Uncertainty Principles in Time-Frequency Analysis – Fractals and Schrödinger Evolutions

Abstract

In this thesis we study two uncertainty principles from the perspective of timefrequency analysis. The first part, to which a significant portion is dedicated, is concerned with deriving an analog in the joint time-frequency representation of the pre-established fractal uncertainty principle for a function and its Fourier transform. This study is motivated by, and the subsequent results substantiate, the fundamental idea that such analogs should exist, where the uncertainty principles cannot be avoided by a change of representation. The second part is focused on Hardy’s uncertainty principle in the joint representation, where we utilize this statement to reproduce and derive new uniqueness results for the solution of the Schr¨odinger equation. This showcases that uncertainty principles in the joint representation are not only an interesting object of study as analogous statements, but, within their own right, might contain applications beyond their initial scope of time-frequency analysis.