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.