DIVERGENCE PHENOMENA IN FOURIER SERIES AND THEIR IMPLICATIONS

Abstract

This project investigates the convergence and divergence behavior of Fourier series by integrating both their foundational mathematical structure and the critical counterexamples that shaped modern harmonic analysis. The first part of the work develops the theoretical basis of Fourier expansions, including the orthogonality of trigonometric and complex exponential systems, the computation of Fourier coefficients, and the principal modes of convergence pointwise, uniform, and L² convergence. Classical results such as Dirichlet’s theorem, Fej´er’s theorem, and Parseval’s identity are presented to illustrate how smoothness, periodicity, and energy distribution govern the stability of Fourier representations.Building upon this foundation, the second part examines the divergence phenomena that reveal inherent limitations in harmonic approximation. Beginning with du Bois-Reymond’s pioneering constructions of continuous functions whose Fourier series diverge at one or all points, the study then turns to Kolmogorov’s landmark demonstration of an L¹ function whose Fourier series diverges almost everywhere, thereby exposing the insufficiency of integrability alone. The discussion culminates with Carleson’s theorem and its extension to L spaces, which establish that square-integrability ensures almost everywhere convergence and thus delineate the precise boundary between stable and pathological behavior in Fourier series.Together, these chapters provide a comprehensive and systematic account of how convergence, divergence, and function-space regularity interact in Fourier analysis. They underscore both the power and the limitations of Fourier series, offering critical insight into the structural principles that continue to influence contemporary harmonic analysis and its applications.