Korovkin-Type Theorems for Positive Linear Operators Based on the Statistical Derivative of Deferred Cesàro Summability

Abstract

In this paper, we introduce and investigate the concept of statistical derivatives within the framework of the deferred Cesàro summability technique, supported by illustrative examples. Using this approach, we establish a novel Korovkin-type theorem for a specific set of exponential test functions, namely 1, $e^{-\upsilon }$ and $e^{-2\upsilon }$, which are defined on the Banach space $\mathcal{C}[0,\infty )$. Our results significantly extend several well-known Korovkin-type theorems. Additionally, we analyze the rate of convergence associated with the statistical derivatives under deferred Cesàro summability. To support our theoretical findings, we provide compelling numerical examples, followed by graphical representations generated using MATLAB software, to visually illustrate and enhance the understanding of the convergence behavior of the operators.

1. Introduction and Preliminaries

In mathematical analysis, fundamental concepts such as limits, continuity, and derivatives have been rigorously explored for centuries, forming the foundation of calculus and analysis (see [1]). These fundamental ideas have been extended and adapted in various ways to address more complex problems and introduce new mathematical tools. One such extension is the study of Cesàro limits, a variation that has led to significant advancements in the broader field of summability theory, particularly for sequences and series (see [2]). Building on these developments, researchers have continued to investigate new concepts and techniques to deepen our understanding of summability and its applications (see [3]). Among the recent contributions, Connor and Grosse-Erdmann [4] made notable strides by exploring Cesàro continuity, statistical limits, and statistical continuity. These studies have expanded the theoretical landscape, providing valuable insights into the behavior of functions and sequences in terms of their convergence properties under various summability methods.

Despite these advancements, certain areas within summability theory remain underexplored. Specifically, the notions of the deferred Cesàro derivative and the deferred Cesàro derivative in the statistical sense have yet to be fully examined in the literature. These concepts hold the potential to further refine the understanding of summability, offering a more nuanced approach to analyzing the convergence of functions, particularly in the context of statistical and Cesàro summability.

This study aims to bridge the existing gap by introducing the concepts of the deferred Cesàro derivative and the statistical deferred Cesàro derivative within the framework of the deferred Cesàro summability technique, supported by illustrative examples. Through this methodology, we introduce a new Korovkin-type theorem applicable to a particular class of exponential test functions, specifically 1, $e^{-\upsilon }$ and $e^{-2\upsilon }$, within the Banach space $\mathcal{C}[0,\infty )$. Our findings offer a substantial generalization of several established Korovkin-type results. Additionally, we investigate the convergence rate of statistical derivatives under the framework of deferred Cesàro summability. To validate our theoretical results, we present numerical examples alongside MATLAB-generated graphical illustrations, providing a clearer insight into the convergence behavior of the operators.

2. Preliminaries

Let $(C,1)$ be the mean [2] of the sequence $\upsilon =\left({\upsilon }_{\rho }\right)$; thus, we have

$${\sigma }_{\rho }={\displaystyle \frac{1}{\rho +1}}\sum _{i=0}^{\upsilon }{\upsilon }_i.$$

Subsequently, the deferred Cesàro method for sequences is given as follows:

Let $\left(a_{\rho }\right)\mathrm{and}\left(b_{\rho }\right)\in {\mathbb{Z}}^{0+}$ be such that

$$\begin{array}{c}\hfill a_{\rho }<b_{\rho },\rho \in \mathbb{N}\mathrm{and}\underset{\rho \to \infty }{lim}{b}_{\rho }=\infty .\end{array}$$

The deferred Cesàro $D(a_{\rho },b_{\rho })$-mean (see Agnew [5]) of a sequence $\left({\upsilon }_{\rho }\right)$ is given by

$$\begin{array}{cc}\hfill D(a_{\rho },b_{\rho })& ={\displaystyle \frac{{\upsilon }_{a_{\rho }+1}+{\upsilon }_{a_{\rho }+2}+{\upsilon }_{a_{\rho }+3}+\cdots +{\upsilon }_{b_{\rho }}}{b_{\rho }-a_{\rho }}}\hfill \\ \hfill & ={\displaystyle \frac{1}{b_{\rho }-a_{\rho }}}\sum _{i=a_{\rho }+1}^{b_{\rho }}{\upsilon }_i.\hfill \end{array}$$

In recent years, statistical convergence has gained considerable importance, often being preferred over usual convergence due to its broader applicability. This development can be traced back to the foundational contributions of Fast [6] and Steinhaus [7], who expanded the theoretical framework of statistical convergence. Nowadays, this concept plays a vital part in various areas of mathematics, as well as analytical statistics. It has significant applications in disciplines such as data science, industrial mathematics, computational algebra, and cryptography. For further details, readers may refer to [8,9].

Suppose $\mathcal{H}\subseteq \mathbb{N}$, setting

$$\begin{array}{c}\hfill {\mathcal{H}}_{\rho }:=\left\{\rho :\rho \leqq \tau \mathrm{and}\rho \in \mathbb{N}\right\}.\end{array}$$

(1)

Thus, the asymptotic density $d\left(\mathcal{H}\right)$ of $\mathcal{H}$ is specified by

$$\begin{array}{cc}\hfill d\left(\mathcal{H}\right)& =\underset{\rho \to \infty }{lim}{\displaystyle \frac{|{\mathcal{H}}_{\rho }|}{\rho }}=a,\hfill \end{array}$$

(2)

where $|{\mathcal{H}}_{\rho }|$ represents the number of elements in the set ${\mathcal{H}}_{\rho }$.

Definition 1.

A given sequence $\left({\upsilon }_{\rho }\right)$ converges statistically to a finite number κ if, for all $ϵ>0$,

$$\begin{array}{c}\hfill {\mathcal{H}}_{ϵ}=\left\{\rho \in \mathbb{N}\mathrm{and}\left|{\upsilon }_{\rho }-\kappa \right|\geqq ϵ\right\}\end{array}$$

ensures zero asymptotic density (see [6,7]). That is, for every$ϵ>0$,

$$\begin{array}{c}\hfill d\left({\mathcal{H}}_{ϵ}\right)={\displaystyle \frac{\left|{\mathcal{H}}_{ϵ}\right|}{\rho }}=0\mathrm{as}\rho \to \infty .\end{array}$$

If the limit exist, we denote this as

$$\begin{array}{c}\hfill \mathrm{stat}lim{\upsilon }_{\rho }=\kappa .\end{array}$$

Definition 2.

