Error bounds for spectral enhancement which are based on variable Hilbert scale inequalities

TL;DR

This paper develops error bounds for spectral enhancement techniques using variable Hilbert scale inequalities, addressing the limitations of standard convergence theory and demonstrating the approach with practical examples.

Contribution

It introduces a novel method for deriving error bounds in spectral enhancement using variable Hilbert scales, applicable when standard source conditions are not met.

Findings

Error bounds of order $O(\, ext{data error}^{1- ext{small function}})$ are derived.

The method is demonstrated with the Eddington correction formula.

Applied to Voigt spectra, the bounds involve a logarithmic term, $ ext{O}(1/\sqrt{| extlog ext{error}|})$.

Abstract

Spectral enhancement -- which aims to undo spectral broadening -- leads to integral equations which are ill-posed and require special regularisation techniques for their solution. Even when an optimal regularisation technique is used, however, the errors in the solution -- which originate in data approximation errors -- can be substantial and it is important to have good bounds for these errors in order to select appropriate enhancement methods. A discussion of the causes and nature of broadening provides regularity or source conditions which are required to obtain bounds for the regularised solution of the spectral enhancement problem. The source conditions do only in special cases satisfy the requirements of the standard convergence theory for ill-posed problems. Instead we have to use variable Hilbert scales and their interpolation inequalities to get error bounds. The error bounds in this case turn out to be of the form $O(\epsilon^{1-\eta(\epsilon)})$ where $\epsilon$ is the data error and $\eta(\epsilon)$ is a function which tends to zero when $\epsilon$ tends to zero. The approach is demonstrated with the Eddington correction formula and applied to a new spectral reconstruction technique for Voigt spectra. In this case $\eta(\epsilon)=O(1/\sqrt{|\log\epsilon|})$ is found.