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Inversion formulae for the $ \mathrm{\cosh}$-weighted Hilbert transform


Authors: M. Bertola, A. Katsevich and A. Tovbis
Journal: Proc. Amer. Math. Soc. 141 (2013), 2703-2718
MSC (2010): Primary 44A12, 44A15
DOI: https://doi.org/10.1090/S0002-9939-2013-11642-4
Published electronically: April 4, 2013
MathSciNet review: 3056561
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Abstract: In this paper we develop formulae for inverting the so-called $ \cosh $-weighted Hilbert transform $ H_\mu $, which arises in Single Photon Emission Computed 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_\mu $ 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\ge 2$. We prove that $ H_\mu $ is a Fredholm operator of index $ -1$ when it acts between the $ L^p$ spaces, $ p\in (1,\infty )$, $ p\not =2$. Finally, in the case where $ p=2$ we find the range condition for $ H_\mu $, which is similar to that for the FHT $ H_0$. Our work is based on the method of the Riemann-Hilbert problem.


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Additional Information

M. Bertola
Affiliation: Department of Mathematics, Concordia University, Montreal, Canada
Email: bertola@mathstat.concordia.ca

A. Katsevich
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816-1364
Email: Alexander.Katsevich@ucf.edu

A. Tovbis
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816-1364
Email: Alexander.Tovbis@ucf.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11642-4
Received by editor(s): October 27, 2011
Published electronically: April 4, 2013
Additional Notes: 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
Communicated by: Walter Craig
Article copyright: © Copyright 2013 American Mathematical Society