The Gibbs phenomenon

Abstract

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.