Spectral Analysis of the Truncated Hilbert Transform with Overlap

We study a restriction of the Hilbert transform as an operator HT from L (a2, a4) to L(a1, a3) for real numbers a1 < a2 < a3 < a4. The operator HT arises in tomographic reconstruction from limited data, more precisely in the method of differentiated back-projection (DBP). There, the reconstruction requires recovering a family of one-dimensional functions f supported on compact intervals [a2, a4] from its Hilbert transform measured on intervals [a1, a3] that might only overlap, but not cover [a2, a4]. We show that the inversion of HT is ill-posed, which is why we investigate the spectral properties of HT . We relate the operator HT to a self-adjoint two-interval Sturm-Liouville problem, for which we prove that the spectrum is discrete. The Sturm-Liouville operator is found to commute with HT , which then implies that the spectrum of H∗ THT is discrete. Furthermore, we express the singular value decomposition of HT in terms of the solutions to the Sturm-Liouville problem. The singular values of HT accumulate at both 0 and 1, implying that HT is not a compact operator. We conclude by illustrating the properties obtained for HT numerically.