  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.
