PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 141, Number 8, August 2013, Pages 2703–2718 S 0002-9939(2013)11642-4 Article electronically published on April 4, 2013 INVERSION FORMULAE FOR THE cosh-WEIGHTED HILBERT TRANSFORM M. BERTOLA, A. KATSEVICH, AND A. TOVBIS (Communicated by Walter Craig) Abstract. In this paper we develop formulae for inverting the so-called cosh- weighted Hilbert transform H μ , which arises in Single Photon Emission Com- puted Tomography (SPECT). The formulae are theoretically exact, require a minimal amount of data, and are similar to the classical inversion formulae for the finite Hilbert transform (FHT) 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 ∈ (1, 2) and trivial – for p ≥ 2. We prove that H μ is a Fredholm operator of index −1 when it acts between the L p spaces, p ∈ (1, ∞), p = 2. Finally, in the case where 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 the Riemann-Hilbert problem. 1. Introduction A relationship between the cone-beam transform of a function f and the Hilbert transform of f along lines was found by Gelfand and Graev in [GG91]. In com- bination with a formula for the finite Hilbert transform (FHT) inversion, the two results led to a development of new, accurate, and flexible algorithms for image reconstruction in transmission tomography (CT); for some of the references see [ZP04, NCP04, PNC05, YZYW05]. The key property of FHT inversion is that it can be performed using an efficient convolution-type formula and requires only a minimal amount of data. More precisely, for each line L one needs to compute the Hilbert transform of f from the CT data only for points in I := L ∩ supp f (assuming this intersection is an interval). Since the amount of CT data required is minimal (i.e., any algorithm that uses data on a subinterval of I is severely un- stable), reconstruction algorithms based on FHT inversion have the potential to reduce the x-ray dose to the patients. It was shown recently by Rullgard that in the case of Single Photon Emission Computed Tomography (SPECT) with constant attenuation there is a relation between the attenuated projections of f and a modified Hilbert transform of f along lines [Rul04]. This relation is analogous to the one in transmission CT. Let μ be the constant attenuation coefficient of the medium. It is known that by using simple weighting of the attenuated projections of f , one can compute the Received by the editors October 27, 2011. 2010 Mathematics Subject Classification. Primary 44A12, 44A15. The work of the first author was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC). The work of the second author was supported in part by NSF grants DMS-0806304 and DMS- 1115615. c 2013 American Mathematical Society 2703 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use