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 Title
 FRAMES IN HILBERT C*MODULES.
 Creator

Jing, Wu, Han, Deguang, University of Central Florida
 Abstract / Description

Since the discovery in the early 1950's, frames have emerged as an important tool in signal processing, image processing, data compression and sampling theory etc. Today, powerful tools from operator theory and Banach space theory are being introduced to the study of frames producing deep results in frame theory. In recent years, many mathematicians generalized the frame theory from Hilbert spaces to Hilbert C*modules and got significant results which enrich the theory of frames. Also...
Show moreSince the discovery in the early 1950's, frames have emerged as an important tool in signal processing, image processing, data compression and sampling theory etc. Today, powerful tools from operator theory and Banach space theory are being introduced to the study of frames producing deep results in frame theory. In recent years, many mathematicians generalized the frame theory from Hilbert spaces to Hilbert C*modules and got significant results which enrich the theory of frames. Also there is growing evidence that Hilbert C*modules theory and the theory of wavelets and frames are tightly related to each other in many aspects. Both research fields can benefit from achievements of the other field. Our purpose of this dissertation is to work on several basic problems on frames for Hilbert C*modules. We first give a very useful characterization of modular frames which is easy to be applied. Using this characterization we investigate the modular frames from the operator theory point of view. A condition under which the removal of element from a frame in Hilbert C*modules leaves a frame or a nonframe set is also given. In contrast to the Hilbert space situation, Riesz bases of Hilbert C*modules may possess infinitely many alternative duals due to the existence of zerodivisors and not every dual of a Riesz basis is again a Riesz basis. We will present several such examples showing that the duals of Riesz bases in Hilbert $C^*$modules are much different and more complicated than the Hilbert space cases. A complete characterization of all the dual sequences for a Riesz basis, and a necessary and sufficient condition for a dual sequence of a Riesz basis to be a Riesz basis are also given. In the case that the underlying C*algebra is a commutative W*algebra, we prove that the set of the Parseval frame generators for a unitary group can be parameterized by the set of all the unitary operators in the double commutant of the unitary group. Similar result holds for the set of all the general frame generators where the unitary operators are replaced by invertible and adjointable operators. Consequently, the set of all the Parseval frame generators is pathconnected. We also prove the existence and uniqueness of the best Parseval multiframe approximations for multiframe generators of unitary groups on Hilbert C*modules when the underlying C*algebra is commutative. For the dilation results of frames we show that a complete Parseval frame vector for a unitary group on Hilbert C*module can be dilated to a complete wandering vector. For any dual frame pair in Hilbert C*modules, we prove that the pair are orthogonal compressions of a Riesz basis and its canonical dual basis for some larger Hilbert C*module. For the perturbation of frames and Riesz bases in Hilbert C*modules we prove that the CasazzaChristensen general perturbation theorem for frames in Hilbert spaces remains valid in Hilbert C*modules. In the Hilbert space setting, under the same perturbation condition, the perturbation of any Riesz basis remains a Riesz basis. However, this no longer holds for Riesz bases in Hilbert C*modules. We also give a complete characterization on all the Riesz bases for Hilbert C*modules such that the perturbation (under CasazzaChristensen's perturbation condition) of a Riesz basis still remains a Riesz basis.
Show less  Date Issued
 2006
 Identifier
 CFE0001182, ucf:46859
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0001182
 Title
 I'M BEING FRAMED: PHASE RETRIEVAL AND FRAME DILATION IN FINITEDIMENSIONAL REAL HILBERT SPACES.
 Creator

Greuling, Jason L, Han, Deguang, University of Central Florida
 Abstract / Description

Research has shown that a frame for an ndimensional real Hilbert space offers phase retrieval if and only if it has the complement property. There is a geometric characterization of general frames, the HanLarsonNaimark Dilation Theorem, which gives us the necessary and suffcient conditions required to dilate a frame for an ndimensional Hilbert space to a frame for a Hilbert space of higher dimension k. However, a frame having the complement property in an ndimensional real Hilbert space...
Show moreResearch has shown that a frame for an ndimensional real Hilbert space offers phase retrieval if and only if it has the complement property. There is a geometric characterization of general frames, the HanLarsonNaimark Dilation Theorem, which gives us the necessary and suffcient conditions required to dilate a frame for an ndimensional Hilbert space to a frame for a Hilbert space of higher dimension k. However, a frame having the complement property in an ndimensional real Hilbert space does not ensure that its dilation will offer phase retrieval. In this thesis, we will explore and provide what necessary and suffcient conditions must be satisfed to dilate a phase retrieval frame for an ndimensional real Hilbert space to a phase retrieval frame for a kdimensional real Hilbert.
Show less  Date Issued
 2018
 Identifier
 CFH2000319, ucf:45868
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFH2000319
 Title
 OPTIMAL DUAL FRAMES FOR ERASURES AND DISCRETE GABOR FRAMES.
 Creator