Let $\left(a_{\rho }\right)\mathrm{and}\left(b_{\rho }\right)\in {\mathbb{Z}}^{0+}$. A sequence $\left({\upsilon }_{\rho }\right)$ is deferred to as statistically convergent $\left({\mathrm{stat}}_{\mathrm{D}}\right)$ to κ if, for all $ϵ>0$,

$$\begin{array}{c}\hfill {\mathcal{H}}_{ϵ}=\left\{a_{\rho }\leqq b_{\rho }\mathrm{and}\left|{\upsilon }_{\rho }-\kappa \right|\geqq ϵ\right\}\end{array}$$

ensures zero asymptotic density. That is, for all $ϵ>0$,

$$\begin{array}{c}\hfill d\left({\mathcal{H}}_{ϵ}\right)={\displaystyle \frac{\left|{\mathcal{H}}_{ϵ}\right|}{\rho }}=0\mathrm{as}\rho \to \infty .\end{array}$$

We write

$$\begin{array}{c}\hfill {\mathrm{stat}}_{\mathrm{D}}\underset{\rho \to \infty }{lim}{\upsilon }_{\rho }=\kappa .\end{array}$$

In the latter half of the twentieth century, numerous researchers made significant contributions to the development of statistical convergence. Šalát [10] investigated statistically convergent sequences of real numbers, particularly their boundedness properties. Fridy [6] formalized the Cauchy criterion for statistical convergence and established fundamental results using summability techniques. Maddox [11] expanded this research by exploring statistical convergence in locally convex spaces, leading to several important insights. Building upon these advancements, Fridy and Orhan [12] introduced the notion of lacunary statistical summability for sequences of real numbers, which paved the way for further significant developments in the field.

The concept of Cesàro summability in the statistical sense, along with its applications, was originally introduced by Móricz [13]. Mohiuddine et al. [14] expanded on this foundation by presenting significant results on statistical summability means, supported by illustrative examples, and by proving related Korovkin-type theorems. Karakaya and Chishti [15] further contributed to the field by enhancing the understanding of statistical convergence through weighted summability means. Subsequently, Mursaleen et al. [16] refined this approach by introducing modifications that led to the formulation of fundamental limit theorems. More recently, Baliarsingh et al. [17] extended the concept by proposing an advanced framework for uncertain sequences using statistical deferred A-convergence, establishing several inclusion theorems. For further insights in this area, scholars and researchers may consult recently published works, such as [18,19,20,21,22,23].

3. Statistical Deferred Cesàro Derivative

The derivative of a function $\eta :\mathbb{R}\to \mathbb{R}$ at a point ${\upsilon }_0$ can be defined using two fundamental formulas, both of which involve a positive sequence $\left({\upsilon }_{\rho }\right)$ such that ${lim}_{\rho \to \infty }{\upsilon }_{\rho }=0$. These formulas are given as follows:

$$\begin{array}{c}\hfill {\eta }^{\prime }\left({\upsilon }_0\right)=\underset{\rho \to \infty }{lim}{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta \left({\upsilon }_0\right)}{{\upsilon }_{\rho }}}\mathrm{and}{\eta }^{\prime }\left({\upsilon }_0\right)=\underset{\rho \to \infty }{lim}{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta ({\upsilon }_0-{\upsilon }_{\rho })}{2{\upsilon }_{\rho }}}.\end{array}$$

The initial formula, referred to as Newton’s difference quotient, represents the slope of a secant line to the graph of $\eta$. Meanwhile, the second formula, known as the symmetric difference quotient, calculates the slope of a chord on the graph of $\eta$. For additional information, see [24].

Using a similar methodology, we now extend the definition to encompass the derivative of the deferred Cesàro summability mean.

Definition 3.

A function $\eta :\mathbb{R}\to \mathbb{R}$ is said to possess a deferred Cesàro derivative $\kappa \in \mathbb{R}$ at a point ${\upsilon }_0\in \mathbb{R}$ if

$$\begin{array}{c}\hfill \underset{\rho \to \infty }{lim}{\displaystyle \frac{1}{b_{\rho }-a_{\rho }}}\sum _{i=a_{\rho }+1}^{b_{\rho }}{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_i)-\eta \left({\upsilon }_0\right)}{{\upsilon }_i}}=\kappa ,\end{array}$$

where ${\upsilon }_0>0$ and ${lim}_{\rho \to \infty }{\upsilon }_{\rho }=0$.

An equivalent formulation of Definition 3 is presented as follows:

Definition 4.

A function $\eta :\mathbb{R}\to \mathbb{R}$ is said to possess a deferred Cesàro derivative $\kappa \in \mathbb{R}$ at a point ${\upsilon }_0\in \mathbb{R}$ if

$$\begin{array}{c}\hfill \underset{\rho \to \infty }{lim}{\displaystyle \frac{1}{b_{\rho }-a_{\rho }}}\sum _{i=a_{\rho }+1}^{b_{\rho }}{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_i)-\eta ({\upsilon }_0-{\upsilon }_i)}{2{\upsilon }_i}}=\kappa \end{array}$$

holds, whenever ${\upsilon }_0>0$ and ${lim}_{\rho \to \infty }{\upsilon }_{\rho }=0$.

Next, we introduce the definition and an illustrative example of the statistical derivative for the deferred Cesàro mean as follows:

Definition 5.

A function $\eta :\mathbb{R}\to \mathbb{R}$ is said to possess a statistical deferred Cesàro derivative $\kappa \in \mathbb{R}$ at a point ${\upsilon }_0\in \mathbb{R}$ if

$$\begin{array}{c}\hfill \underset{\rho \to \infty }{lim}{\displaystyle \frac{1}{b_{\rho }-a_{\rho }}}\left|\left\{a_{\rho }\leqq b_{\rho }\mathrm{and}\left|{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta \left({\upsilon }_0\right)}{{\upsilon }_{\rho }}}-\kappa \right|\geqq ϵ\right\}\right|=0\end{array}$$

holds, whenever ${\upsilon }_0>0$ and ${lim}_{\rho \to \infty }{\upsilon }_{\rho }=0$.

In this context, we express it as follows:

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\eta \left({\upsilon }_{\rho }\right)=\kappa .\end{array}$$

Example 1.

Consider the function $\eta \left(\upsilon \right)={\upsilon }^2$, and let ${\upsilon }_0=1$. Define the sequences $\left(a_{\rho }\right)=\rho$, $\left(b_{\rho }\right)=2\rho$, and $\left({\upsilon }_{\rho }\right)={\displaystyle \frac{1}{\rho }}$. We aim to verify whether η has a statistical deferred Cesàro derivative κ at ${\upsilon }_0=1$. For $\eta \left(\upsilon \right)={\upsilon }^2$, the difference quotient is given by

$${\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta \left({\upsilon }_0\right)}{{\upsilon }_{\rho }}}={\displaystyle \frac{{\left(1+\frac{1}{\rho }\right)}^2-1^2}{\frac{1}{\rho }}}=2+{\displaystyle \frac{1}{\rho }}.$$

As $\rho \to \infty$, the difference quotient converges to $\kappa =2$. In view of the statistical deferred Cesàro derivative,

$$\begin{array}{c}\hfill \underset{\rho \to \infty }{lim}{\displaystyle \frac{1}{b_{\rho }-a_{\rho }}}\left|\left\{a_{\rho }\leqq b_{\rho }\mathrm{and}\left|{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta \left({\upsilon }_0\right)}{{\upsilon }_{\rho }}}-\kappa \right|\geqq ϵ\right\}\right|=0,\end{array}$$

for a given $ϵ>0$, we have

$$\left|{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta \left({\upsilon }_0\right)}{{\upsilon }_{\rho }}}-\kappa \right|={\displaystyle \frac{1}{\rho }},$$

which becomes arbitrarily small as $\rho \to \infty$. Hence, the set of ρ for which the inequality

$$\left|{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta \left({\upsilon }_0\right)}{{\upsilon }_{\rho }}}-\kappa \right|\geqq ϵ,$$

holds becomes negligible in the deferred Cesàro sense as $\rho \to \infty$. Thus, $\eta \left(\upsilon \right)={\upsilon }^2$ has a statistical deferred Cesàro derivative $\kappa =2$ at ${\upsilon }_0=1$.

