| Abstract: |
Statistical convergence, introduced by Fast and Steinhaus in 1951, extends classical pointwise convergence by focusing on the behavior of the majority of sequence elements while disregarding a negligible set. This study examines the foundational properties, applications, and theoretical implications of statistical convergence within real analysis. Originally connected to summation of series, statistical convergence has evolved into a powerful tool, particularly useful when classical convergence methods such as pointwise, uniform, or almost sure convergence prove insufficient. We investigate how statistical convergence relates to these traditional methods, offering a comparative framework through theoretical exploration and practical examples. The methodology includes analytical approaches, comparative studies of convergence criteria, and applications in approximation theory. Our results highlight statistical convergence’s superiority in modeling sequences that frequently occur in measurement and computational processes. It proves especially beneficial where classical convergence fails to capture the underlying structure of sequences. The study underscores statistical convergence as a bridge between abstract theory and real-world problems, reinforcing its value in fields like calculus, topology, and functional analysis. Ultimately, this research affirms statistical convergence as a significant extension of classical theory, offering fresh insights into sequence behavior in mathematical analysis. |
