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

Abstract

Abstract This paper analyzes the inverse problem of deautoconvolution in the multi-dimensional case with respect to solution uniqueness and ill-posedness. Deautoconvolution 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 Mikusiński, 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.