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.
Categories: math.CA