Lopez, Jerry, Han, Deguang, University of Central Florida
 Abstract / Description

Since their discovery in the early 1950's, frames have emerged as an important tool in areas such as signal processing, image processing, data compression and sampling theory, just to name a few. Our purpose of this dissertation is to investigate dual frames and the ability to find dual frames which are optimal when coping with the problem of erasures in data transmission. In addition, we study a special class of frames which exhibit algebraic structure, discrete Gabor frames. Much work...
Show moreSince their discovery in the early 1950's, frames have emerged as an important tool in areas such as signal processing, image processing, data compression and sampling theory, just to name a few. Our purpose of this dissertation is to investigate dual frames and the ability to find dual frames which are optimal when coping with the problem of erasures in data transmission. In addition, we study a special class of frames which exhibit algebraic structure, discrete Gabor frames. Much work has been done in the study of discrete Gabor frames in $\mathbb^n$, but very little is known about the $\ell^2(\mathbb)$ case or the $\ell^2(\mathbb^d)$ case. We establish some basic Gabor frame theory for $\ell^2(\mathbb)$ and then generalize to the $\ell^2(\mathbb^d)$ case.
Show less  Date Issued
 2009
 Identifier
 CFE0002614, ucf:48274
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0002614
 Title
 Variational inclusions with general overrelaxed proximal point and variationallike inequalities with densely pseudomonotonicity.
 Creator

Nguyen, George, Mohapatra, Ram, Han, Deguang, Shuai, Zhisheng, Xu, Mengyu, University of Central Florida
 Abstract / Description

This dissertation focuses on the existence and uniqueness of the solutions of variational inclusion and variational inequality problems and then attempts to develop efficient algorithms to estimate numerical solutions for the problems. The dissertation consists a total of five chapters. Chapter 1 is an introduction to variational inequality problems, variational inclusion problems, monotone operators, and some basic definitions and preliminaries from convex analysis. Chapter 2 is a studyof a...
Show moreThis dissertation focuses on the existence and uniqueness of the solutions of variational inclusion and variational inequality problems and then attempts to develop efficient algorithms to estimate numerical solutions for the problems. The dissertation consists a total of five chapters. Chapter 1 is an introduction to variational inequality problems, variational inclusion problems, monotone operators, and some basic definitions and preliminaries from convex analysis. Chapter 2 is a studyof a general class of nonlinear implicit inclusion problems. The objective of this study is to explore how to omit the Lipschitz continuity condition by using an alternating approach to the proximal point algorithm to estimate the numerical solution of the implicit inclusion problems. In chapter 3 we introduce generalized densely relaxed ? ? ? pseudomonotone operators and generalized relaxed ? ? ? proper quasimonotone operators as well as relaxed ? ? ? quasimonotone operators. Using these generalized monotonicity notions, we establish the existence results for the generalized variationallike inequality in the general setting of Banach spaces. In chapter 4, we use the auxiliary principle technique to introduce a general algorithm for solutions of the densely relaxed pseudomonotone variationallike inequalities. Chapter 5 is the chapter concluding remarks and scope for future work.
Show less  Date Issued
 2019
 Identifier
 CFE0007693, ucf:52410
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007693
 Title
 Frames and Phase Retrieval.
 Creator

Juste, Ted, Han, Deguang, Sun, Qiyu, Dutkay, Dorin, Wang, Dingbao, University of Central Florida
 Abstract / Description

