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SCALING OF SPECTRA OF CANTOR-TYPE MEASURES AND SOME NUMBER THEORETIC CONSIDERATIONS
Title
SCALING OF SPECTRA OF CANTOR-TYPE MEASURES AND SOME NUMBER THEORETIC CONSIDERATIONS.
Creator
Kraus, Isabelle, Dutkay, Dorin, University of Central Florida
Abstract / Description
We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number m generates a complete or incomplete Fourier basis for a Cantor-type measure with scale g.
Date Issued
2017
Identifier
CFH2000169, ucf:45948
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFH2000169
FRACTAL SPECTRAL MEASURES IN TWO DIMENSIONS
Title
FRACTAL SPECTRAL MEASURES IN TWO DIMENSIONS.
Creator
Alrud, Bengt, Dutkay, Dorin, University of Central Florida
Abstract / Description
We study spectral properties for invariant measures associated to affine iterated function systems. We present various conditions under which the existence of a Hadamard pair implies the existence of a spectrum for the fractal measure. This solves a conjecture proposed by Dorin Dutkay and Palle Jorgensen, in several special cases in dimension 2.
Date Issued
2011
Identifier
CFE0003873, ucf:48732
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0003873
Frames and Phase Retrieval
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 X-ray 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 phase-retrievable 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 X-ray 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 phase-retrievable frames as a way of measuring aframe's redundancy with respect to its phase retrieval property. We show that, in the d-dimensional real Hilbert space case, exact phase-retrievable 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 phase-retrievability by studying maximal phase-retrievable subspaces with respect to a given frame. These maximal PR-subspaces 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 phase-retrievable frames. In the basis case, we prove that if M is a k-dimensional PR-subspace then |supp(x)| ? k for every nonzero vector x ? M . Moreover, if1 ? k (<) [(d + 1)/2], then a k-dimensional PR-subspace is maximal if and only if there exists a vector x ? M such that |supp(x)| = k.Chapter 5 is devoted to investigating phase-retrievable operator-valued frames. We obtain some characterizations of phase-retrievable frames for general operator systems acting on both finite and infinite dimensional Hilbert spaces; thus generalizing known results for vector-valued 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 recov-ering of the phase-retrieval 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 phase-retrieval with Gabor (or STFT) measurements.
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Date Issued
2019
Identifier
CFE0007660, ucf:52503
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0007660
Tiling the Integers
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 Coven-Meyerowitz properties, then everyspectral set in R is a tile. We will present some of the main results related to integer tiles and give a self-contained description of the theory with detailed proofs.
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Date Issued
2014
Identifier
CFE0005199, ucf:50642
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005199
Extensions of S-spaces
Title
Extensions of S-spaces.
Creator
Losert, Bernd, Richardson, Gary, Mikusinski, Piotr, Dutkay, Dorin, Brennan, Joseph, Marinescu, Dan, University of Central Florida
Abstract / Description
Given a convergence space X, a continuous action of a convergence semigroup S on X and a compactification Y of X, under what conditions on X and the action on X is it possible to extend the action to a continuous action on Y. Similarly, given a Cauchy space X, a Cauchy continuous action of a Cauchy semigroup S on X and a completion Y of X, under what conditions on X and the action on X is it possible to extend the action to a Cauchy continuous action on Y. We answer the first question for...
Show moreGiven a convergence space X, a continuous action of a convergence semigroup S on X and a compactification Y of X, under what conditions on X and the action on X is it possible to extend the action to a continuous action on Y. Similarly, given a Cauchy space X, a Cauchy continuous action of a Cauchy semigroup S on X and a completion Y of X, under what conditions on X and the action on X is it possible to extend the action to a Cauchy continuous action on Y. We answer the first question for some particular compactifications like the one-point compactification and the star compactification as well as for the class of regular compactifications. We answer the second question for the class of regular strict completions. Using these results, we give sufficient conditions under which the pseudoquotient of a compactification/completion of a space is the compactification/completion of the pseudoquotient of the given space.
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Date Issued
2013
Identifier
CFE0004881, ucf:49661
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0004881
Tiling properties of spectra of measures
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 Coven-Meyerowitz 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.
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Date Issued
2014
Identifier
CFE0005182, ucf:50656
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005182
Integral Representations of Positive Linear Functionals
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.
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Date Issued
2015
Identifier
CFE0005713, ucf:50144
Format
Document (PDF)
PURL
http://purl.flvc.org/ucf/fd/CFE0005713