Inversion formulae for the -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 | References | Similar Articles | Additional Information

Abstract: In this paper we develop formulae for inverting the so-called -weighted Hilbert transform , 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) . We also find the null-space and the range of in with . Similarly to the FHT, the null-space turns out to be one-dimensional in for any and trivial - for . We prove that is a Fredholm operator of index when it acts between the spaces, , . Finally, in the case where we find the range condition for , which is similar to that for the FHT . 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