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- Creators: Cochran, Douglas
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Since Duffin and Schaeffer's introduction of frames in 1952, the concept of a frame has received much attention in the mathematical community and has inspired several generalizations. The focus of this thesis is on the concept of an operator-valued frame (OVF) and a more general concept called herein an operator-valued frame associated with a measure space (MS-OVF), which is sometimes called a continuous g-frame. The first of two main topics explored in this thesis is the relationship between MS-OVFs and objects prominent in quantum information theory called positive operator-valued measures (POVMs). It has been observed that every MS-OVF gives rise to a POVM with invertible total variation in a natural way. The first main result of this thesis is a characterization of which POVMs arise in this way, a result obtained by extending certain existing Radon-Nikodym theorems for POVMs. The second main topic investigated in this thesis is the role of the theory of unitary representations of a Lie group G in the construction of OVFs for the L^2-space of a relatively compact subset of G. For G=R, Duffin and Schaeffer have given general conditions that ensure a sequence of (one-dimensional) representations of G, restricted to (-1/2,1/2), forms a frame for L^{2}(-1/2,1/2), and similar conditions exist for G=R^n. The second main result of this thesis expresses conditions related to Duffin and Schaeffer's for two more particular Lie groups: the Euclidean motion group on R^2 and the (2n+1)-dimensional Heisenberg group. This proceeds in two steps. First, for a Lie group admitting a uniform lattice and an appropriate relatively compact subset E of G, the Selberg Trace Formula is used to obtain a Parseval OVF for L^{2}(E) that is expressed in terms of irreducible representations of G. Second, for the two particular Lie groups an appropriate set E is found, and it is shown that for each of these groups, with suitably parametrized unitary duals, the Parseval OVF remains an OVF when perturbations are made to the parameters of the included representations.
ContributorsRobinson, Benjamin (Author) / Cochran, Douglas (Thesis advisor) / Moran, William (Thesis advisor) / Boggess, Albert (Committee member) / Milner, Fabio (Committee member) / Spielberg, John (Committee member) / Arizona State University (Publisher)
Created2014
Description
The solution of the linear system of equations $Ax\approx b$ arising from the discretization of an ill-posed integral equation with a square integrable kernel is considered. The solution by means of Tikhonov regularization in which $x$ is found to as the minimizer of $J(x)=\{ \|Ax -b\|_2^2 + \lambda^2 \|L x\|_2^2\}$ introduces the unknown regularization parameter $\lambda$ which trades off the fidelity of the solution data fit and its smoothing norm, which is determined by the choice of $L$. The Generalized Discrepancy Principle (GDP) and Unbiased Predictive Risk Estimator (UPRE) are methods for finding $\lambda$ given prior conditions on the noise in the measurements $b$. Here we consider the case of $L=I$, and hence use the relationship between the singular value expansion and the singular value decomposition for square integrable kernels to prove that the GDP and UPRE estimates yield a convergent sequence for $\lambda$ with increasing problem size. Hence the estimate of $\lambda$ for a large problem may be found by down-sampling to a smaller problem, or to a set of smaller problems, and applying these estimators more efficiently on the smaller problems. In consequence the large scale problem can be solved in a single step immediately with the parameter found from the down sampled problem(s).
ContributorsHorst, Michael Jacob (Author) / Renaut, Rosemary (Thesis director) / Cochran, Douglas (Committee member) / Wang, Yang (Committee member) / Barrett, The Honors College (Contributor) / School of Music (Contributor) / School of Mathematical and Statistical Sciences (Contributor)
Created2014-05