Power Series Statistical Convergence and an Abstract Korovkin-Type Approximation Theorem

This paper establishes an abstract Korovkin-type approximation theorem in general spaces, extending the framework of approximation theory to accommodate broader contexts. A critical result supporting this theorem is the proof that any P-statistically convergent sequence contains a classically convergent subsequence over a density 1 set, which plays a foundational role in the analysis. As a conclusion, we investigate the convergence of the r-th order generalization of linear operators, which may lack positivity, and present a Korovkin-type approximation theorem for periodic functions, both utilizing P-statistical convergence. These contributions generalize and improve existing results in approximation theory, providing novel insights and methodologies, supported by practical examples and corollaries.