#### Title

INVERSION FORMULAE FOR THE cosh-WEIGHTED HILBERT TRANSFORM

#### Abbreviated Journal Title

Proc. Amer. Math. Soc.

#### Keywords

CONE-BEAM CT; IMAGE-RECONSTRUCTION; Mathematics, Applied; Mathematics

#### 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. (1, 2) and trivial - for p > = 2. We prove that H-mu is a Fredholm operator of index - 1 when it acts between the L-p spaces, p is an element of (1, infinity), p not equal 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.

#### Journal Title

Proceedings of the American Mathematical Society

#### Volume

141

#### Issue/Number

8

#### Publication Date

1-1-2013

#### Document Type

Article

#### Language

English

#### First Page

2703

#### Last Page

2718

#### WOS Identifier

#### ISSN

0002-9939

#### Recommended Citation

"INVERSION FORMULAE FOR THE cosh-WEIGHTED HILBERT TRANSFORM" (2013). *Faculty Bibliography 2010s*. 3701.

https://stars.library.ucf.edu/facultybib2010/3701

## Comments

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