2{y}^2)}^{\frac{1}{2}}}}{\left({\mathfrak{S}}_{r,\varsigma ,\lambda }({\eta }_2;y)\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le \tilde{M}{\left({\displaystyle \frac{\lambda \left(y\right)}{{\phi }_{1}y+{\phi }_2{y}^2}}\right)}^{{\displaystyle \frac{1}{2}}},\hfill \end{array}$$

This confirms that Theorem 5 holds for $\tau =1$. We now turn our attention to the case $\tau \in (0,1)$, and in light of Hölder’s inequality, we proceed by opting for $p={\displaystyle \frac{2}{\tau }}$ and $q={\displaystyle \frac{2}{2-\tau }}$:

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

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 obtain

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

Thus, the desired result is established. □

We next examine the results of local approximation in the framework of the $b^{th}$ order modulus of continuity, utilizing the Lipschitz-type maximal function introduced by Lenze [31] as follows:

$$\begin{array}{c}\hfill {\tilde{\omega }}_b(⌀;\varkappa )=\underset{t\ne \varkappa ,t\in (0,\infty )}{sup}{\displaystyle \frac{|⌀(t)-⌀(\varkappa \left)\right|}{{|t-\varkappa |}^b}},\varkappa \in [0,\infty )\mathrm{and}b\in (0,1].\end{array}$$

(19)

Theorem 6.

Let $⌀\in {\tilde{C}}_{\tilde{B}}[0,\infty )$ and $b\in (0,1]$. Then, $\forall \varkappa \in [0,\infty )$, one has

$$\begin{array}{c}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)|\le {\tilde{\omega }}_r(⌀;\varkappa ){\left(\lambda \left(\varkappa \right)\right)}^{{\displaystyle \frac{b}{2}}}.\end{array}$$

Proof. 

It can be found that

$$\begin{array}{c}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)|\le {\mathfrak{S}}_{r,\varsigma ,\lambda }\left(\right|⌀\left(t\right)-⌀\left(\varkappa \right)|;\varkappa ).\end{array}$$

In light of Equation (19), we have

$$\begin{array}{c}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)|\le {\tilde{\omega }}_s(⌀;y){\mathfrak{S}}_{r,\varsigma ,\lambda }{\left(\right|t-\varkappa |}^b;\varkappa ).\end{array}$$

Then, according to Hölder’s inequality with $p_1={\displaystyle \frac{2}{b}}$ and $p_2={\displaystyle \frac{2}{2-b}}$, we have

$$\begin{array}{c}\hfill |{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)|\le {\tilde{\omega }}_b(⌀;\varkappa ){\left({\mathfrak{S}}_{r,\varsigma ,\lambda }{\left(\right|t-\varkappa |}^2;\varkappa )\right)}^{{\displaystyle \frac{b}{2}}}.\end{array}$$

This concludes the proof of the stated result. □

4. Bivariate Extension of Szász-Gamma Operators via Adjoint Bernoulli Polynomials

The Szász–Gamma operators are a class of operators combining the Szász basis with the Gamma function, typically used for approximation on unbounded intervals. Our bivariate extension follows this pattern. This section is dedicated to the construction of a bivariate extension of the Szász–Gamma type sequence of operators, formulated via the adjoint Bernoulli polynomials associated with ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$ defined in Equation (4). For related work on bivariate Appell interpolation, we refer to [32].

Let ${\kappa }^2=\left\{({\varkappa }_1,{\varkappa }_2):0\le {\varkappa }_1<\infty ,0\le {\varkappa }_2<\infty \right\}$, and the space of all continuous functions $C\left({\kappa }^2\right)$ on ${\kappa }^2$ be equipped with the norm

$$\begin{array}{c}\hfill {\parallel g\parallel }_{C\left({\kappa }^2\right)}=\underset{({\varkappa }_1,{\varkappa }_2)\in {\kappa }^2}{sup}|⌀({\varkappa }_1,{\varkappa }_2)|.\end{array}$$

