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Spectral properties of the finite Hilbert transform on two adjacent intervals via the method of RiemannHilbert problem
 Date Issued:
 2019
 Abstract/Description:
 In this dissertation, we study a selfadjoint integral operator $\hat{K}$ which is defined in terms of finite Hilbert transforms on two adjacent intervals. These types of transforms arise when one studies the interior problem of tomography. The operator $\hat{K}$ possesses a socalled ``integrable kernel'' and it is known that the spectral properties of $\hat{K}$ are intimately related to a $2\times2$ matrix function $\Gamma(z;\lambda)$ which is the solution to a particular RiemannHilbert problem (in the $z$ plane). We express $\Gamma(z;\lambda)$ explicitly in terms of hypergeometric functions and find the small $\lambda$ asymptotics of $\Gamma(z;\lambda)$. This asymptotic analysis is necessary for the spectral analysis of the finite Hilbert transform on multiple adjacent intervals. We show that $\Gamma(z;\lambda)$ also has a jump in the $\lambda$ plane which allows us to compute the jump of the resolvent of $\hat{K}$. This jump is an important step in showing that the finite Hilbert transforms has simple and purely absolutely continuous spectrum. The well known spectral theory now allows us to construct unitary operators which diagonalize the finite Hilbert transforms. Lastly, we mention some future directions which include the many interval scenario and a bispectral property of $\hat{K}$.
Title:  Spectral properties of the finite Hilbert transform on two adjacent intervals via the method of RiemannHilbert problem. 
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Name(s): 
Blackstone, Elliot, Author Tovbis, Alexander, Committee Chair Katsevich, Alexander, Committee CoChair Tamasan, Alexandru, Committee Member Pang, Sean, Committee Member University of Central Florida, Degree Grantor 

Type of Resource:  text  
Date Issued:  2019  
Publisher:  University of Central Florida  
Language(s):  English  
Abstract/Description:  In this dissertation, we study a selfadjoint integral operator $\hat{K}$ which is defined in terms of finite Hilbert transforms on two adjacent intervals. These types of transforms arise when one studies the interior problem of tomography. The operator $\hat{K}$ possesses a socalled ``integrable kernel'' and it is known that the spectral properties of $\hat{K}$ are intimately related to a $2\times2$ matrix function $\Gamma(z;\lambda)$ which is the solution to a particular RiemannHilbert problem (in the $z$ plane). We express $\Gamma(z;\lambda)$ explicitly in terms of hypergeometric functions and find the small $\lambda$ asymptotics of $\Gamma(z;\lambda)$. This asymptotic analysis is necessary for the spectral analysis of the finite Hilbert transform on multiple adjacent intervals. We show that $\Gamma(z;\lambda)$ also has a jump in the $\lambda$ plane which allows us to compute the jump of the resolvent of $\hat{K}$. This jump is an important step in showing that the finite Hilbert transforms has simple and purely absolutely continuous spectrum. The well known spectral theory now allows us to construct unitary operators which diagonalize the finite Hilbert transforms. Lastly, we mention some future directions which include the many interval scenario and a bispectral property of $\hat{K}$.  
Identifier:  CFE0007602 (IID), ucf:52527 (fedora)  
Note(s): 
20190801 Ph.D. Sciences, Mathematics Doctoral This record was generated from author submitted information. 

Subject(s):  interior problem of tomography  finite Hilbert transform  TitchmarshWeyl theory  diagonalization  large parameter asymptotics  RiemannHilbert problem  nonlinear steepest descent  
Persistent Link to This Record:  http://purl.flvc.org/ucf/fd/CFE0007602  
Restrictions on Access:  public 20190815  
Host Institution:  UCF 