Figure 1 illustrates how the difference quotient, calculated for the sequence $\left({\upsilon }_{\rho }\right)={\displaystyle \frac{1}{\rho }}$, approaches the true derivative value $\kappa =2$ as $\rho \to \infty$. This behavior highlights the concept of the statistical derivative under deferred Cesàro summability, demonstrating how the values stabilize around the limit over increasing index ranges. Moreover, Figure 1 provides a practical verification of the theoretical definition of the statistical derivative of deferred Cesàro summability. By including the true derivative as a reference line $(\kappa =2)$, the figure effectively distinguishes the transient fluctuations of the difference quotient from its eventual convergence. Notably, Example 1 neither exhibits usual convergence nor statistical convergence. Additionally, it is not statistically Cesàro summable. However, it has a statistical deferred Cesàro derivative 2.

An alternative but equivalent formulation of Definition 5 is given as follows:

Definition 6.

A function $\eta :\mathbb{R}\to \mathbb{R}$ is to possess a statistical deferred Cesàro derivative $\kappa \in \mathbb{R}$ at a point ${\upsilon }_0\in \mathbb{R}$ if

$$\begin{array}{c}\hfill \underset{\rho \to \infty }{lim}{\displaystyle \frac{1}{b_{\rho }-a_{\rho }}}\left|\left\{a_{\rho }\leqq b_{\rho }\mathrm{and}\left|{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta ({\upsilon }_0-{\upsilon }_{\rho })}{2{\upsilon }_{\rho }}}-\kappa \right|\geqq ϵ\right\}\right|=0\end{array}$$

holds, whenever ${\upsilon }_0>0$ and ${lim}_{\rho \to \infty }{\upsilon }_{\rho }=0$.

In this scenario, we can express it as follows:

$$\begin{array}{c}\hfill {\mathrm{stat}}_{\mathrm{DCD}}\eta \left({\upsilon }_{\rho }\right)=\kappa .\end{array}$$

Motivated by the aforementioned studies, we introduce and thoroughly examine the concept of statistical derivatives within the context of the deferred Cesàro summability method. Building upon this approach, we derive a new Korovkin-type theorem for given exponential test functions 1, $e^{-\upsilon }$ and $e^{-2\upsilon }$ specified over $\mathcal{C}[0,\infty )$. Our results extend and enhance several well-established Korovkin-type theorems, introducing a more generalized and robust method for analyzing the summability of functions within this class. Through the application of the deferred Cesàro summability technique, we demonstrate how the behavior of the statistical derivatives can be examined more precisely, offering a deeper understanding of their convergence properties. In addition, we investigate the rate of convergence of statistical derivatives under deferred Cesàro summability, providing a thorough analysis of how this method influences the convergence rate for the exponential test functions we consider. To substantiate our theoretical results, we present a compelling numerical example, accompanied by graphical representations generated using MATLAB software. These visualizations clearly illustrate the practical effectiveness of our approach and offer insights into the convergence behavior of the operators.

4. Korovkin-Type Theorem

The Korovkin-type theorem is a fundamental result in approximation theory, providing a criterion for determining when a sequence of operators converges to a given function. The classical Korovkin theorem states that a sequence of operators ${\mathcal{L}}_{\rho }(\eta ;\upsilon )$ operating within the space of continuous functions $\mathcal{C}[a,b]$ converges uniformly to any function $\eta$ if it successfully approximates the three test functions 1, $\upsilon$ and ${\upsilon }^2$ on $[a,b]$. This theorem simplifies the verification of convergence, as proving it for these basic functions is sufficient for the entire function space. Over time, several generalizations have been developed, extending the theorem to different function spaces under weighted settings, statistical summability, and deferred Cesàro methods. These extensions have played a crucial role in functional analysis, numerical analysis, and applied mathematics, making Korovkin-type theorems fundamental in understanding the behavior of approximation processes across different mathematical and engineering applications. More recently, researchers have explored Korovkin-type results within the framework of statistical convergence techniques (see [25,26]).

Balcerzak et al. [27] introduced a stronger result through the concept of equi-statistical convergence, which improves upon statistical uniform convergence. Furthermore, based on equi-statistical convergence, several results have been established in different settings by various researchers (see, for example, [9,22,28,29,30]).

In light of recent advancements in this area, we focus on utilizing the proposed statistical derivative of the deferred Cesàro summability mean to establish a Korovkin-type theorem.

Let $\mathcal{C}[0,\infty )$ be the space consisting of all continuous functions specified in the interval $[0,\infty )$, endowed with the supremum norm ${\parallel ·\parallel }_{\infty }$. It is well established that a value over $\mathcal{C}[0,\infty )$ forms a Banach space. For any function $\eta \in \mathcal{C}[0,\infty )$, its norm is expressed as

$$\begin{array}{c}\hfill {\parallel \eta \parallel }_{\infty }=\underset{\upsilon \in [0,\infty )}{sup}\left\{\right|\eta \left(\upsilon \right)\left|\right\}.\end{array}$$

Moreover, the modulus of continuity $\omega (\delta ,\eta )$ for a function $\eta \in \mathcal{C}[0,\infty )$ is given by

$$\begin{array}{c}\hfill \omega (\delta ,\eta )=\underset{0\leqq \left|h\right|\leqq \delta }{sup}{\parallel \eta (\upsilon +h)-\eta \left(\upsilon \right)\parallel }_{\infty },\eta \in \mathcal{C}[0,\infty ).\end{array}$$

The quantity $\omega (\delta ,\eta )$ characterizes the modulus of continuity (MOC) of the function $\eta$.

Consider a linear operator $\mathfrak{A}:\mathcal{C}[0,\infty )\to \mathcal{C}[0,\infty )$. The operator $\mathfrak{A}$ is said to be positive if, whenever $\eta \geqq 0$, it follows that $\mathfrak{A}\left(\eta \right)\geqq 0$. The value of $\mathfrak{A}\left(\eta \right)$ at a specific point $\upsilon \in [0,\infty )$ is represented as $\mathfrak{A}\left(\eta \right(\upsilon );\upsilon )$, or simply $\mathfrak{A}(\eta ;\upsilon )$ for brevity.

The well-known Korovkin-type theorem [31] is traditionally stated as follows:

Let $\eta \in \mathcal{C}[a,b]$ and ${\mathfrak{A}}_{\rho }:\mathcal{C}[a,b]\to \mathcal{C}[a,b]$ be positive linear operators. Then,

$$\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(\eta ;\upsilon )-\eta \left(\upsilon \right)\parallel }_{\infty }=0$$

if

$$\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }({\eta }_i;\upsilon )-{\eta }_i\left(\upsilon \right)\parallel }_{\infty }=0,$$

where

$$\begin{array}{c}\hfill {\eta }_0\left(\upsilon \right)=1,{\eta }_1\left(\upsilon \right)=\upsilon \mathrm{and}{\eta }_2\left(\upsilon \right)={\upsilon }^2.\end{array}$$

(3)

The theorem on Cesàro summability in a statistical sense, as presented by Mohiuddine et al. [14], can be reformulated as follows:

Consider the operators ${\mathfrak{A}}_{\rho }:\mathcal{C}[0,\infty )\to \mathcal{C}[0,\infty )$. Then,

$$\begin{array}{c}\hfill {\mathrm{C}}_1\left(\mathrm{stat}\right)\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(\eta ;\upsilon )-\eta \left(\upsilon \right)\parallel }_{\infty }=0\end{array}$$

if

$$\begin{array}{c}\hfill {\mathrm{C}}_1\left(\mathrm{stat}\right)\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }({\eta }_i;\upsilon )-{\eta }_i\left(\upsilon \right)\parallel }_{\infty }=0,\end{array}$$

where ${\eta }_i\left(\upsilon \right)$ are mentioned in (3).