Then, for all $⌀\in C\left({\kappa }^2\right)$ and $r_1,r_2\in \mathbb{N},0\le {\varsigma }_1,{\varsigma }_2<\infty$, we establish a bivariant extension of ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$ as follows:

$$\begin{array}{ccc}{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(⌀;{\varkappa }_1,{\varkappa }_2)\hfill & =\hfill & {\displaystyle \sum _{{\varrho }_1=0}^{\infty }}{\displaystyle \sum _{{\varrho }_2=0}^{\infty }}{g}_{{\varrho }_1}\left((r_1+{\varsigma }_1){\varrho }_1\right)g_{{\varrho }_2}\left((r_2+{\varsigma }_2){\varrho }_2\right)\hfill \\ \hfill & \times \hfill & \underset{0}{\overset{\infty }{\int }}\underset{0}{\overset{\infty }{\int }}{b}_{r_1,{\varsigma }_1,{\varrho }_1}^{\lambda }\left({\theta }_1\right)b_{r_2,{\varsigma }_2,{\varrho }_2}^{\lambda }\left({\theta }_2\right)⌀({\theta }_1,{\theta }_2)d{\theta }_{1}d{\theta }_2,\hfill \end{array}$$

(20)

where $\lambda >0$ is a real parameter ensuring convergence of the integral, $g_{{\varrho }_i}\left((r_i+{\varsigma }_i){\varrho }_i\right)$ and $b_{r_i,{\varsigma }_i,{\varrho }_i}^{\lambda }\left({\theta }_i\right)$ is mentioned in Equation (4).

Remark 3.

The operators formulated in Equation (20) are linear and positive, and are intended to approximate a class of Lebesgue measurable functions defined on two variables.

To analyze their rate of convergence and approximation order, we now establish several key lemmas as follows:

Here, we assume two-dimensional test functions as $p_{ij}={\varkappa }_1^i{\varkappa }_2^j$, and two-dimensional central moments as ${\Theta }_{ij}^{{\varkappa }_1,{\varkappa }_2}({\theta }_1,{\theta }_2)={({\theta }_1-{\varkappa }_1)}^i{({\theta }_2-{\varkappa }_2)}^j$, for $i,j\in \{0,1,2\}$.

Lemma 5.

For $⌀\in C\left({\kappa }^2\right)$, ${\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(.;.)$ given in Equation (20), and test functions $p_{ij}(.;.)$, we acquire

$$\begin{array}{ccc}\hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{00};{\varkappa }_1,{\varkappa }_2)& =& 1,\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{10};{\varkappa }_1,{\varkappa }_2)& =& {\varkappa }_1+{\displaystyle \frac{1}{(r_1+{\varsigma }_1)}}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{01};{\varkappa }_1,{\varkappa }_2)& =& {\varkappa }_2+{\displaystyle \frac{1}{(r_2+{\varsigma }_2)}}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,\lambda }(p_{20};{\varkappa }_1,{\varkappa }_2)& =& {\varkappa }_1^2+{\displaystyle \frac{{\varkappa }_1}{(r_1+{\varsigma }_1)}}(2e(\lambda +1)-2\lambda )\hfill \\ & +& {\displaystyle \frac{1}{{(r_1+{\varsigma }_1)}^2}}\left(1+{\displaystyle \frac{2\lambda +1}{e-1}}+{\lambda }^2+\lambda \right),\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,\lambda }(p_{02};{\varkappa }_1,{\varkappa }_2)& =& {\varkappa }_2^2+{\displaystyle \frac{{\varkappa }_2}{(r_2+{\varsigma }_2)}}(2e(\lambda +1)-2\lambda )\hfill \\ & +& {\displaystyle \frac{1}{{(r_2+{\varsigma }_2)}^2}}\left(1+{\displaystyle \frac{2\lambda +1}{e-1}}+{\lambda }^2+\lambda \right).\hfill \end{array}$$

Proof. 

To establish the lemma stated above, we begin by revisiting the concept of linear positive operators and Lemma 2 as follows:

$$\begin{array}{ccc}\hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{00};{\varkappa }_1,{\varkappa }_2)& =& {\mathfrak{S}}_{r_1,{\varsigma }_1,\lambda }(p_0;{\varkappa }_1){\mathfrak{S}}_{r_2,{\varsigma }_2,\lambda }(p_0;{\varkappa }_2),\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{10};{\varkappa }_1,{\varkappa }_2)& =& {\mathfrak{S}}_{r_1,{\varsigma }_1,\lambda }(p_1;{\varkappa }_1){\mathfrak{S}}_{r_2,{\varsigma }_2,\lambda }(p_0;{\varkappa }_2),\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{01};{\varkappa }_1,{\varkappa }_2)& =& {\mathfrak{S}}_{r_1,{\varsigma }_1,\lambda }(p_0;{\varkappa }_1){\mathfrak{S}}_{r_2,{\varsigma }_2,\lambda }(p_1;{\varkappa }_2),\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{20};{\varkappa }_1,{\varkappa }_2)& =& {\mathfrak{S}}_{r_1,{\varsigma }_1,\lambda }(p_2;{\varkappa }_1){\mathfrak{S}}_{r_2,{\varsigma }_2,\lambda }(p_0;{\varkappa }_2),\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{02};{\varkappa }_1,{\varkappa }_2)& =& {\mathfrak{S}}_{r_1,{\varsigma }_1,\lambda }(p_0;{\varkappa }_2){\mathfrak{S}}_{r_2,{\varsigma }_2,\lambda }(p_1;{\varkappa }_2).\hfill \end{array}$$

