## 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_  
\[\mathfrak{A}_{\rho}(\eta;v)=(1+v_{\rho})\mathcal{V}_{\rho}(\eta;v) \tag{17}\]  
_where_  
\[\mathcal{V}_{\rho}(\eta;v)=\sum_{\mu=0}^{\infty}\eta\bigg{(}\frac{\mu}{\rho} \bigg{)}\binom{\rho-\mu+1}{\mu}v^{\mu}.(1+v)^{-\rho-\mu},\]  
_and \(\eta(v_{\rho})\) is mentioned in Example 1. Now,_  
\[\mathfrak{A}_{\rho}(1;v)=[1+v_{\rho}]1=1+v_{\rho};\]\[\mathfrak{A}_{\rho}(e^{-s};v) =[1+v_{\rho}](1+v-ve^{-\frac{1}{\rho}})^{-\rho}\] \[=[1+v_{\rho}]v(1+v-ve^{-\frac{1}{\rho}})^{-\rho}(1-\rho v(1-e^{- \frac{1}{\rho}})(1+v-ve^{-\frac{1}{\rho}})^{-1});\]  
\[\mathfrak{A}_{\rho}(e^{-2s};v) =[1+v_{\rho}](1+v^{2}-v^{2}e^{-\frac{1}{\rho}})^{-\rho}\] \[=[1+v_{\rho}]v(1+v^{2}-v^{2}e^{-\frac{1}{\rho}})^{-\rho}(1-2\rho v ^{2}(1-e^{-\frac{1}{\rho}})(1+v^{2}-v^{2}e^{-\frac{1}{\rho}})^{-1}).\]  
_Thus, we obtain_  
\[\mathrm{DCD}_{\mathrm{stat}}\lim_{\rho\to\infty}\|\mathfrak{A}_{\rho}(1;v)-1 \|_{\infty}=0, \tag{18}\]  
\[\mathrm{DCD}_{\mathrm{stat}}\lim_{\rho\to\infty}\|\mathfrak{A}_{\rho}(e^{-s}; v)-e^{-v}\|_{\infty}=0, \tag{19}\]  
\[\mathrm{DCD}_{\mathrm{stat}}\lim_{\rho\to\infty}\|\mathfrak{A}_{\rho}(e^{-2s}; v)-e^{-2v}\|_{\infty}=0. \tag{20}\]  
_That is, the sequence \(\mathfrak{A}_{\rho}(\eta;v)\) meets the requirements outlined in Conditions (5) to (7)._  
_Consequently, by applying Theorem 1, we obtain_  
\[\mathrm{DCD}_{\mathrm{stat}}\lim_{\rho\to\infty}\|\mathfrak{A}_{\rho}(\eta;v)- \eta(v)\|_{\infty}=0. \tag{21}\]  
_Thus, the function \(\eta(v_{\rho})\) possesses a statistical deferred Cesaro derivative. Given that \(\eta(v_{\rho})\) is not convergent in the usual sense, and also not Cesaro 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 \(1\), \(e^{-v}\) and \(e^{-2v}\) is illustrated in Figures 2-4._  
_The significance of Figures 2-4 lies in their demonstration of the statistical deferred Cesaro derivative of positive linear operators. Specifically, they illustrate the following key aspects:_  
_Figures 2-4 confirm that the function \(\eta(v)=v^{2}\) possesses a statistical deferred Cesaro derivative converging to \(\kappa=2\) at the point \(v_{0}=1\). This validates the proposed method under the statistical deferred Cesaro derivative. The figures visually represent the convergence of the operators \(\mathfrak{A}_{\rho}(\eta;v)\) to \(\eta(v)\) by means of the deferred Cesaro derivative in a statistical sense. The individual operators \(\mathfrak{A}_{\rho}(1;v)\), \(\mathfrak{A}_{\rho}(e^{-s};v)\) and \(\mathfrak{A}_{\rho}(e^{-2s};v)\) 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, Figures 2-4 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, Figures 2-4 play a crucial role in visualizing and verifying the effectiveness of statistical deferred Cesaro derivatives in approximation theory._Figure 4: Convergence behavior of positive linear operator for \(f_{2}=e^{-2x}\).  
Figure 3: Convergencebehavior of positive linear operator for \(f_{1}=e^{-x}\).  
Figure 2: Convergencebehavior of positive linear operator for \(f_{0}=1\).