## 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 \((v_{\rho})_{\rho\in\mathbb{N}}\), as introduced in Example 1. Next, as_  
\[\mathrm{DCD}_{\mathrm{stat}}\lim_{\rho\to\infty}v_{\rho}\to 0\ \mathrm{on}\ \ [0, \infty),\]  
_we have_  
\[\mathrm{DCD}_{\mathrm{stat}}\lim_{\rho\to\infty}\|\mathfrak{A}_{ \rho}(\eta_{i},v)-\eta_{i}(v)\|_{\infty}=0. \tag{24}\]  
_Therefore, by applying Theorem 1, we can express it as follows:_  
\[\mathrm{DCD}_{\mathrm{stat}}\lim_{\rho\to\infty}\|\mathfrak{A}_{ \rho}(\eta,v)-\eta(v)\|_{\infty}=0, \tag{25}\]  
_where_  
\[\eta_{0}(v)=1,\quad\eta_{1}(v)=e^{-v}\quad\mathrm{and}\quad\eta_{ 2}(v)=e^{-2v}.\]_Since the sequence \((v_{\rho})\) 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 \((v_{\rho})_{\rho\in\mathbb{N}}\) is given in Example 2. As_  
\[\mathrm{DCD}_{\mathrm{stat}}\lim_{\rho\to\infty}v_{\rho}=0\ \ \mathrm{on}\ \ [0, \infty),\]  
_it follows that (24) is satisfied. By utilizing (24) along with Theorem 1, we confirm the validity of Condition (25). Given that the sequence \((v_{\rho})\) does not exhibit statistical Cesaro summability, Theorem 2.1 from [PERSON] et al. [14] cannot be applied to the operator defined in (17). Therefore, Theorem 1 constitutes a notable generalization of Theorem 2.1 by [PERSON] 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 Cesaro 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._  
**Author Contributions:** Formal analysis, H.M.S. and U.M.; investigation, B.B.J.; methodology, S.K.P.; supervision, H.M.S. and U.M.; writing-original draft, B.B.J.; writing-review and editing, S.K.P. All authors have read and agreed to the published version of the manuscript.  
This research received no external funding.  
Data Availability Statement: Data is contained within the article.  
The authors declare no conflict of interest.