The main problem in sampling theory is to reconstruct a function from its values (samples) on some discrete subset <font face=symbol>W</font> of its domain. However, taking samples on an appropriate sampling set <font face=symbol>W</font> 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. <br />
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<sup>n</sup>f,g) for 0 <font face=symbol>£</font> n<L(g) and g <font face=symbol>Ã</font> G,
where G is a countable (finite or infinite) set of vectors in H, and the function L:G<font face=symbol>®</font> {1,2,...,<font face=symbol>Â¥</font>} represents the "sampling level." Then the main problem becomes to recover the unknown vector f <font face=symbol>Ã</font> H from the above measurements. <br />
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<sup>*</sup>-algebras, spectral theory of normal operators, and frame theory.
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| Identifer | oai:union.ndltd.org:VANDERBILT/oai:VANDERBILTETD:etd-05172017-130101 |
| Date | 17 May 2017 |
| Creators | Petrosyan, Armenak |
| Contributors | David Smith, Gieri Simonett, Alexander Powell, Douglas Hardin, Akram Aldroubi |
| Publisher | VANDERBILT |
| Source Sets | Vanderbilt University Theses |
| Language | English |
| Detected Language | English |
| Type | text |
| Format | application/pdf |
| Source | http://etd.library.vanderbilt.edu/available/etd-05172017-130101/ |
| Rights | unrestricted, I hereby certify that, if appropriate, I have obtained and attached hereto a written permission statement from the owner(s) of each third party copyrighted matter to be included in my thesis, dissertation, or project report, allowing distribution as specified below. I certify that the version I submitted is the same as that approved by my advisory committee. I hereby grant to Vanderbilt University or its agents the non-exclusive license to archive and make accessible, under the conditions specified below, my thesis, dissertation, or project report in whole or in part in all forms of media, now or hereafter known. I retain all other ownership rights to the copyright of the thesis, dissertation or project report. I also retain the right to use in future works (such as articles or books) all or part of this thesis, dissertation, or project report. |
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