We establish the following theorem by employing the concept of the statistical derivative within the framework of the deferred Cesàro mean, under distinct test functions 1, $e^{-\upsilon }$ and $e^{-2\upsilon }$.

Theorem 1.

Let ${\mathfrak{A}}_{\rho }:\mathcal{C}[0,\infty )\to \mathcal{C}[0,\infty )$ be linear positive operators. Then, for each $\eta \in \mathcal{C}[0,\infty )$,

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(\eta ;\upsilon )-\eta \left(\upsilon \right)\parallel }_{\infty }=0\end{array}$$

(4)

if and only if

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(1;\upsilon )-1\parallel }_{\infty }=0,\end{array}$$

(5)

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(e^{-s};\upsilon )-e^{-\upsilon }\parallel }_{\infty }=0,\end{array}$$

(6)

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(e^{-2s};\upsilon )-e^{-2\upsilon }\parallel }_{\infty }=0.\end{array}$$

(7)

Proof. 

As each of the functions ${\eta }_i\left(x\right)=\{1,e^{-\upsilon },e^{-2\upsilon }\}$, where $i=\{0,1,2\}$ belongs to $\mathcal{C}(Z=[0,\infty \left)\right)$ and is continuous, the implication (4) ⟹ (5) to (7) is evident.

To finalize the proof of the theorem, we begin by assuming that Conditions (5) to (7) are satisfied. Consider a function $\eta \in \mathcal{C}\left(Z\right)$. Then, there exists a constant $\mathcal{K}>0$ with

$$\begin{array}{c}\hfill \left|\eta \right(\upsilon \left)\right|\le \mathcal{K},\forall \upsilon \in Z=[0,\infty ).\end{array}$$

Thus, for $s,\upsilon \in Z$,

$$\begin{array}{c}\hfill \left|\eta \right(\xi )-\eta (\upsilon \left)\right|\le 2\mathcal{K}.\end{array}$$

It is evident that for any $ϵ>0$, ∃$\delta >0$ with

$$\begin{array}{c}\hfill \left|\eta \right(\xi )-\eta (\upsilon \left)\right|<ϵ\end{array}$$

(8)

whenever

$$\begin{array}{c}\hfill |e^{-\xi }-e^{-\upsilon }|<\delta (\forall \xi ,\upsilon \in Z).\end{array}$$

Let us write $\chi =\chi (\xi ,\upsilon )={(e^{-\xi }-e^{-\upsilon })}^2$. For $|e^{-\xi }-e^{-\upsilon }|\leqq \delta$; thus, we have

$$\begin{array}{c}\hfill |\eta \left(\xi \right)-\eta \left(\upsilon \right)|<{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}\chi (\xi ,\upsilon ).\end{array}$$

(9)

From Equations (8) and (9),

$$\begin{array}{c}\hfill |\eta \left(\xi \right)-\eta \left(\upsilon \right)|<ϵ+{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}\chi (\xi ,\upsilon )\end{array}$$

This implies that

$$\begin{array}{c}\hfill -ϵ-{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}\chi (\xi ,\upsilon )\leqq \eta \left(\xi \right)-\eta \left(\upsilon \right)\leqq ϵ+{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}\chi (\xi ,\upsilon ).\end{array}$$

(10)

Since ${\mathfrak{A}}_{\rho }(1,\upsilon )$ is both monotonic and linear, applying ${\mathfrak{A}}_{\rho }(1,\upsilon )$ to this inequality yields the following:

$$\begin{array}{c}\hfill {\mathfrak{A}}_{\rho }(1,\upsilon )\left(-ϵ-{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}\chi (\xi ,\upsilon )\right)\leqq {\mathfrak{A}}_{\rho }(1,\upsilon )(\eta \left(\xi \right)-\eta \left(\upsilon \right))\leqq {\mathfrak{A}}_{\rho }(1,\upsilon )\left(ϵ+{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}\chi (\xi ,\upsilon )\right).\end{array}$$

(11)

Since $\upsilon$ is fixed, $\eta \left(\upsilon \right)$ remains a constant value. Consequently,

$$\begin{array}{cc}\hfill -ϵ{\mathfrak{A}}_{\rho }(1,\upsilon )-{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}{\mathfrak{A}}_{\rho }(\chi ,\upsilon )& \leqq {\mathfrak{A}}_{\rho }(\eta ,\upsilon )-\eta \left(\upsilon \right){\mathfrak{A}}_{\rho }(1,\upsilon )\hfill \\ \hfill & \leqq ϵ{\mathfrak{A}}_{\rho }(1,\upsilon )+{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}{\mathfrak{A}}_{\rho }(\chi ,\upsilon ).\hfill \end{array}$$

(12)

However,

$$\begin{array}{c}\hfill {\mathfrak{A}}_{\rho }(\eta ,\upsilon )-\eta \left(\upsilon \right)=[{\mathfrak{A}}_{\rho }(\eta ,\upsilon )-\eta \left(\upsilon \right){\mathfrak{A}}_{\rho }(1,\upsilon )]+\eta \left(\upsilon \right)[{\mathfrak{A}}_{\rho }(1,\upsilon )-1].\end{array}$$

(13)

Using (12) and (13), we have

$$\begin{array}{c}\hfill {\mathfrak{A}}_{\rho }(\eta ,\upsilon )-\eta \left(\upsilon \right)<ϵ{\mathfrak{A}}_{\rho }(1,x)+{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}{\mathfrak{A}}_{\rho }(\chi ,\upsilon )+\eta \left(\upsilon \right)[{\mathfrak{A}}_{\rho }(1,\upsilon )-1].\end{array}$$

(14)

We now estimate ${\mathfrak{A}}_{\rho }(\chi ,\upsilon )$ as

$$\begin{array}{cc}\hfill {\mathfrak{A}}_{\rho }(\chi ;\upsilon )& ={\mathfrak{A}}_{\rho }({(e^{-\xi }-e^{-\upsilon })}^2;\upsilon )={\mathfrak{A}}_{\rho }(e^{-2\xi }-2e^{-\upsilon }{e}^{-\xi }+e^{-2\upsilon },\upsilon )\hfill \\ \hfill & ={\mathfrak{A}}_{\rho }(e^{-2\xi };\upsilon )-2e^{-\upsilon }{\mathfrak{A}}_{\rho }(e^{-\xi };\upsilon )+e^{-2\xi }{\mathfrak{A}}_{\rho }(1;\upsilon )\hfill \\ \hfill & =[{\mathfrak{A}}_{\rho }(e^{-2\xi };\upsilon )-e^{-2\upsilon }]-2e^{-\upsilon }[{\mathfrak{A}}_{\rho }(e^{-\xi };\upsilon )-e^{-\upsilon }]+e^{-2\upsilon }[{\mathfrak{A}}_{\rho }(1;\upsilon )-1].\hfill \end{array}$$

