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Spectral properties of the finite Hilbert transform on two adjacent intervals via the method of Riemann-Hilbert problem

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Date Issued:
2019
Abstract/Description:
In this dissertation, we study a self-adjoint 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 so-called ``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 Riemann-Hilbert 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 Riemann-Hilbert 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 self-adjoint 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 so-called ``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 Riemann-Hilbert 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): 2019-08-01
Ph.D.
Sciences, Mathematics
Doctoral
This record was generated from author submitted information.
Subject(s): interior problem of tomography -- finite Hilbert transform -- Titchmarsh-Weyl theory -- diagonalization -- large parameter asymptotics -- Riemann-Hilbert problem -- nonlinear steepest descent
Persistent Link to This Record: http://purl.flvc.org/ucf/fd/CFE0007602
Restrictions on Access: public 2019-08-15
Host Institution: UCF

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