Using the regularization theory for improperly posed problems, we discuss object restoration beyond the diffraction limit in the presence of noise. Only the case of one-dimensional coherent objects is considered. We focus attention n the estimation of the error on the restored objects, and we show that, in most realistic cases, it is at best proportional to an inverse power of |In epsilon|, where epsilon is the error on the data (logarithmic continuity). Finally we suggest the extension of this result to other inverse problems.