The Hausdorf moment problem (HMP) over the unit interval in an $L^2$-setting is a classical example of an ill-posed inverse problem. Since various applications can be rewritten in terms of the HMP, it has gathered significant attention in the literature. From the point of view of regularization it is of special interest because of the occurrence of a non-compact forward operator with non-closed range. Consequently, HMP constitutes one of few examples of a linear ill-posed problem of type~I in the sense of Nashed. In this paper we highlight this property and its consequences, for example, the existence of a infinite-dimensional subspace of stability. On the other hand, we show conditional stability estimates for the HMP in Sobolev spaces that indicate severe ill-posedness for the full recovery of a function from its moments, because H\"{o}lder-type stability can be excluded. However, the associated recovery %of the linear functional that characterizes of the function value at the rightmost point of the unit interval is stable of H\"{o}lder-type in an $H^1$-setting. We moreover discuss stability estimates for the truncated HMP, where the forward operator becomes compact. Some numerical case studies illustrate the theoretical results and complete the paper.
翻译:以 $L $2$ 设定单位间隔的Hausdorf 时点问题( HMP) 是一个典型的错误问题。 由于各种应用程序都可以用 HMP 重写, 它在文献中引起极大关注。 从正规化的角度来看, 它特别值得注意, 因为出现了一个非契约的前方操作器, 且不封闭范围。 因此, HMP 是纳希德意义上的 ~ I 型线性错误问题的少数例子之一。 在本文中, 我们突出这一属性及其后果, 例如, 存在一个无限的稳定性子空间。 另一方面, 我们展示了Sobolev 空间中 HMP 的有条件稳定性估计, 这表明它从时间上完全恢复功能存在严重错误, 因为 H\ { o}lder 类型稳定可以排除。 但是, 在单位间隔最右端的函数值的线性功能恢复% 。 例如, 我们强调该属性及其后果的无限维维度子空间的存在。 另一方面, 我们展示了在 $H Q $ 1 格式的模型分析结果。