Based on the equalities derived above and Lemma 2, proof of Lemma 5 readily follows. □

Lemma 6.

For ${\Theta }_{ij}={({\theta }_1-{\varkappa }_1)}^i{({\theta }_2-{\varkappa }_2)}^j$ for $i,j=0,1,2,$ we yield

$$\begin{array}{ccc}\hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }({\Theta }_{00};{\varkappa }_1,{\varkappa }_2)& =& 1,\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }({\Theta }_{10};{\varkappa }_1,{\varkappa }_2)& =& {\displaystyle \frac{1}{(r_1+{\varsigma }_1)}}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }({\Theta }_{01};{\varkappa }_1,{\varkappa }_2)& =& {\displaystyle \frac{1}{(r_2+{\varsigma }_2)}}\left(\lambda +{\displaystyle \frac{1}{e-1}}\right),\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }({\Theta }_{20};{\varkappa }_1,{\varkappa }_2)& =& {\displaystyle \frac{{\varkappa }_1}{(r_1+{\varsigma }_1)}}\left(2e(\lambda +1)-2\lambda -2\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)\right)\hfill \\ & +& {\displaystyle \frac{1}{{(r_1+{\varsigma }_1)}^2}}\left(1+{\displaystyle \frac{2\lambda +1}{e-1}}+{\lambda }^2+\lambda \right),\hfill \\ \hfill {\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }({\Theta }_{02};{\varkappa }_1,{\varkappa }_2)& =& {\displaystyle \frac{{\varkappa }_2}{(r_2+{\varsigma }_2)}}\left(2e(\lambda +1)-2\lambda -2\left(\lambda +{\displaystyle \frac{1}{e-1}}\right)\right)\hfill \\ & +& {\displaystyle \frac{1}{{(r_2+{\varsigma }_2)}^2}}\left(1+{\displaystyle \frac{2\lambda +1}{e-1}}+{\lambda }^2+\lambda \right).\hfill \end{array}$$

Proof. 

Taking into account Lemma 5 along with the linearity of operators, the required result can be demonstrated with ease. □

Definition 2.

Consider $\kappa =[0,\infty )\subset \backslash +⫋$ as given intervals and $B(\kappa \times \kappa )=\{⌀:\kappa \times \kappa \to \backslash +⫋:⌀isdefinedandboundedon\kappa \times \kappa \}.$ Then, for $g\in B(\kappa \times \kappa )$, the total modulus of continuity is defined as follows: ${\omega }_{total}(⌀;·,*):C\left({\kappa }^2\right)\to \backslash +⫋$ provided that $({\tilde{\delta }}_1,{\tilde{\delta }}_2)\in \kappa \times \kappa$ and is defined by

$${\omega }_{total}(⌀;{\tilde{\delta }}_1,{\tilde{\delta }}_2)=\underset{|x_1-x_1^{\prime }|\le {\tilde{\delta }}_1,|y_1-y_1^{\prime }|\le {\tilde{\delta }}_2}{sup}\left\{\right|⌀(x_1,y_1)-⌀(x_1^{\prime },y_1^{\prime })|:(x_1,y_1),$$

$(x_1^{\prime },y_1^{\prime })\in \kappa \times \kappa \},$ is referred to as the total modulus of continuity associated with function ⌀.

We now examine the convergence rate of operators defined in Equation (20). To support this investigation, we recall the following result established by Volkov [33]:

Theorem 7.