Phase retrieval tackles the problem of recovering a signal after loss of phase. The phase problem shows up in many different settings such as Xray crystallography, speech recognition, quantum information theory, and coherent diffraction imaging. In this dissertation we present some results relating to three topics on phase retrieval. Chapters 1 and 2 contain the relevant background materials. In chapter 3, we introduce the notion of exact phaseretrievable frames as a way of measuring aframe...
Show morePhase retrieval tackles the problem of recovering a signal after loss of phase. The phase problem shows up in many different settings such as Xray crystallography, speech recognition, quantum information theory, and coherent diffraction imaging. In this dissertation we present some results relating to three topics on phase retrieval. Chapters 1 and 2 contain the relevant background materials. In chapter 3, we introduce the notion of exact phaseretrievable frames as a way of measuring aframe's redundancy with respect to its phase retrieval property. We show that, in the ddimensional real Hilbert space case, exact phaseretrievable frames can be of any lengths between 2d ? 1 and d(d + 1)/2, inclusive. The complex Hilbert space case remains open.In chapter 4, we investigate phaseretrievability by studying maximal phaseretrievable subspaces with respect to a given frame. These maximal PRsubspaces can have different dimensions. We are able to identify the ones with the largest dimension and this can be considered as a generalizationof the characterization of real phaseretrievable frames. In the basis case, we prove that if M is a kdimensional PRsubspace then supp(x) ? k for every nonzero vector x ? M . Moreover, if1 ? k (<) [(d + 1)/2], then a kdimensional PRsubspace is maximal if and only if there exists a vector x ? M such that supp(x) = k.Chapter 5 is devoted to investigating phaseretrievable operatorvalued frames. We obtain some characterizations of phaseretrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces; thus generalizing known results for vectorvalued frames, fusion frames, and frames of Hermitian matrices.Finally, in Chapter 6, we consider the problem of characterizing projective representations that admit frame vectors with the maximal span property, a property that allows for an algebraic recovering of the phaseretrieval problem. We prove that every irreducible projective representation of a finite abelian group admits a frame vector with the maximal span property. All such vectors can be explicitly characterized. These generalize some of the recent results about phaseretrieval with Gabor (or STFT) measurements.
Show less  Date Issued
 2019
 Identifier
 CFE0007660, ucf:52503
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007660
 Title
 Applications of Compressive Sensing To Surveillance Problems.
 Creator

Huff, Christopher, Mohapatra, Ram, Sun, Qiyu, Han, Deguang, University of Central Florida
 Abstract / Description

In many surveillance scenarios, one concern that arises is how to construct an imager that is capable of capturing the scene with high fidelity. This could be problematic for two reasons: first, the optics and electronics in the camera may have difficulty in dealing with so much information; secondly, bandwidth constraints, may pose difficulty in transmitting information from the imager to the user efficiently for reconstruction or realization. In this thesis, we will discuss a mathematical...
Show moreIn many surveillance scenarios, one concern that arises is how to construct an imager that is capable of capturing the scene with high fidelity. This could be problematic for two reasons: first, the optics and electronics in the camera may have difficulty in dealing with so much information; secondly, bandwidth constraints, may pose difficulty in transmitting information from the imager to the user efficiently for reconstruction or realization. In this thesis, we will discuss a mathematical framework that is capable of skirting the two aforementioned issues. This framework is rooted in a technique commonly referred to as compressive sensing. We will explore two of the seminal works in compressive sensing and will present the key theorems and definitions from these two papers. We will then survey three different surveillance scenarios and their respective compressive sensing solutions. The original contribution of this thesis is the development of a distributed compressive sensing model.
Show less  Date Issued
 2012
 Identifier
 CFE0004317, ucf:49473
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0004317
 Title
 Tiling with Polyominoes, Polycubes, and Rectangles.
 Creator

Saxton, Michael, Reid, Michael, Lee, Junho, Han, Deguang, University of Central Florida
 Abstract / Description

In this paper we study the hierarchical structure of the 2d polyominoes. We introduce a new infinite family of polyominoes which we prove tiles a strip. We discuss applications of algebra to tiling. We discuss the algorithmic decidability of tiling the infinite plane Z x Z given a finite set of polyominoes. We will then discuss tiling with rectangles. We will then get some new, and some analogous results concerning the possible hierarchical structure for the 3d polycubes.
 Date Issued
 2015
 Identifier
 CFE0005995, ucf:50791
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005995
 Title
 Nonparametric and Empirical Bayes Estimation Methods.
 Creator

Benhaddou, Rida, Pensky, Marianna, Han, Deguang, Swanson, Jason, Ni, Liqiang, University of Central Florida
 Abstract / Description

