The Hausdorff Moment Problem in the light of ill-posedness of type I

TL;DR

This paper investigates the ill-posedness of the Hausdorff moment problem, highlighting its classification as a type I ill-posed problem, and discusses stability estimates and numerical illustrations for various cases.

Contribution

It characterizes the Hausdorff moment problem as a type I ill-posed problem and analyzes stability properties, including Sobolev space estimates and the impact of truncation.

Findings

HMP is a linear ill-posed problem of type I with non-compact forward operator.

Full recovery of functions from moments is severely ill-posed, excluding Hölder stability.

Recovery of the function value at the right endpoint is Hölder-stable in H^1-space.

Abstract

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.