On uniqueness and ill-posedness for the deautoconvolution problem in the multi-dimensional case

This paper analyzes the inverse problem of deautoconvolution in the multidimensional case with respect to solution uniqueness and ill-posedness. Deauto-convolution means here the reconstruction of a real-valued L-2-function with support in the n-dimensional unit cube [0, 1](n) from observations of its autoconvolution either in the full data case (i.e. on [0, 2](n)) or in the limited data case (i.e. on [0,1](n)). Based on multi-dimensional variants of the Titchmarsh convolution theorem due to Lions and Mikusi'nski, we prove in the full data case a twofoldness assertion, and in the limited data case uniqueness of non-negative solutions for which the origin belongs to the support. The latter assumption is also shown to be necessary for any uniqueness statement in the limited data case. A glimpse of rate results for regularized solutions completes the paper.