## 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 \(\mathcal{L}_{\rho}(\eta;v)\) 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 \(1\), \(v\) and \(v^{2}\) 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 Cesaro 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]). [PERSON] 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]). Figure 1: Statistical deferred Cesaro derivative of \((\frac{1}{\rho})\). In light of recent advancements in this area, we focus on utilizing the proposed statistical derivative of the deferred Cesaro 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 \(\|\cdot\|_{\infty}\). 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  
\[\|\eta\|_{\infty}=\sup_{v\in[0,\infty)}\{|\eta(v)|\}.\]  
Moreover, the modulus of continuity \(\omega(\delta,\eta)\) for a function \(\eta\in\mathcal{C}[0,\infty)\) is given by  
\[\omega(\delta,\eta)=\sup_{0\leq\|h\|\leq\delta}\|\eta(v+h)-\eta(v)\|_{\infty}, \ \ \eta\in\mathcal{C}[0,\infty).\]  
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 \(\eta\geq 0\), it follows that \(\mathfrak{A}(\eta)\geqq 0\). The value of \(\mathfrak{A}(\eta)\) at a specific point \(v\in[0,\infty)\) is represented as \(\mathfrak{A}(\eta(v);v)\), or simply \(\mathfrak{A}(\eta;v)\) 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,  
\[\lim_{\rho\to\infty}\|\mathfrak{A}_{\rho}(\eta;v)-\eta(v)\|_{\infty}=0\]  
if  
\[\lim_{\rho\to\infty}\|\mathfrak{A}_{\rho}(\eta_{i};v)-\eta_{i}(v)\|_{\infty}=0,\]  
where  
\[\eta_{0}(v)=1,\ \ \eta_{1}(v)=v\ \ \text{and}\ \ \eta_{2}(v)=v^{2}. \tag{3}\]  
The theorem on Cesaro summability in a statistical sense, as presented by [PERSON] et al. [14], can be reformulated as follows:  
Consider the operators \(\mathfrak{A}_{\rho}:\mathcal{C}[0,\infty)\to\mathcal{C}[0,\infty)\). Then,  
\[\text{C}_{1}(\text{stat})\ \lim_{\rho\to\infty}\|\mathfrak{A}_{\rho}(\eta;v)- \eta(v)\|_{\infty}=0\]  
if  
\[\text{C}_{1}(\text{stat})\ \lim_{\rho\to\infty}\|\mathfrak{A}_{\rho}(\eta_{i};v)- \eta_{i}(v)\|_{\infty}=0,\]  
where \(\eta_{i}(v)\) are mentioned in (3).