Date of Award
Doctor of Philosophy (PhD)
The uncertainty principle is one of the most fundamental concepts in harmonic analysis. It has many facets and appears in many different forms. In the first two chapters of the thesis, we state the classical uncertainty principle and the Balian-Low theorem, and introduce the concept of framed Hilbert spaces as a general setting for studying problems of uncertainty principle type. These spaces could be viewed as a special type of reproducing kernel Hilbert spaces which include many function spaces that play an important role in harmonic analysis.
The thesis mainly consists of Chapters 3-5 whose common theme is the uncertainty principle. In Chapter 3, we define a very general interpolation set called d-approximate interpolation set in framed Hilbert spaces. And we prove a necessary density condition for those sets similar to the one for usual interpolation sets. In Chapter 4, we focus on the problem of estimating the sampling constant in the space of multiband-limited functions. Comparing with the old results, by imposing suboptimal conditions on the sampling set, we obtain a much-improved sampling constant which depends linearly on the length of the multiband. In Chapter 5, we give a very general version of the Balian-Low theorem, which does not necessarily require the orthonormal basis to be the time-frequency shifts of a single function.
Li, Haodong, "Uncertainty Principles in Framed Hilbert Spaces" (2020). All Dissertations. 2748.