Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms [Formula: see text] and [Formula: see text] . These operators arise when one studies the interior problem of tomography. The diagonalization of [Formula: see text] has been previ...

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Autores principales: Bertola, Marco, Blackstone, Elliot, Katsevich, Alexander, Tovbis, Alexander
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Springer International Publishing 2020
Materias:
Article
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7357778/
https://www.ncbi.nlm.nih.gov/pubmed/32684912
http://dx.doi.org/10.1007/s13324-020-00371-6
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author Bertola, Marco
Blackstone, Elliot
Katsevich, Alexander
Tovbis, Alexander
author_facet Bertola, Marco
Blackstone, Elliot
Katsevich, Alexander
Tovbis, Alexander
author_sort Bertola, Marco
collection PubMed
description In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms [Formula: see text] and [Formula: see text] . These operators arise when one studies the interior problem of tomography. The diagonalization of [Formula: see text] has been previously obtained, but only asymptotically when [Formula: see text] . We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes [Formula: see text] explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.
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spelling pubmed-73577782020-07-16 Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach Bertola, Marco Blackstone, Elliot Katsevich, Alexander Tovbis, Alexander Anal Math Phys Article In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms [Formula: see text] and [Formula: see text] . These operators arise when one studies the interior problem of tomography. The diagonalization of [Formula: see text] has been previously obtained, but only asymptotically when [Formula: see text] . We implement a novel approach based on the method of matrix Riemann–Hilbert problems (RHP) which diagonalizes [Formula: see text] explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates. Springer International Publishing 2020-06-10 2020 /pmc/articles/PMC7357778/ /pubmed/32684912 http://dx.doi.org/10.1007/s13324-020-00371-6 Text en © The Author(s) 2020 Open AccessThis article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
spellingShingle Article
Bertola, Marco
Blackstone, Elliot
Katsevich, Alexander
Tovbis, Alexander
Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach
title Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach
title_full Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach
title_fullStr Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach
title_full_unstemmed Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach
title_short Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann–Hilbert approach
title_sort diagonalization of the finite hilbert transform on two adjacent intervals: the riemann–hilbert approach
topic Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC7357778/
https://www.ncbi.nlm.nih.gov/pubmed/32684912
http://dx.doi.org/10.1007/s13324-020-00371-6
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