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arXiv · 1909.08870

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

Marco Bertola·Elliot Blackstone·Alexander Katsevich·Alexander Tovbis
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Abstract

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

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Marco Bertola, Elliot Blackstone, Alexander Katsevich, Alexander Tovbis. 2019-09-19. Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann-Hilbert approach. https://arxiv.org/abs/1909.08870

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