Title

Inversion Formulae For The Cosh-Weighted Hilbert Transform

Abstract

In this paper we develop formulae for inverting the so-called coshweighted Hilbert transform Hμ, 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) H0. We also find the null-space and the range of H0μ in Lp with p > 1. Similarly to the FHT, the null-space turns out to be one-dimensional in Lp for any p ∈ (1, 2) and trivial - for p ≥ 2. We prove that Hμ is a Fredholm operator of index -1 when it acts between the Lp spaces, p ∈ (1,∞), p ≠= 2. Finally, in the case where p = 2 we find the range condition for H0μ, which is similar to that for the FHT H0. Our work is based on the method of the Riemann-Hilbert problem. © 2012 American Mathematical Society.

Publication Date

8-1-2013

Publication Title

Proceedings of the American Mathematical Society

Volume

141

Issue

8

Number of Pages

2703-2718

Document Type

Article

Personal Identifier

scopus

DOI Link

https://doi.org/10.1090/S0002-9939-2013-11642-4

Socpus ID

84878176525 (Scopus)

Source API URL

https://api.elsevier.com/content/abstract/scopus_id/84878176525

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