A generalization of power series statistical convergence and its applications to positive linear operators

Abstract In this study, we investigate a summability method known as P q t ${P}^{{q}_{t}}$ -statistical convergence, which generalizes P -statistical convergence, where P is a power series method. By incorporating a strictly increasing sequence q t $\left\{{q}_{t}\right\}$ , we provide a more flexible framework for analyzing the convergence behaviour of sequences that may not converge using classical or P -statistical methods. Through a specific counterexample, we demonstrate that P q t ${P}^{{q}_{t}}$ -statistical convergence is stronger than P -statistical convergence. We present a Vitali-type convergence theorem in the framework of P q t ${P}^{{q}_{t}}$ -summability, utilizing a truncation technique to characterize the relationship between strong and statistical convergence. Furthermore, we establish and prove a Korovkin-type approximation theorem for positive linear operators acting on the space of continuous functions C U $C\left(U\right)$ . Finally, we define the rate of P q t ${P}^{{q}_{t}}$ -statistical convergence using the modulus of continuity and provide quantitative results that characterize the approximation’s effectiveness. The validity of the main approximation result is supported by a graphical representation.