In short, the Gibbs phenomenon describes the oscillating behaviour followed by an overshoot or undershoot of a Fourier partial sum expansion compared to the original function near jump discontinuities. In particular, the fact that this overshoot or undershoot will never decrease below 9% of the height of the “jump”.
First, some historical remarks about the discovery of the Gibbs phenomenon are given and several fundamental elements of Fourier analysis are introduced. By thoroughly analyzing the Saw-tooth wave function, an example of a function that exhibits the Gibbs phenomenon, we provide some figures in order to visualize this phenomenon. Furthermore, we address and fill in the gaps of the existing mathematical proof to advance the theoretical understanding of the Gibbs phenomenon. This analysis is then generalized to apply to a broader class of functions with jump discontinuities, by first considering a function with a jump discontinuity at 0 and then proving the occurrence of the phenomenon at a general point of jump discontinuity.
Furthermore, a literature research is conducted to evaluate different methods for resolving the Gibbs phenomenon. This thesis includes an in-depth analysis of filtering methods and spectral reprojection methods. Specifically, the Gegenbauer reconstruction method is described and its theoretical foundations in removing the Gibbs phenomenon is examined.
Finally, a real-life application of the Gibbs phenomenon is given by looking at its occurrence in Magnetic Resonance Imaging.
Overall, this thesis not only advances the theoretical understanding of the Gibbs phenomenon but also provides an insight into its resolution and application in Magnetic Resonance Imaging.

This thesis examines the approximation capabilities of path signatures within rough path spaces, focusing on both local and global universality. To this end, we provide a self-contained introduction to Rough Path theory, highlighting the interplay between additive and multiplicative functionals. This leads to the renowned Lyons' Extension theorem and the definition of rough path spaces. We also re-examine the concept of universality from a kernel-theoretic perspective, culminating in the classical universal approximation result for signatures over a compact subset of paths. To broaden the scope beyond compact domains, we introduce the framework of weighted spaces and elaborate on the notion of global universality. Specifically, we formally define globally universal kernels and prove sufficient conditions for their existence. The associated reproducing kernel Hilbert space is shown to approximate a wide range of functions over the entire domain, which may be non-(locally) compact. In particular, we apply these theoretical tools to rough path spaces, thereby cementing the global universality of signatures.