In the present dissertation, we investigate two different nonparametric models; empirical Bayes model and functional deconvolution model. In the case of the nonparametric empirical Bayes estimation, we carried out a complete minimax study. In particular, we derive minimax lower bounds for the risk of the nonparametric empirical Bayes estimator for a general conditional distribution. This result has never been obtained previously. In order to attain optimal convergence rates, we use a wavelet...
Show moreIn the present dissertation, we investigate two different nonparametric models; empirical Bayes model and functional deconvolution model. In the case of the nonparametric empirical Bayes estimation, we carried out a complete minimax study. In particular, we derive minimax lower bounds for the risk of the nonparametric empirical Bayes estimator for a general conditional distribution. This result has never been obtained previously. In order to attain optimal convergence rates, we use a wavelet series based empirical Bayes estimator constructed in Pensky and Alotaibi (2005). We propose an adaptive version of this estimator using Lepski's method and show that the estimator attains optimal convergence rates. The theory is supplemented by numerous examples. Our study of the functional deconvolution model expands results of Pensky and Sapatinas (2009, 2010, 2011) to the case of estimating an $(r+1)$dimensional function or dependent errors. In both cases, we derive minimax lower bounds for the integrated square risk over a wide set of Besov balls and construct adaptive wavelet estimators that attain those optimal convergence rates. In particular, in the case of estimating a periodic $(r+1)$dimensional function, we show that by choosing Besov balls of mixed smoothness, we can avoid the ''curse of dimensionality'' and, hence, obtain higher than usual convergence rates when $r$ is large. The study of deconvolution of a multivariate function is motivated by seismic inversion which can be reduced to solution of noisy twodimensional convolution equations that allow to draw inference on underground layer structures along the chosen profiles. The common practice in seismology is to recover layer structures separately for each profile and then to combine the derived estimates into a twodimensional function. By studying the twodimensional version of the model, we demonstrate that this strategy usually leads to estimators which are less accurate than the ones obtained as twodimensional functional deconvolutions. Finally, we consider a multichannel deconvolution model with longrange dependent Gaussian errors. We do not limit our consideration to a specific type of longrange dependence, rather we assume that the eigenvalues of the covariance matrix of the errors are bounded above and below. We show that convergence rates of the estimators depend on a balance between the smoothness parameters of the response function, the smoothness of the blurring function, the long memory parameters of the errors, and how the total number of observations is distributed among the channels.
Show less  Date Issued
 2013
 Identifier
 CFE0004814, ucf:49737
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0004814
 Title
 Tiling the Integers.
 Creator

Li, Shasha, Dutkay, Dorin, Han, Deguang, Sun, Qiyu, University of Central Florida
 Abstract / Description

A set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set. Counterexamples based on ?nite Abelian groups show that Fuglede conjecture is false inhigh dimensions. A solution for the Fuglede conjecture in Z or all the groups ZN would provide a solution for the Fuglede conjecture in R. Focusing on tiles in dimension one, we will concentrate on the analysis of tiles in the ?nite groups ZN. Based on the Coven Meyerowitz conjecture, it has...
Show moreA set tiles the integers if and only if the integers can be written as a disjoint union of translates of that set. Counterexamples based on ?nite Abelian groups show that Fuglede conjecture is false inhigh dimensions. A solution for the Fuglede conjecture in Z or all the groups ZN would provide a solution for the Fuglede conjecture in R. Focusing on tiles in dimension one, we will concentrate on the analysis of tiles in the ?nite groups ZN. Based on the Coven Meyerowitz conjecture, it has been proved that if any spectral set in Z satis?es the the CovenMeyerowitz properties, then everyspectral set in R is a tile. We will present some of the main results related to integer tiles and give a selfcontained description of the theory with detailed proofs.
Show less  Date Issued
 2014
 Identifier
 CFE0005199, ucf:50642
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005199
 Title
 LatticeValued TFilters and Induced Structures.
 Creator

Reid, Frederick, Richardson, Gary, Brennan, Joseph, Han, Deguang, Lang, SheauDong, University of Central Florida
 Abstract / Description

A complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a Tconvergence structure which is defined in terms of Tfilters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of...
Show moreA complete lattice is called a frame provided meets distribute over arbitrary joins. The implication operation in this context plays a central role. Intuitively, it measures the degree to which one element is less than or equal to another. In this setting, a category is defined by equipping each set with a Tconvergence structure which is defined in terms of Tfilters. This category is shown to be topological, strongly Cartesian closed, and extensional. It is well known that the category of topological spaces and continuous maps is neither Cartesian closed nor extensional.Subcategories of compact and of complete spaces are investigated. It is shown that each Tconvergence space has a compactification with the extension property provided the frame is a Boolean algebra. TCauchy spaces are defined and sufficient conditions for the existence of a completion are given. Tuniform limit spaces are also defined and their completions are given in terms of the TCauchy spaces they induce. Categorical properties of these subcategories are also investigated. Further, for a fixed Tconvergence space, under suitable conditions, it is shown that there exists an order preserving bijection between the set of all strict, regular, Hausdorff compactifications and the set of all totally bounded TCauchy spaces which induce the fixed space.
Show less  Date Issued
 2019
 Identifier
 CFE0007520, ucf:52586
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0007520
 Title
 Spectrally Uniform Frames and Spectrally Optimal Dual Frames.
 Creator

