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Title page for ETD etd-05192011-215511


Type of Document Dissertation
Author Wang, Haichao
URN etd-05192011-215511
Title Topic on Shift-Invariant Spaces with Extra Invariance
Degree PhD
Department Mathematics
Advisory Committee
Advisor Name Title
Aldroubi,Akram Committee Chair
Hardin, Doug Committee Member
Powell, Alex Committee Member
Wilkes, Mitchell Committee Member
Zheng, Dechao Committee Member
Keywords
  • Fourier transform
  • Least square
  • Shift-invariant spaces
  • Uncertainty Principle
Date of Defense 2011-04-25
Availability unrestricted
Abstract
We consider shift-invariant spaces with extra invariance, namely translation-invarianr or 1/nZ--invariant for some n > I. In general a shift invariant space does not possess .any invariance other than translation by integers. As additional invariance is desirable in signal processing and wavelet analysis, one may circumvent the slow spatial-decay property by seeking principle shift-invariant spaces invariant under the translation by 1/nZ for large integer n. In fact, such principle shift-invariant spaces with integrable generators do exist. The most surprising result is that there is a uncertainty principle of Balian-Low type under the additional invariance of principal shift-invariant space and the time-frequency localization of its generator. Similar results can also be obtained for finitely generated shift-invariant spaces.

In some signal and image processing applications, it is assumed that signals live in some principal shift­ invariant spaces. As the observation data is usually corrupted due to various reasons, it is natural to look for principal shift-invariant spaces (even with additional invariance) that best approximate a given data set in the sense of the least squares. We will also construct principal shift-invariant spaces with additional invariance that best approximate a given data set.

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