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

TL;DR

This paper investigates the uniqueness and ill-posedness of the multi-dimensional deautoconvolution problem, establishing conditions for solution uniqueness and analyzing the problem's stability and regularization aspects.

Contribution

It extends the analysis of deautoconvolution to multiple dimensions, providing new uniqueness results based on the support and data availability.

Findings

Uniqueness in full data case proven using multi-dimensional Titchmarsh theorem.

Uniqueness of non-negative solutions with support at the origin in limited data case.

Discussion of regularization and rate results for solutions.

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 Mikusi\'{n}ski, 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.