Using (14), we obtain

$$\begin{array}{cc}\hfill {\mathfrak{A}}_{\rho }(\eta ,\upsilon )-\eta \left(\upsilon \right)& <ϵ{\mathfrak{A}}_{\rho }(1,\upsilon )+{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}\{[{\mathfrak{A}}_{\rho }(e^{-2\xi };\upsilon )-e^{-2\xi }]-2e^{-\upsilon }[{\mathfrak{A}}_{\rho }(e^{-\xi };\upsilon )-e^{-\upsilon }]\hfill \\ \hfill & +e^{-2\xi }[{\mathfrak{A}}_{\rho }(1;\upsilon )-1]\}+\eta \left(\upsilon \right)[{\mathfrak{A}}_{\rho }(1;\upsilon )-1].\hfill \\ \hfill & =ϵ[{\mathfrak{A}}_{\rho }(1;\upsilon )-1]+ϵ+{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}\{[{\mathfrak{A}}_{\rho }(e^{-2\xi };\upsilon )-e^{-2\upsilon }]-2e^{-\upsilon }[{\mathfrak{A}}_{\rho }(e^{-\xi };\upsilon )-e^{-\upsilon }]\hfill \\ \hfill & +e^{-2\upsilon }[{\mathfrak{A}}_{\rho }(1,\upsilon )-1]\}+\eta \left(\upsilon \right)[{\mathfrak{A}}_{\rho }(1,\upsilon )-1].\hfill \end{array}$$

Since $ϵ$ is arbitrary, we write

$$\begin{array}{cc}\hfill |{\mathfrak{A}}_{\rho }(\eta ,\upsilon )-\eta \left(\upsilon \right)|& \leqq ϵ+\left(ϵ+{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}+\mathcal{K}\right)|{\mathfrak{A}}_{\rho }(1,\upsilon )-1|\hfill \\ \hfill & +{\displaystyle \frac{4\mathcal{K}}{{\delta }^2}}|{\mathfrak{A}}_{\rho }(e^{-\xi };\upsilon )-e^{-\upsilon }|+{\displaystyle \frac{2K}{{\delta }^2}}|{\mathfrak{A}}_{\rho }(e^{-2\xi };\upsilon )-e^{-2\upsilon }|\hfill \\ \hfill & \leqq \mathcal{B}\left(|{\mathfrak{A}}_{\rho }(1;\upsilon )-1|+|{\mathfrak{A}}_{\rho }(e^{-\xi };\upsilon )-e^{-\upsilon }|+|{\mathfrak{A}}_{\rho }(e^{-2\xi };\upsilon )-e^{-2\upsilon }|\right)\hfill \end{array}$$

(15)

where

$$\begin{array}{c}\hfill \mathcal{B}=max\left(ϵ+{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}+\mathcal{K},{\displaystyle \frac{4\mathcal{K}}{{\delta }^2}},{\displaystyle \frac{2\mathcal{K}}{{\delta }^2}}\right).\end{array}$$

Next, for a given $r>0$, ∃$ϵ>0$, with $ϵ<r$. Consequently,

$$\begin{array}{c}\hfill {\Psi }_{\rho }(\upsilon ,r)=\left\{a_{\rho }\leqq b_{\rho }\mathrm{and}\left|{\displaystyle \frac{{\mathfrak{A}}_{\rho }(\eta ,\upsilon )-\eta \left(\upsilon \right)}{{\upsilon }_{\rho }}}-\kappa \right|\geqq r\right\}\end{array}$$

and

$$\begin{array}{c}\hfill {\Psi }_{i,\rho }(\upsilon ,r)=\left\{a_{\rho }\leqq b_{\rho }\mathrm{and}\left|{\displaystyle \frac{{\mathfrak{A}}_{\rho }({\eta }_i,\upsilon )-{\eta }_i\left(\upsilon \right)}{{\upsilon }_{\rho }}}-\kappa \right|\geqq {\displaystyle \frac{r-ϵ}{3K}}\right\}\end{array}$$

hold true, whenever ${\upsilon }_0>0$ and ${\upsilon }_{\rho }\to 0$ as $\rho \to \infty$.

Thus, we obtain

$$\begin{array}{c}\hfill {\Psi }_{\rho }(\upsilon ,r)\le \sum _{i=0}^2{\Psi }_{i,\rho }(\upsilon ,r).\end{array}$$

Clearly,

$$\begin{array}{c}\hfill {\displaystyle \frac{\parallel {\Psi }_{\rho }{(\upsilon ,r)\parallel }_{\mathcal{C}\left(Z\right)}}{b_{\rho }-a_{\rho }}}\leqq \sum _{i=0}^2{\displaystyle \frac{\parallel {\Psi }_{i,\rho }{(\upsilon ,r)\parallel }_{\mathcal{C}\left(Z\right)}}{b_{\rho }-a_{\rho }}}.\end{array}$$

(16)

By considering the assumptions related to the implications in (5) to (7) and utilizing Definition 5, it follows that the right-hand side (RHS) of (16) is equal to 0 (zero) as $\rho \to \infty$. As a result, we obtain

$$\begin{array}{c}\hfill \underset{\rho \to \infty }{lim}{\displaystyle \frac{\parallel {\Psi }_{\rho }{(\upsilon ,r)\parallel }_{\mathcal{C}\left(Z\right)}}{b_{\rho }-a_{\rho }}}=0(r>0).\end{array}$$

Thus, the validity of implication (4) is established, thereby concluding the proof of Theorem 1. □

Corollary 1.

Let ${\mathfrak{A}}_{\rho }:\mathcal{C}[0,\infty )\to \mathcal{C}[0,\infty )$ be linear positive operators. Then, for each $\eta \in \mathcal{C}[0,\infty )$,

$$\begin{array}{c}\hfill {\mathrm{stat}}_{\mathrm{DCD}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(\eta ;\upsilon )-\eta \left(\upsilon \right)\parallel }_{\infty }=0\end{array}$$

if and only if

$$\begin{array}{c}\hfill {\mathrm{stat}}_{\mathrm{DCD}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(1;\upsilon )-1\parallel }_{\infty }=0,\end{array}$$

$$\begin{array}{c}\hfill {\mathrm{stat}}_{\mathrm{DCD}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(e^{-s};\upsilon )-e^{-\upsilon }\parallel }_{\infty }=0,\end{array}$$

$$\begin{array}{c}\hfill {\mathrm{stat}}_{\mathrm{DCD}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(e^{-2s};\upsilon )-e^{-2\upsilon }\parallel }_{\infty }=0.\end{array}$$

Proof. 

The proof of Corollary 1 follows a similar approach to that of Theorem 1. Hence, the detailed steps are omitted for brevity. □

5. Numerical Example

Based on Theorem 1, we provide an example below utilizing specific positive linear polynomials known as Baskakov operators.

Example 2.

