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1. Leshen, Sara. Balian-Low Type Results for Gabor Schauder Bases.

Degree: PhD, Mathematics, 2019, Vanderbilt University

URL: http://etd.library.vanderbilt.edu/available/etd-03202019-161148/ ;

The uncertainty principle implies that a function and its Fourier transform cannot both be well-localized. The Balian-Low theorem is a version of the uncertainty principle for generators of Gabor orthonormal bases. This dissertation proves a new Balian-Low type theorem for compactly supported generators of Gabor Schauder bases. Moreover, we show that the classical Balian-Low theorem for orthonormal bases does not hold for Schauder bases.
*Advisors/Committee Members: Alexander Powell (chair), Akram Aldroubi (committee member), Doug Hardin (committee member), Gieri Simonett (committee member), David Smith (committee member).*

Subjects/Keywords: Schauder bases; uncertainty principle; time-frequency analysis

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Leshen, S. (2019). Balian-Low Type Results for Gabor Schauder Bases. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://etd.library.vanderbilt.edu/available/etd-03202019-161148/ ;

Chicago Manual of Style (16^{th} Edition):

Leshen, Sara. “Balian-Low Type Results for Gabor Schauder Bases.” 2019. Doctoral Dissertation, Vanderbilt University. Accessed October 17, 2019. http://etd.library.vanderbilt.edu/available/etd-03202019-161148/ ;.

MLA Handbook (7^{th} Edition):

Leshen, Sara. “Balian-Low Type Results for Gabor Schauder Bases.” 2019. Web. 17 Oct 2019.

Vancouver:

Leshen S. Balian-Low Type Results for Gabor Schauder Bases. [Internet] [Doctoral dissertation]. Vanderbilt University; 2019. [cited 2019 Oct 17]. Available from: http://etd.library.vanderbilt.edu/available/etd-03202019-161148/ ;.

Council of Science Editors:

Leshen S. Balian-Low Type Results for Gabor Schauder Bases. [Doctoral Dissertation]. Vanderbilt University; 2019. Available from: http://etd.library.vanderbilt.edu/available/etd-03202019-161148/ ;

2. Petrosyan, Armenak. Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators.

Degree: PhD, Mathematics, 2017, Vanderbilt University

URL: http://etd.library.vanderbilt.edu/available/etd-05172017-130101/ ;

The main problem in sampling theory is to reconstruct a function from its values (samples) on some discrete subset W of its domain. However, taking samples on an appropriate sampling set W is not always practical or even possible - for example, associated measuring devices may be too expensive or scarce.
In the dynamical sampling problem, it is assumed that the sparseness of the sampling locations can be compensated by involving dynamics. For example, when f is the initial state of a physical process (say, change of temperature or air pollution), we can sample its values at the same sampling locations as time progresses, and try to recover f from the combination of these spatio-temporal samples.
In our recent work, we have taken a more operator-theoretic approach to the dynamical sampling problem. We assume that the unknown function f is a vector in some Hilbert space H and A is a bounded linear operator on H. The samples are given in the form (A^{n}f,g) for 0 Â£ n<L(g) and g Ã G,
where G is a countable (finite or infinite) set of vectors in H, and the function L:GÂ® {1,2,...,Â¥} represents the "sampling level." Then the main problem becomes to recover the unknown vector f Ã H from the above measurements.
The dynamical sampling problem has potential applications in plenacoustic sampling, on-chip sensing, data center temperature sensing, neuron-imaging, and satellite remote sensing, and more generally to Wireless Sensor Networks (WSN). It also has connections and applications to other areas of mathematics including C^{*}-algebras, spectral theory of normal operators, and frame theory.
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*Advisors/Committee Members: David Smith (committee member), Gieri Simonett (committee member), Alexander Powell (committee member), Douglas Hardin (committee member), Akram Aldroubi (chair).*

Subjects/Keywords: Feichtinger conjecture; Reconstruction; Shift-Invariant Spaces; Sub-Sampling; Frames; Dynamical Sampling; Sampling Theory

Record Details Similar Records

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APA · Chicago · MLA · Vancouver · CSE | Export to Zotero / EndNote / Reference Manager

APA (6^{th} Edition):

Petrosyan, A. (2017). Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators. (Doctoral Dissertation). Vanderbilt University. Retrieved from http://etd.library.vanderbilt.edu/available/etd-05172017-130101/ ;

Chicago Manual of Style (16^{th} Edition):

Petrosyan, Armenak. “Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators.” 2017. Doctoral Dissertation, Vanderbilt University. Accessed October 17, 2019. http://etd.library.vanderbilt.edu/available/etd-05172017-130101/ ;.

MLA Handbook (7^{th} Edition):

Petrosyan, Armenak. “Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators.” 2017. Web. 17 Oct 2019.

Vancouver:

Petrosyan A. Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators. [Internet] [Doctoral dissertation]. Vanderbilt University; 2017. [cited 2019 Oct 17]. Available from: http://etd.library.vanderbilt.edu/available/etd-05172017-130101/ ;.

Council of Science Editors:

Petrosyan A. Dynamical Sampling and Systems of Vectors from Iterative Actions of Operators. [Doctoral Dissertation]. Vanderbilt University; 2017. Available from: http://etd.library.vanderbilt.edu/available/etd-05172017-130101/ ;