We have applied the Pixon Based Image Restoration Method (Puetter, this volume; Piña &Puetter 1993, 1992; Puetter &Piña 1993) to real and simulated HST Spectra. Both the Richardson-Lucy (Richardson 1972, Lucy 1974) and Jansson (1984) algorithms have already been used extensively for spectroscopic applications (Gilliland 1990, Morris et al. 1991). All three methods are nonlinear. An undesirable characteristic of most restoration techniques is the noise amplification, over-resolution and ringing. In spectra, these problems are manifested as narrow features. Ringing is especially bad in the vicinity of emission lines, where the slope of the continuum changes significantly. Since our emphasis is on the restoration of absorption lines, we have worked only with normalized spectra. This approach can be useful when the continuum does not change rapidly over the length of the LSF. The advantage of this technique is that we can impose upper and lower bounds, namely positivity and continuum. We can therefore restrict the algorithm from unnecessarily amplifying the noise in the spectrum. Our selection of the proper continuum will be a factor of error in our analysis and caution has to be used to minimize its effects. In all cases, the inclusion of the two bounds improves the restored spectrum quality significantly, compared to the unconstrained implementation.