Current Search: Dutkay, Dorin (x)
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 Title
 SCALING OF SPECTRA OF CANTORTYPE 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 Cantortype measure with scale g.
 Date Issued
 2017
 Identifier
 CFH2000169, ucf:45948
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFH2000169
 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
 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
 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
 Extensions of Sspaces.
 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 onepoint 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.
Show less  Date Issued
 2013
 Identifier
 CFE0004881, ucf:49661
 Format
 Document (PDF)
 PURL
 http://purl.flvc.org/ucf/fd/CFE0004881
 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