Consider that the operators ${\mathfrak{A}}_{\rho }:\mathcal{C}[0,\infty )\to \mathcal{C}[0,\infty )$ are specified by

$$\begin{array}{c}\hfill {\mathfrak{A}}_{\rho }(\eta ;\upsilon )=(1+{\upsilon }_{\rho }){\mathcal{V}}_{\rho }(\eta ;\upsilon )\end{array}$$

(17)

where

$$\begin{array}{c}\hfill {\mathcal{V}}_{\rho }(\eta ;\upsilon )=\sum _{\mu =0}^{\infty }\eta \left({\displaystyle \frac{\mu }{\rho }}\right)\left({\displaystyle \genfrac{}{}{0pt}{}{\rho -\mu +1}{\mu }}\right){\upsilon }^{\mu }.{(1+\upsilon )}^{-\rho -\mu },\end{array}$$

and $\eta \left({\upsilon }_{\rho }\right)$ is mentioned in Example 1.

• Now,

$$\begin{array}{c}\hfill {\mathfrak{A}}_{\rho }(1;\upsilon )=[1+{\upsilon }_{\rho }]1=1+{\upsilon }_{\rho };\end{array}$$

$$\begin{array}{cc}\hfill {\mathfrak{A}}_{\rho }(e^{-s};\upsilon )& =[1+{\upsilon }_{\rho }]{(1+\upsilon -\upsilon e^{-{\displaystyle \frac{1}{\rho }}})}^{-\rho }\hfill \\ \hfill & =[1+{\upsilon }_{\rho }]\upsilon {(1+\upsilon -\upsilon e^{-{\displaystyle \frac{1}{\rho }}})}^{-\rho }(1-\rho \upsilon (1-e^{-{\displaystyle \frac{1}{\rho }}}){(1+\upsilon -\upsilon e^{-{\displaystyle \frac{1}{\rho }}})}^{-1});\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\mathfrak{A}}_{\rho }(e^{-2s};\upsilon )& =[1+{\upsilon }_{\rho }]{(1+{\upsilon }^2-{\upsilon }^2{e}^{-{\displaystyle \frac{1}{\rho }}})}^{-\rho }\hfill \\ \hfill & =[1+{\upsilon }_{\rho }]\upsilon {(1+{\upsilon }^2-{\upsilon }^2{e}^{-{\displaystyle \frac{1}{\rho }}})}^{-\rho }(1-2\rho {\upsilon }^2(1-e^{-{\displaystyle \frac{1}{\rho }}}){(1+{\upsilon }^2-{\upsilon }^2{e}^{-{\displaystyle \frac{1}{\rho }}})}^{-1}).\hfill \end{array}$$

Thus, we obtain

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(1;\upsilon )-1\parallel }_{\infty }=0,\end{array}$$

(18)

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(e^{-s};\upsilon )-e^{-\upsilon }\parallel }_{\infty }=0,\end{array}$$

(19)

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(e^{-2s};\upsilon )-e^{-2\upsilon }\parallel }_{\infty }=0.\end{array}$$

(20)

That is, the sequence ${\mathfrak{A}}_{\rho }(\eta ;\upsilon )$ meets the requirements outlined in Conditions (5) to (7).

Consequently, by applying Theorem 1, we obtain

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(\eta ;\upsilon )-\eta \left(\upsilon \right)\parallel }_{\infty }=0.\end{array}$$

(21)

Thus, the function $\eta \left({\upsilon }_{\rho }\right)$ possesses a statistical deferred Cesàro derivative. Given that $\eta \left({\upsilon }_{\rho }\right)$ is not convergent in the usual sense, and also not Cesàro summable in a statistical sense, it follows that the results presented in [14,32] do not apply to the operators given in (17). On the other hand, our Theorem 1 remains applicable. The convergence behavior of these operators for the given exponential test functions 1, $e^{-\upsilon }$ and $e^{-2\upsilon }$ is illustrated in Figure 2, Figure 3 and Figure 4.

The significance of Figure 2, Figure 3 and Figure 4 lies in their demonstration of the statistical deferred Cesàro derivative of positive linear operators. Specifically, they illustrate the following key aspects:

Figure 2, Figure 3 and Figure 4 confirm that the function $\eta \left(\upsilon \right)={\upsilon }^2$ possesses a statistical deferred Cesàro derivative converging to $\kappa =2$ at the point ${\upsilon }_0=1$. This validates the proposed method under the statistical deferred Cesàro derivative. The figures visually represent the convergence of the operators ${\mathfrak{A}}_{\rho }(\eta ;\upsilon )$ to $\eta \left(\upsilon \right)$ by means of the deferred Cesàro derivative in a statistical sense. The individual operators ${\mathfrak{A}}_{\rho }(1;\upsilon )$, ${\mathfrak{A}}_{\rho }(e^{-s};\upsilon )$ and ${\mathfrak{A}}_{\rho }(e^{-2s};\upsilon )$ are shown to satisfy the conditions required for the Korovkin−type theorem. The results establish that the proposed theorem holds for the given sequence of operators. It confirms that earlier classical and statistical summability approaches (such as those in [14,32]) do not work for this setting, while the newly introduced concept successfully applies. Consequently, Figure 2, Figure 3 and Figure 4 support the claim that Theorem 1 provides a non-trivial advancement of the usual Korovkin−type theorem. The approach used in the study is stronger than conventional summability techniques and offers a more generalized framework. Thus, Figure 2, Figure 3 and Figure 4 play a crucial role in visualizing and verifying the effectiveness of statistical deferred Cesàro derivatives in approximation theory.

6. Rate of Deferred Cesàro Derivative

In this section, we examine the rate of deferred Cesàro derivatives in a statistical sense for the operators $\mathfrak{A}(\eta ;\upsilon )$ defined over $\mathcal{C}[0,\infty )$ by employing the modulus of continuity (MOC).

Next, we introduce the definition of the statistical deferred Cesàro derivative.

Definition 7.

Let $\left(u_{\rho }\right)$ be a positive decreasing sequence. A function $\eta :\mathbb{R}\to \mathbb{R}$ possesses a statistical deferred Cesàro derivative κ with the rate $o\left(u_{\rho }\right)$ if, for all $ϵ>0$,

$$\begin{array}{c}\hfill \underset{\rho \to \infty }{lim}{\displaystyle \frac{1}{u_{\rho }(b_{\rho }-a_{\rho })}}\left|\left\{a_{\rho }\leqq b_{\rho }\mathrm{and}\left|{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta \left({\upsilon }_0\right)}{{\upsilon }_{\rho }}}-\kappa \right|\geqq ϵ\right\}\right|=0\end{array}$$

holds, whenever ${\upsilon }_0>0$ and ${\upsilon }_{\rho }\to 0$ as $\rho \to \infty$.

In this scenario, we express it as

$$\begin{array}{c}\hfill \eta \left({\upsilon }_{\rho }\right)-\kappa ={\mathrm{DCD}}_{\mathrm{stat}}-o\left(u_{\rho }\right).\end{array}$$

To prove Theorem 2, we first require the following lemma:

Lemma 1.

