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Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann-Hilbert approach
Marco Bertola1,2, Elliot Blackstone3, Alexander Katsevich4
1Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec H3G 1M8 Canada.
Analysis and Mathematical Physics
|July 21, 2020
Summary
This study introduces a new matrix Riemann-Hilbert problem (RHP) method to explicitly diagonalize bounded self-adjoint operators relevant to tomography. This approach overcomes previous asymptotic limitations, providing exact solutions and error estimates.
Area of Science:
- Mathematical Physics
- Tomography
- Operator Theory
Background:
- The interior problem of tomography involves bounded self-adjoint linear operators.
- Previous work achieved only asymptotic diagonalization of these operators for large parameters.
- Exact diagonalization is crucial for precise analysis and applications.
Purpose of the Study:
- To explicitly diagonalize bounded self-adjoint linear operators related to finite Hilbert transforms.
- To develop a novel method for solving the interior problem of tomography.
- To provide exact solutions and error estimates for the studied operators.
Main Methods:
- Utilizing the method of matrix Riemann-Hilbert problems (RHP).
- Developing a novel approach for operator diagonalization.
- Analyzing the asymptotics of related RHP solutions.
Main Results:
- Explicit diagonalization of the bounded self-adjoint linear operators.
- Asymptotic analysis of solutions to a related RHP.
- Derivation of error estimates for the obtained solutions.
Conclusions:
- The matrix RHP method provides an effective tool for explicit operator diagonalization.
- This work advances the understanding and solution of the interior problem in tomography.
- The findings offer precise analytical tools for related mathematical and physical problems.
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