In this study, we investigate the Korovkin-type theorems depending upon some type of convergence such as alpha convergence, semi-alpha convergence and the notion of exhaustiveness. Since it is known that the convergence types mentioned above are between point-wise convergence and uniform convergence, it will be observed that the conditions can be alleviated in the Korovkin theorem.

In this paper, we investigate various types of convergence for sequences of functions and examine the relationships among these types. Our findings contribute to a deeper understanding of the structural properties of function sequences and their convergence behaviors.

In this study, we scrutinize the Korovkin-type theorems based on various forms of convergence, such as almost uniform convergence, semi-uniform convergence, and the concept of semi-exhaustiveness. Since it is known that the convergence types mentioned above are between point-wise and uniform convergence, it will be noticed that the circumstances can be mitigated in the Korovkin theorem.

In this paper we introduced the notions of Cauchy alpha convergence and uniform alpha convergence for sequences of functions between metric spaces. Using these new concepts, we investigated properties between uniform convergence and alpha convergence and provided figures illustrating the relationships among these different types of convergence based on the structure of the functions and their domains. Finally, we established some Korovkin-type approximation theorems involving these new concepts.

In this current study, we introduce a new operator called the P-Bernstein operator derived through the utilization of ?P-factorial? (Pell factorial) and ?Pellnomial? (Pell binomial). Subsequently, we investigate the fundamental properties of the P-Bernstein basis polynomials. Furthermore, we establish some connections between the P-Bernstein Polynomials and Pell Numbers.