Let $\left(u_{\rho }\right)$ and $\left({\alpha }_{\rho }\right)$ be two decreasing sequences (positive). Let $\eta \left({\upsilon }_{\rho }\right)$ and $\eta \left(z_{\rho }\right)$ be two other sequences satisfying

$$\begin{array}{c}\hfill \eta \left({\upsilon }_{\rho }\right)-\kappa ={\mathrm{DCD}}_{\mathrm{stat}}-o\left(u_{\rho }\right)\end{array}$$

and

$$\begin{array}{c}\hfill \eta \left(z_{\rho }\right)-\kappa ={\mathrm{DCD}}_{\mathrm{stat}}-o\left({\alpha }_{\rho }\right).\end{array}$$

Then, the following conditions hold:

(i) 

$(\eta \left({\upsilon }_{\rho }\right)+\eta \left(z_{\rho }\right))-({\kappa }_1+{\kappa }_2)={\mathrm{DCD}}_{\mathrm{stat}}-o\left({\beta }_{\rho }\right)$;

(ii) 

$(\eta \left({\upsilon }_{\rho }\right)-{\kappa }_1)(\eta \left(z_{\rho }\right)-{\kappa }_2)={\mathrm{DCD}}_{\mathrm{stat}}-o\left(u_{\rho }{\alpha }_{\rho }\right)$;

(iii) 

$\lambda (\eta \left({\upsilon }_{\rho }\right)-{\kappa }_1)={\mathrm{DCD}}_{\mathrm{stat}}-o\left(u_{\rho }\right)$;

(iv) 

$\sqrt{|{\upsilon }_{\rho }-{\kappa }_1|}={\mathrm{DCD}}_{\mathrm{stat}}-o\left(u_{\rho }\right)$

• where

$$\begin{array}{c}\hfill {\beta }_{\rho }=max\{u_{\rho },{\alpha }_{\rho }\}.\end{array}$$

Proof. 

To deduce Condition (i), let $ϵ>0$ and $\upsilon \in [0,\infty )$. Moreover, we consider the following sets:

$$\begin{array}{c}\hfill {\mathcal{Q}}_{\rho }(\upsilon ,ϵ)=\left|\left\{a_{\rho }\leqq b_{\rho }\mathrm{and}\left|\left({\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta \left({\upsilon }_0\right)}{{\upsilon }_{\rho }}}+{\displaystyle \frac{\eta (z_0+z_{\rho })-\eta \left(z_0\right)}{z_{\rho }}}\right)-({\kappa }_1+{\kappa }_2)\right|\geqq ϵ\right\}\right|,\end{array}$$

$$\begin{array}{c}\hfill {\mathcal{Q}}_{1,\rho }(\upsilon ,ϵ)=\left|\left\{a_{\rho }\leqq b_{\rho }\mathrm{and}\left|{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta \left({\upsilon }_0\right)}{{\upsilon }_{\rho }}}-{\kappa }_1\right|\geqq {\displaystyle \frac{ϵ}{2}}\right\}\right|,\end{array}$$

and

$$\begin{array}{c}\hfill {\mathcal{Q}}_{2,\rho }(\upsilon ,ϵ)=\left|\left\{a_{\rho }\leqq b_{\rho }\mathrm{and}\left|{\displaystyle \frac{\eta ({\upsilon }_0+{\upsilon }_{\rho })-\eta \left({\upsilon }_0\right)}{{\upsilon }_{\rho }}}-{\kappa }_2\right|\geqq {\displaystyle \frac{ϵ}{2}}\right\}\right|.\end{array}$$

It is straightforward to observe that

$$\begin{array}{ccc}& \hfill & \hfill {\mathcal{Q}}_{\rho }(\upsilon ,ϵ)\subseteq {\mathcal{Q}}_{1,\rho }(\upsilon ,ϵ)\cup {\mathcal{Q}}_{2,\rho }(\upsilon ,ϵ).\end{array}$$

Moreover, since

$$\begin{array}{c}\hfill {\beta }_{\rho }=max\{u_{\rho },{\alpha }_{\rho }\},\end{array}$$

using Condition (4) of Theorem 1 yields

$$\begin{array}{c}\hfill {\displaystyle \frac{\parallel {\mathcal{Q}}_{\rho }{(\upsilon ,ϵ)\parallel }_{\infty }}{{\beta }_{\rho }(b_{\rho }-a_{\rho })}}\le {\displaystyle \frac{\parallel {\mathcal{Q}}_{1,\rho }{(\upsilon ,ϵ)\parallel }_{\infty }}{u_{\rho }(b_{\rho }-a_{\rho })}}+{\displaystyle \frac{\parallel {\mathcal{Q}}_{2,\rho }{(\upsilon ,ϵ)\parallel }_{\infty }}{{\alpha }_{\rho }(b_{\rho }-a_{\rho })}}.\end{array}$$

By applying Conditions (5) to (7) from Theorem 1, we derive

$$\begin{array}{c}\hfill {\displaystyle \frac{\parallel {\mathcal{Q}}_{\rho }{(\upsilon ,ϵ)\parallel }_{\infty }}{{\beta }_{\rho }(b_{\rho }-a_{\rho })}}=0,\end{array}$$

and this establishes Condition (i).

As the proofs for Conditions (ii) to (iv) follow similar reasoning, we will omit them here. □

We now revisit the concept of the MOC for a function $\eta \in C[0,\infty )$, which is specified by

$$\begin{array}{c}\hfill \omega (\eta ,\delta )=\underset{|z-\upsilon |\leqq \delta :\upsilon ,z\in Z}{sup}|\eta \left(z\right)-\eta \left(\upsilon \right)|(\delta >0).\end{array}$$

That is,

$$\begin{array}{c}\hfill |\eta \left(z\right)-\eta \left(\upsilon \right)|\leqq \omega (\eta ,\delta )\left({\displaystyle \frac{|\upsilon -z|}{\delta }}+1\right).\end{array}$$

(22)

Theorem 2.

Consider the linear positive operators ${\mathfrak{A}}_{\rho }:\mathcal{C}[0,\infty )\to \mathcal{C}[0,\infty )$. Then,

$$\begin{array}{c}\hfill \parallel {\mathfrak{A}}_{\rho }{(\eta ,\upsilon )-\eta \parallel }_{\infty }={\mathrm{DCD}}_{\mathrm{stat}}-o\left({\beta }_{\rho }\right)\end{array}$$

(23)

holds for every $\eta \in \mathcal{C}\left(Z\right)$ and where ${\beta }_{\rho }=max\{u_{\rho },{\alpha }_{\rho }\}$, and $Z=\mathcal{C}[0,\infty )$, provided that the below conditions hold:

(i) 

$\parallel {\mathfrak{A}}_{\rho }{(1,\upsilon )-1\parallel }_{\infty }={\mathrm{DCD}}_{\mathrm{stat}}-o\left(u_{\rho }\right)$,

(ii) 

$\omega (\eta ,{\zeta }_{\rho })={\mathrm{DCD}}_{\mathrm{stat}}-o\left({\alpha }_{\rho }\right)$

• where

$$\begin{array}{c}\hfill {\zeta }_{\rho }={\left({\mathfrak{A}}_{\rho }({\chi }^2;\upsilon )\right)}^{1/2}\mathrm{and}\chi (z,\upsilon )={(e^{-z}-e^{-\upsilon })}^2.\end{array}$$

Proof. 