Let κ be the compact interval of a real line and $L_{r_1,r_2}^{{\varsigma }_1,{\varsigma }_2}:C(I\times J)\to C(\kappa \times \kappa ),(r_1,r_2)\in \mathbb{N}\times \mathbb{N},{\varsigma }_1,{\varsigma }_2\in [0,\infty )\times [0,\infty )$ be positive linear operators. If

$$\begin{array}{ccc}\hfill \underset{r_1,r_2\to \infty }{lim}{L}_{r_1,r_2}^{{\varsigma }_1,{\varsigma }_2}\left(p_{ij}\right)& =& p_{ij}({\varkappa }_1,{\varkappa }_2),(i,j)\in \{(0,0),(1,0),(0,1)\}\hfill \end{array}$$

and

$$\begin{array}{ccc}\hfill \underset{r_1,r_2\to \infty }{lim}{L}_{r_1,r_2}^{{\varsigma }_1,{\varsigma }_2}(p_{20}\left(({\varkappa }_1,{\varkappa }_2)\right)+p_{02}\left(({\varkappa }_1,{\varkappa }_2)\right))& =& p_{20}\left(({\varkappa }_1,{\varkappa }_2)\right)+p_{02}\left(({\varkappa }_1,{\varkappa }_2)\right),\hfill \end{array}$$

uniformly on ${\kappa }^2$; then, sequence $\left(L_{r_1,r_2}^{{\varsigma }_1,{\varsigma }_2}f\right)$ converges to ⌀ uniformly on ${\kappa }^2$ for any $⌀\in C\left({\kappa }^2\right)$.

Theorem 8.

Let $p_{ij}({\varkappa }_1,{\varkappa }_2)={\varkappa }_1^i{\varkappa }_2^j(0\le i+j\le 2,i,j\in \mathbb{N})$ be test functions restricted on ${\kappa }^2$. If

$$\begin{array}{c}\hfill \underset{r_1,r_2\to \infty }{lim}{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{ij};{\varkappa }_1,{\varkappa }_2)=p_{ij}({\varkappa }_1,{\varkappa }_2),\end{array}$$

and

$$\begin{array}{c}\hfill \underset{r_1,r_2\to \infty }{lim}{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{20}+p_{02};{\varkappa }_1,{\varkappa }_2)=p_{20}({\varkappa }_1,{\varkappa }_2)+p_{02}({\varkappa }_1,{\varkappa }_2),\end{array}$$

uniformly on ${\kappa }^2$, then

$$\begin{array}{c}\hfill \underset{r_1,r_2\to \infty }{lim}{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(⌀;{\varkappa }_1,{\varkappa }_2)=⌀({\varkappa }_1,{\varkappa }_2),\end{array}$$

uniformly for every $⌀\in C\left({\kappa }^2\right)$.

Proof. 

According to Lemma 5, it is clear that for $i=j=0$

$$\begin{array}{c}\hfill \underset{r_1,r_2\to \infty }{lim}{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{00};{\varkappa }_1,{\varkappa }_2)=p_{00}({\varkappa }_1,{\varkappa }_2).\end{array}$$

For $i=1$, $j=0$, one yields

$$\begin{array}{ccc}\hfill \underset{r_1,r_2\to \infty }{lim}{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{10};{\varkappa }_1,{\varkappa }_2)& =& {\varkappa }_1,\hfill \\ \hfill \underset{r_1,r_2\to \infty }{lim}{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{10};{\varkappa }_1,{\varkappa }_2)& =& p_{10}({\varkappa }_1,{\varkappa }_2).\hfill \end{array}$$

Similarly,

$$\begin{array}{ccc}\hfill \underset{r_1,r_2\to \infty }{lim}{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{01};{\varkappa }_1,{\varkappa }_2)& =& {\varkappa }_2,\hfill \\ \hfill \underset{r_1,r_2\to \infty }{lim}{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{01};{\varkappa }_1,{\varkappa }_2)& =& p_{01}({\varkappa }_1,{\varkappa }_2),\hfill \end{array}$$

and in light of lemma 5, we obtain

$$\begin{array}{ccc}\hfill \underset{r_1,r_2\to \infty }{lim}{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(p_{20}+p_{02};{\varkappa }_1,{\varkappa }_2)& =& {\varkappa }_1^2+{\varkappa }_2^2,\hfill \\ & =& p_{20}({\varkappa }_1,{\varkappa }_2)+p_{02}({\varkappa }_1,{\varkappa }_2).\hfill \end{array}$$

In light of Theorems 7 and 8 this proceeds directly. □

In the final result, we address the approximation order associated with operators ${\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(.;.)$ in Equation (20) as follows:

Theorem 9 ([34]).

