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

Author(s): Bertola M; Blackstone E; Katsevich A; Tovbis A;

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms H L : L 2 ( [ b L , 0 ] ) → L 2 ( [ 0 , b R ] ) and H R : L 2 ( [ 0 , b R ] ) → L 2 ( [ b L , 0 ] ) . These operato...

Article GUID: 32684912
Title:Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann-Hilbert approach
Authors:Bertola M,  Blackstone E,  Katsevich A,  Tovbis A, 
Link:https://pubmed.ncbi.nlm.nih.gov/32684912/
DOI:10.1007/s13324-020-00371-6
Category:Anal Math Phys
PMID:32684912
Dept Affiliation: MATHSTATS
1 Department of Mathematics and Statistics, Concordia University, 1455 de Maisonneuve W., Montréal, Québec H3G 1M8 Canada.
2 SISSA, International School for Advanced Studies, Via Bonomea 265, Trieste, Italy.
3 Department of Mathematics, KTH Royal Institute of Technology, Lindstedtsvägen 25, 114 28 Stockholm, Sweden.
4 Department of Mathematics, University of Central Florida, P.O. Box 161364, 4000 Central Florida Blvd, Orlando, FL 32816-1364 USA.

Description:

In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms H L : L 2 ( [ b L , 0 ] ) → L 2 ( [ 0 , b R ] ) and H R : L 2 ( [ 0 , b R ] ) → L 2 ( [ b L , 0 ] ) . These operators arise when one studies the interior problem of tomography. The diagonalization of H R , H L has been previously obtained, but only asymptotically when b L ≠ - b R . We implement a novel approach based on the method of matrix Riemann-Hilbert problems (RHP) which diagonalizes H R , H L explicitly. We also find the asymptotics of the solution to a related RHP and obtain error estimates.