Let $\eta \in C[0,\infty )$ and $\upsilon \in [0,\infty )$. By using (22), we have

$$\begin{array}{cc}\hfill |{\mathfrak{A}}_{\rho }(\eta ;\upsilon )-\eta \left(\upsilon \right)|& \leqq {\mathfrak{A}}_{\rho }\left(\right|\eta \left(z\right)-\eta \left(\upsilon \right)|;\upsilon )+\left|\eta \left(\upsilon \right)\right||{\mathfrak{A}}_{\rho }(1;\upsilon )-1|\hfill \\ \hfill & \leqq {\mathfrak{A}}_{\rho }\left({\displaystyle \frac{|e^{-\upsilon }-e^{-\upsilon }|}{{\zeta }_{\rho }}}+1;\upsilon \right)\omega (\eta ,{\zeta }_{\rho })+|\eta \left(\upsilon \right)\left|\right|{\mathfrak{A}}_{\rho }(1;\upsilon )-1|\hfill \\ \hfill & \leqq {\mathfrak{A}}_{\rho }\left(1+{\displaystyle \frac{1}{{\zeta }_{\rho }^2}}{(e^{-\upsilon }-e^{-z})}^2;\upsilon \right)\omega (\eta ,{\zeta }_{\rho })+|\eta \left(\upsilon \right)\left|\right|{\mathfrak{A}}_{\rho }(1;\upsilon )-1|\hfill \\ \hfill & \leqq \left({\mathfrak{A}}_{\rho }(1;\upsilon )+{\displaystyle \frac{1}{{\zeta }_{\rho }^2}}{\mathfrak{A}}_{\rho }({\chi }_{\upsilon };\upsilon )\right)\omega (\eta ,{\zeta }_{\rho })+|\eta \left(\upsilon \right)\left|\right|{\mathfrak{A}}_{\rho }(1;\upsilon )-1|.\hfill \end{array}$$

Now, by setting ${\zeta }_{\rho }={\left({\mathfrak{A}}_{\rho }({\chi }^2;\upsilon )\right)}^{1/2}$, we obtain

$$\begin{array}{cc}\hfill \parallel {\mathfrak{A}}_{\rho }{(\eta ;\upsilon )-\eta \left(\upsilon \right)\parallel }_{\infty }& \leqq 2\omega (\eta ,{\zeta }_{\rho })+\omega (\eta ,{\zeta }_{\rho })\parallel {\mathfrak{A}}_{\rho }(1;\upsilon ){-1\parallel }_{\infty }+\parallel \eta \left(\upsilon \right)\parallel \parallel {\mathfrak{A}}_{\rho }(1;\upsilon ){-1\parallel }_{\infty }\hfill \\ \hfill & \leqq \mathfrak{K}\{\omega (\eta ,{\zeta }_{\rho })+\omega (\eta ,{\zeta }_{\rho })\parallel {\mathfrak{A}}_{\rho }(1;\upsilon )-1{\parallel }_{\infty }+\parallel {\mathfrak{A}}_{\rho }(1;\upsilon )-1{\parallel }_{\infty }\},\hfill \end{array}$$

where $\mathfrak{K}=\{\parallel \eta {\parallel }_{\infty },2\}$.

Consequently, we have

$$\begin{array}{cc}& {∥{\displaystyle \frac{1}{b_{\rho }-a_{\rho }}}\sum _{k=a_{\rho }+1}^{b_{\rho }}{\mathfrak{A}}_{\rho }(\eta ;\upsilon )-\eta \left(\upsilon \right)∥}_{\infty }\hfill \\ \hfill & \leqq \mathfrak{K}\left\{\omega (\eta ,{\zeta }_{\rho }){\displaystyle \frac{1}{b_{\rho }-a_{\rho }}}+\omega (\eta ,{\zeta }_{\rho }){∥{\displaystyle \frac{1}{b_{\rho }-a_{\rho }}}\sum _{\mu =a_{\rho }+1}^{b_{\rho }}{\mathfrak{A}}_{\rho }(\eta ;\upsilon )-\eta \left(\upsilon \right)∥}_{\infty }\right\}\hfill \\ \hfill & +\mathfrak{K}\left\{{∥{\displaystyle \frac{1}{b_{\rho }-a_{\rho }}}\sum _{\mu =a_{\rho }+1}^{b_{\rho }}{\mathfrak{A}}_{\rho }(\eta ;\upsilon )-\eta \left(\upsilon \right)∥}_{\infty }\right\}.\hfill \end{array}$$

By applying Lemma 1 and Theorem 2 (Conditions (i) and (ii)), we derive the result stated in (23) of Theorem 2, thereby completing its proof. □

7. Conclusions

In the concluding section of our study, we offer further remarks and insights on the various findings we have presented.

Remark 1.

Consider the sequence ${\left({\upsilon }_{\rho }\right)}_{\rho \in \mathbb{N}}$, as introduced in Example 1. Next, as

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\upsilon }_{\rho }\to 0\mathrm{on}[0,\infty ),\end{array}$$

we have

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }({\eta }_i,\upsilon )-{\eta }_i\left(\upsilon \right)\parallel }_{\infty }=0.\end{array}$$

(24)

Therefore, by applying Theorem 1, we can express it as follows:

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\parallel {\mathfrak{A}}_{\rho }(\eta ,\upsilon )-\eta \left(\upsilon \right)\parallel }_{\infty }=0,\end{array}$$

(25)

where

$$\begin{array}{c}\hfill {\eta }_0\left(\upsilon \right)=1,{\eta }_1\left(\upsilon \right)=e^{-\upsilon }\mathrm{and}{\eta }_2\left(\upsilon \right)=e^{-2\upsilon }.\end{array}$$

Since the sequence $\left({\upsilon }_{\rho }\right)$ does not converge ordinarily, it also does not exhibit uniform convergence in the traditional sense. Consequently, the traditional Korovkin theorem cannot be applied to the operators specified under (17). This example effectively demonstrates that Theorem 1 provides a significant extension of the usual Korovkin-type theorem (refer to [31]).

Remark 2.

Consider that the sequence ${\left({\upsilon }_{\rho }\right)}_{\rho \in \mathbb{N}}$ is given in Example 2. As

$$\begin{array}{c}\hfill {\mathrm{DCD}}_{\mathrm{stat}}\underset{\rho \to \infty }{lim}{\upsilon }_{\rho }=0\mathrm{on}[0,\infty ),\end{array}$$

it follows that (24) is satisfied. By utilizing (24) along with Theorem 1, we confirm the validity of Condition (25). Given that the sequence $\left({\upsilon }_{\rho }\right)$ does not exhibit statistical Cesàro summability, Theorem 2.1 from Mohiuddine et al. [14] cannot be applied to the operator defined in (17). Therefore, Theorem 1 constitutes a notable generalization of Theorem 2.1 by Mohiuddine et al. [14] (refer to [31,32]). Based on the findings presented earlier, we infer that the proposed method proves to be effective for the operators defined in (17). This approach provides a stronger framework compared to both the conventional and statistical formulations of previously established Korovkin-type theorems (see [14,31,32]).

Remark 3.

Korovkin-type theorems for positive linear operators based on the statistical derivative of deferred Cesàro summability have promising applications in data science and computational mathematics. In data science, these theorems can enhance machine learning algorithms by improving approximation techniques for noisy or incomplete data. They can also optimize numerical methods for data smoothing and function approximation, which are crucial in predictive analytics. In computational mathematics, these results can refine convergence analysis in numerical solutions of differential equations and optimization problems. Furthermore, they may contribute to signal processing and image reconstruction, where stable approximations of functions are essential for denoising and feature extraction.