If CLEAN converged, the Clean component list is a plausible solution of the measurement equation (within the noise), but it is not unique... Hence, because of convolution by the clean beam, the clean image is not a solution. However, besides allowing a reasonable definition of the image unit in case of incomplete convergence, there are two reasons to convolve by a clean beam. First, convolution by the clean beam smears out artifacts due to extrapolation beyond the measured area of the uv plane. This is an a posteriori regularization. Second, the clean components are forced to reside on the grid defined by the image. This discrete representation has a number of limitations (e.g. necessity of negative clean components, limited accuracy due to the finite size of the component list), which are reduced by convolution by the clean beam, because the clean image then has finite resolution and can be properly represented on a discrete grid provided the Nyquist sampling is preserved.
An important property of CLEAN is that (to first order) only the inner quarter of the dirty image can be properly cleaned. This is easily understood when dirty beam and dirty images are computed on the same grid size, since a source at one edge of the inner quarter requires knowledge of the dirty beam sidelobes beyond the map size to be deconvolved from the opposite edge. However, this also remains true if one computes the dirty beam on a twice larger grid than the dirty image: more than the inner quarter can be deconvolved, but because of aliasing, the map edges can never be.
Finally, CLEAN offers a very simple way to impose further constraints on the class of solution which is acceptable, by allowing definition of a support. This can be the standard (simple or multiple) Clean Box available in many non interactive implementations, or a user defined mask in interactive implementations. The search region can even be modified from iteration to iteration to help clean convergence. Such a flexible support is available inside the MAPPING program. Note however that the Clean Box or support should not be too limited: cleaning the noise is necessary too (as well as incorporating negative Clean component).