THE CONVERGENCE OF FUNCTIONAL SERIES: POINTWISE VS. UNIFORM CONVERGENCE AND THEIR IMPLICATIONS IN FUNCTIONAL ANALYSIS

Abstract

 The convergence of functional series is a fundamental concept in higher mathematics that underpins many areas of analysis, including approximation theory and differential equations. This paper explores the distinctions between pointwise and uniform convergence, highlighting their definitions, criteria for identification, and practical implications. Pointwise convergence ensures that the series converges at each individual point in the domain, but it may fail to preserve continuity or differentiability of the limit function. In contrast, uniform convergence guarantees stronger properties, such as the preservation of continuity and the interchangeability of limits with integration or differentiation. Through counterexamples and theorems like Weierstrass's M-test, Abel's theorem, and Dini's theorem, we demonstrate how uniform convergence enhances reliability in applications. The discussion extends to real-world uses in numerical analysis, signal processing via Fourier series, and solving partial differential equations, emphasizing the need for uniform convergence to avoid errors in computations.