Spectral analysis of the truncated Hilbert transform with overlap

We study a restriction of the Hilbert transform as an operator $H_T$ from $L^2(a_2,a_4)$ to $L^2(a_1,a_3)$ for real numbers $a_1 < a_2 < a_3 < a_4$. The operator $H_T$ 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 $[a_2,a_4]$ from its Hilbert transform measured on intervals $[a_1,a_3]$ that might only overlap, but not cover $[a_2,a_4]$. We show that the inversion of $H_T$ is ill-posed, which is why we investigate the spectral properties of $H_T$. We relate the operator $H_T$ 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 $H_T$, which then implies that the spectrum of $H_T^* H_T$ is discrete. Furthermore, we express the singular value decomposition of $H_T$ in terms of the solutions to the Sturm-Liouville problem. The singular values of $H_T$ accumulate at both $0$ and $1$, implying that $H_T$ is not a compact operator. We conclude by illustrating the properties obtained for $H_T$ numerically.