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Diagonalization of the finite Hilbert transform on two adjacent intervals: the Riemann-Hilbert approach. (English) Zbl 1452.44004

Summary: In this paper we study the spectra of bounded self-adjoint linear operators that are related to finite Hilbert transforms \(\mathcal{H}_L:L^2([b_L,0])\rightarrow L^2([0,b_R])\) and \(\mathcal{H}_R:L^2([0,b_R])\rightarrow L^2([b_L,0])\). 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.

MSC:

44A15 Special integral transforms (Legendre, Hilbert, etc.)
47A10 Spectrum, resolvent

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References:

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