Let $L:C\left({\kappa }^2\right)\to B\left({\kappa }^2\right)$ be a positive linear operator. For any $⌀\in C\left({\kappa }^2\right)$, any $(z_1,z_2)\in {\kappa }^2$ and any ${\tilde{\delta }}_1,{\tilde{\delta }}_2>0$, the following inequality,

$$\begin{array}{ccc}\hfill |(L⌀)(z_1,z_2)-⌀(z_1,z_2)|& \le & |L{p}_{0,0}(z_1,z_2)-1\left|\right|⌀(z_1,z_2)|+[L{p}_{0,0}(z_1,z_2)\hfill \\ & +& {\tilde{\delta }}_1^{-1}\sqrt{L{p}_{0,0}(z_1,z_2){\left(L(·-z_1)\right)}^2(z_1,z_2)}\hfill \\ & +& {\tilde{\delta }}_2^{-1}\sqrt{L{p}_{0,0}(z_1,z_2){\left(L(*-z_2)\right)}^2(z_1,z_2)}\hfill \end{array}$$

$$\begin{array}{ccc}& +& {\tilde{\delta }}_1^{-1}{\tilde{\delta }}_2^{-1}\sqrt{{\left(L{p}_{0,0}\right)}^2(z_1,z_2){\left(L(·-z_1)\right)}^2(z_1,z_2){\left(L(*-z_2)\right)}^2(z_1,z_2)}]\hfill \\ & \times & {\omega }_{total}(⌀;{\tilde{\delta }}_1,{\tilde{\delta }}_2),\hfill \end{array}$$

holds.

Theorem 10.

For $⌀\in C\left({\kappa }^2\right)$ and $({\varkappa }_1,{\varkappa }_2)\in {\kappa }^2$, $(r_1,r_2)\in \mathbb{N}\times \mathbb{N}$ and ${\tilde{\delta }}_1,{\tilde{\delta }}_2>0$, we yield

$$\begin{array}{c}\hfill |{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }(⌀;{\varkappa }_1,{\varkappa }_2)-⌀({\varkappa }_1,{\varkappa }_2)|\le 4{\omega }_{total}(⌀;{\tilde{\delta }}_1,{\tilde{\delta }}_2),\end{array}$$

where

$$\begin{array}{cc}\hfill {\tilde{\delta }}_1& =\sqrt{{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }\left({({\theta }_1-{\varkappa }_1)}^2;{\varkappa }_1,{\varkappa }_2\right)},\hfill \\ \hfill {\tilde{\delta }}_2& =\sqrt{{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }\left({({\theta }_2-{\varkappa }_2)}^2;{\varkappa }_1,{\varkappa }_2)\right)}.\hfill \end{array}$$

Proof. 

From Theorem 9, we have

$$\begin{array}{ccc}& & |({\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }⌀)({\varkappa }_1,{\varkappa }_2)-⌀({\varkappa }_1,{\varkappa }_2)|\hfill \\ & \le & [1++{\tilde{\delta }}_1^{-1}\sqrt{{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }\left({({\theta }_1-{\varkappa }_1)}^2;{\varkappa }_1,{\varkappa }_2\right)}\hfill \\ & +& {\tilde{\delta }}_2^{-1}\sqrt{{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }\left({({\theta }_2-{\varkappa }_2)}^2;{\varkappa }_1,{\varkappa }_2\right)}\hfill \\ & +& {\tilde{\delta }}_1^{-1}{\tilde{\delta }}_2^{-1}\sqrt{{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }\left({({\theta }_1-{\varkappa }_1)}^2;{\varkappa }_1,{\varkappa }_2\right){\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }\left({({\theta }_2-{\varkappa }_2)}^2;{\varkappa }_1,{\varkappa }_2\right)}]\hfill \\ & \times & {\omega }_{total}(f;{\tilde{\delta }}_1,{\tilde{\delta }}_2).\hfill \end{array}$$

Opting for ${\tilde{\delta }}_1=\sqrt{{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }\left({({\theta }_1-{\varkappa }_1)}^2;{\varkappa }_1,{\varkappa }_2\right)}$ and ${\tilde{\delta }}_2=\sqrt{{\mathfrak{S}}_{r_1,r_2,{\varsigma }_1,{\varsigma }_2,\lambda }\left({({\theta }_2-{\varkappa }_2)}^2;{\varkappa }_1,{\varkappa }_2)\right)}$, we obtain the required result. □

5. Numerical Validation

In order to assess the effectiveness and approximation behavior of the newly developed operators ${\mathfrak{S}}_{r,\varsigma ,\lambda }$, we present a detailed numerical study. The analysis is conducted by applying these operators to a representative test function $⌀\left(\varkappa \right)={\displaystyle \frac{1}{\pi }}\left(\varkappa -e\right)\left(\varkappa -{\displaystyle \frac{1}{3}}\right)$, which is defined on the interval [0,2]. The resulting numerical values and error estimates provide insight into the convergence characteristics and practical performance of the proposed operators.

