论文标题
在多维情况下，关于Deautoconvolution问题的独特性和不良性
On uniqueness and ill-posedness for the deautoconvolution problem in the multi-dimensional case
论文作者
论文摘要
本文分析了在多维情况下，关于解决方案的唯一性和不良性的逆向问题的反问题。在这里，Deautoconvolution在这里意味着重建一个实用值的$ l^2 $功能，并从$ n $二维的单位Cube $ [0,1]^n $中的自动卷入率（即在$ [0,2]^n $）中或在有限的数据案例中（即在$ [0,2]^n $中）（即，在有限的数据案例中）（I.E.E. 1,1）。基于由于狮子和Mikusiński引起的Titchmarsh卷积定理的多维变体，我们在完整的数据案例中证明了双重性的主张，并且在有限的数据案例案例案例唯一性中，非阴性溶液的唯一性属于其来源属于支持。在有限的数据案例中，后一个假设对于任何唯一性语句都是必需的。正则解决方案的率结果瞥见了纸张。
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.
