Date of Award

December 2020

Document Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematical Sciences

Committee Member

Mishko Mitkovski

Committee Member

Shitao Liu

Committee Member

Martin Schmoll

Committee Member

Jeong-Rock Yoon

Abstract

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.

Share

COinS
 
 

To view the content in your browser, please download Adobe Reader or, alternately,
you may Download the file to your hard drive.

NOTE: The latest versions of Adobe Reader do not support viewing PDF files within Firefox on Mac OS and if you are using a modern (Intel) Mac, there is no official plugin for viewing PDF files within the browser window.