Abstract

We develop formulae for inverting the so-called cosh-weighted Hilbert transform H_μ, which arises in Single Photon Emission Computed Tomography (SPECT). The formulae are theoretically exact, require the minimal amount of data, and are similar to the classical inversion formulae for the finite Hilbert transform (FHT) H = H_0. We also find the null-space and the range of H_μ in L_p with p > 1. Similarly to the FHT , the null-space turns out to be one-dimensional in L^p for any p in (1,2), and trivial for p ≥ 2. We prove that H_μ is a Fredholm operator when it acts between the L_p spaces, p in (1,∞), p not equal to 2. Finally, in the case p = 2 we find the range condition for H_μ, which is similar to that for the FHT H _0. Our work is based on the method of Riemann-Hilbert problem. This is joint work with M. Bertola and A. Katsevich, accepted in Proceedings of the AMS