Convergence Types of Function Sequences and Their Applications in Approximation Theory

Sequences of functions arise naturally in real analysis and are especially useful in obtaining approximations to a given function and defining new functions from known ones. This research aims to investigate relationships between various forms of convergence and their adaptations to Korovkin’s theorem. Initially, an exhaustive examination of diverse convergence types will be investigated, encompassing meticulous scrutiny of their definitional nuances, distinctive properties, and interrelations. Subsequently we will elucidate the adaptation of these convergence modalities to the Korovkin’s theorem, thereby broadening the purview of the approximation theorem. Furthermore, the inquiry extends to the algebraic and topological attributes intrinsic to transformations instantiated on operators. This scholarly pursuit aims not only to augment our theoretical comprehension of convergence types and their ramifications but also to foster innovative insights that contribute substantially to the scholarly discourse within the approximation theory.

MERSİN ÜNİVERSİTESİ

Alberta University, Canada 25.01.2025 – 03.11.2025