Deautoconvolution in the two-dimensional case

There is extensive mathematical literature on the inverse problem of deautoconvolution for a function with support in unit interval $[0,1] \subset \mathbb{R}$, but little known about multidimensional situation. This article tries to fill this gap analytical and numerical studies reconstruction real two variables over square from observations its autoconvolution $[0,2]^2 \mathbb{R}^2$ (full data case) or $[0,1]^2$ (limited case). In an $L^2$-setting, twofoldness uniqueness assertions are proven 2D. Moreover, ill-posedness characterized illustrated. Extensive numericalcase give overview behaviour stable approximate solutions two-dimensional obtained by Tikhonov-type regularization different penalties iteratively regularized Gauss–Newton method.