Pehlivan, Saliha, Han, Deguang, Mohapatra, Ram, Sun, Qiyu, Tatari, Mehmet, University of Central Florida
 Abstract / Description

Frames have been useful in signal transmission due to the built in redundancy. In recent years, theerasure problem in data transmission has been the focus of considerable research in the case theerror estimate is measured by operator (or matrix) norm. Sample results include the characterizationof oneerasure optimal Parseval frames, the connection between twoerasure optimal Parsevalframes and equiangular frames, and some characterization of optimal dual frames.If iterations are allowed in...
Show moreFrames have been useful in signal transmission due to the built in redundancy. In recent years, theerasure problem in data transmission has been the focus of considerable research in the case theerror estimate is measured by operator (or matrix) norm. Sample results include the characterizationof oneerasure optimal Parseval frames, the connection between twoerasure optimal Parsevalframes and equiangular frames, and some characterization of optimal dual frames.If iterations are allowed in the reconstruction process of the signal vector, then spectral radiusmeasurement for the error operators is more appropriate then the operator norm measurement.We obtain a complete characterization of spectrally oneuniform frames (i.e., oneerasure optimalframes with respect to the spectral radius measurement) in terms of the redundancy distributionof the frame. Our characterization relies on the connection between spectrally optimal frames andthe linear connectivity property of the frame. We prove that the linear connectivity property isequivalent to the intersection dependence property, and is also closely related to the wellknownconcept of kindependent set. For spectrally twouniform frames, it is necessary that the framemust be linearly connected. We conjecture that it is also necessary that a twouniform frame mustbe nindependent. We confirmed this conjecture for the case when N = n+1, n+2, where N is thenumber of vectors in a frame for an ndimensional Hilbert space. Additionally we also establishseveral necessary and sufficient conditions for the existence of an alternate dual frame to make the iterated reconstruction to work.
Show less  Date Issued
 2013
 Identifier
 CFE0005111, ucf:50747
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005111
 Title
 Tiling properties of spectra of measures.
 Creator

Haussermann, John, Dutkay, Dorin, Han, Deguang, Sun, Qiyu, Dogariu, Aristide, University of Central Florida
 Abstract / Description

We investigate tiling properties of spectra of measures, i.e., sets ? in R with an orthogonal basis in L2 with respect to some finite Borel measure on R. Such measures include Lebesgue measure on bounded Borel subsets, finite atomic measures and some fractal Hausdorff measures. We show that various classes of such spectra of measures have translational tiling properties. This lead to some surprising tiling properties for spectra of fractal measures, the existence of complementing sets and...
Show moreWe investigate tiling properties of spectra of measures, i.e., sets ? in R with an orthogonal basis in L2 with respect to some finite Borel measure on R. Such measures include Lebesgue measure on bounded Borel subsets, finite atomic measures and some fractal Hausdorff measures. We show that various classes of such spectra of measures have translational tiling properties. This lead to some surprising tiling properties for spectra of fractal measures, the existence of complementing sets and spectra for finite sets with the CovenMeyerowitz property, the existence of complementing Hadamard pairs in the case ofHadamard pairs of size 2,3,4 or 5. In the context of the Fuglede conjecture, we prove that any spectral set is a tile, if the period of the spectrum is 2,3,4 or 5.
Show less  Date Issued
 2014
 Identifier
 CFE0005182, ucf:50656
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005182
 Title
 Integral Representations of Positive Linear Functionals.
 Creator

Siple, Angela, Mikusinski, Piotr, Atanasiu, Dragu, Dutkay, Dorin, Han, Deguang, Lee, Junho, Brennan, Joseph, Huo, Qun, University of Central Florida
 Abstract / Description

In this dissertation we obtain integral representations for positive linear functionals on commutative algebras with involution and semigroups with involution. We prove Bochner and Plancherel type theorems for representations of positive functionals and show that, under some conditions, the Bochner and Plancherel representations are equivalent. We also consider the extension of positive linear functionals on a Banach algebra into a space of pseudoquotients and give under conditions in which...
Show moreIn this dissertation we obtain integral representations for positive linear functionals on commutative algebras with involution and semigroups with involution. We prove Bochner and Plancherel type theorems for representations of positive functionals and show that, under some conditions, the Bochner and Plancherel representations are equivalent. We also consider the extension of positive linear functionals on a Banach algebra into a space of pseudoquotients and give under conditions in which the space of pseudoquotients can be identified with all Radon measures on the structure space. In the final chapter we consider a system of integrated Cauchy functional equations on a semigroup, which generalizes a result of Ressel and offers a different approach to the proof.
Show less  Date Issued
 2015
 Identifier
 CFE0005713, ucf:50144
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0005713