Mathematics seminar: A manifestation of the uncertainty principle through Fourier coefficients

Speaker: Adem Limani, Lunds universitet

Abstract: The uncertainty principle in harmonic analysis captures a limitation of how well-localized a function and its Fourier transform can be, simultaneously. In terms of signals, sharp localization requires the presence of high-frequency components, while strong decay of Fourier coefficients suppresses them. Besides the fact that many deep results in mathematical analysis reflect this interplay, this principle has profound implications in quantum physics, most famously through the form of Heisenberg’s uncertainty principle. In this talk, we shall discuss yet another manifestation of the uncertainty principle, which arose from the twentieth-century programme on uniqueness problems of Fourier series. More specifically, we shall identify thresholds connecting two phenomena: localization measured in terms of meagre support, and weak regularity measured by weighted average decay of Fourier coefficients.