We present new results on the singular value decomposition (SVD) of the truncated Hilbert transform (THT). The THT problem consists in recovering a function f(x) with support on an interval [a2, a4] from the knowledge of its Hilbert transform over an interval [a1, a3] which overlaps with the support of f, i.e. a1 < a2 < a3 < a4. This problem has applications in 2D and 3D tomography for the reconstruction of a region of interest using the differential back-projection. Recent work by Al-Aifari and Katsevich demonstrates that the spectrum of the singular values of the THT has two accumulation points in 0 and in 1. For the interior problem, Katsevich and Tovbis have given a characterization of the asymptotic behaviour of the singular values. Building on these results, we derive here the asymptotic behaviour of the singular values of the THT close to 1 and close to 0, and show that the two limits are connected by a simple coordinate transformation. A comparison with the SVD of a discretized version of the problem shows that the asymptotic expressions for the singular values and singular functions are already accurate for small indices.
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