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

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.