Example 1.

To examine the numerical performance and approximation accuracy of the newly introduced operators ${\mathfrak{S}}_{r,\varsigma ,\lambda }$, we consider the test function $⌀\left(\varkappa \right)={\displaystyle \frac{1}{\pi }}\left(\varkappa -e\right)\left(\varkappa -{\displaystyle \frac{1}{3}}\right),$ defined on the interval $[0,2]$. For various values of the parameter r, the operators ${\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )$ are evaluated and compared with the exact function values in Figure 1. The absolute error is computed as $E_r\left(\varkappa \right)=\left|{\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )-⌀\left(\varkappa \right)\right|$ in Figure 2. The numerical values of the approximations and the corresponding errors are summarized in Table 1, while graphical representations are provided to illustrate the convergence behavior and effectiveness of the proposed operators.

Figure 1. Convergence of the operators ${\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )$ to the function $⌀\left(\varkappa \right)={\displaystyle \frac{1}{\pi }}(\varkappa -e)\left(\varkappa -{\displaystyle \frac{1}{3}}\right)$.

Figure 2. Absolute error of the operators ${\mathfrak{S}}_{r,\varsigma ,\lambda }(⌀;\varkappa )$ to the function $⌀\left(\varkappa \right)={\displaystyle \frac{1}{\pi }}(\varkappa -e)\left(\varkappa -{\displaystyle \frac{1}{3}}\right)$.

Table 1. Numerical values for different parameter choices.

x	$E_{40}$	$E_{50}$	$E_{60}$	$E_{70}$
0.10000	0.05590	0.04514	0.03785	0.03259
0.20000	0.05189	0.04192	0.03516	0.03027
0.30000	0.04788	0.03869	0.03246	0.02796
0.40000	0.04387	0.03547	0.02977	0.02564
0.50000	0.03986	0.03225	0.02707	0.02333
0.60000	0.03585	0.02902	0.02438	0.02101
0.70000	0.03185	0.02580	0.02168	0.01870
0.80000	0.02784	0.02258	0.01899	0.01638
0.90000	0.02383	0.01935	0.01629	0.01407
1.00000	0.01982	0.01613	0.01360	0.01175
1.10000	0.01581	0.01291	0.01090	0.00944
1.20000	0.01180	0.00968	0.00821	0.00712
1.30000	0.00779	0.00646	0.00551	0.00481
1.40000	0.00378	0.00324	0.00282	0.00249
1.50000	0.00023	0.00002	0.00013	0.00018
1.60000	0.00424	0.00321	0.00257	0.00214
1.70000	0.00825	0.00643	0.00526	0.00445
1.80000	0.01226	0.00965	0.00796	0.00677
1.90000	0.01626	0.01288	0.01065	0.00908
2.00000	0.02027	0.01610	0.01335	0.01140

It is observed that as the value of r increases, the corresponding absolute error $E_r\left(\varkappa \right)$ decreases, indicating an improvement in the approximation accuracy of the operators.

These theoretical properties are consistent with recent advancements in operator approximation theory [21], where similar modifications to classical operators have yielded improved convergence rates and numerical stability.

These results confirm the theoretical advantages of incorporating adjoint Bernoulli polynomials with gamma functions in operator design, providing enhanced approximation capabilities compared to classical approaches.

6. Conclusions

In this work, we establish a new relation between adjoint Bernoulli’s polynomials and gamma function as a sequence of positive linear operators denoted by ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$. Further, we investigate convergence results of these operators, i.e., ${\mathfrak{S}}_{r,\varsigma ,\lambda }(.;.)$, in several functional spaces employing the Korovkin theorem, the Lipschitz condition, the Voronovskaja-type theorem, the first- and second-order of the modulus of continuity, Peetre’s K-functional, etc. Lastly, we examine the uniform rate and approximation order for the bivariate extension of these operators in several functional spaces. The advantages of these operators over the linear positive operators are as follows:

• These operators serve as a bridge study of operator theory and special functions research.
• These operators are suitable for discussing the approximation behaviour of functions in a Lebesgue sense.