Open questions asked to analysis and numerics concerning the Hausdorff moment problem

TL;DR

This paper investigates the ill-posedness of the Hausdorff moment problem, analyzing the decay of singular values of the associated operator and discussing discrepancies between numerical results and theoretical smoothness properties.

Contribution

It clarifies the apparent contradiction between numerical exponential decay of singular values and the limited smoothness of the kernel, exploring the problem's ill-posedness nature.

Findings

Numerical computations show exponential decay of singular values.

The kernel of the operator has limited smoothness.

Discussion on the potential polynomial decay of singular values.

Abstract

We address facts and open questions concerning the degree of ill-posedness of the composite Hausdorff moment problem aimed at the recovery of a function $x \in L^2(0,1)$ from elements of the infinite dimensional sequence space $\ell^2$ that characterize moments applied to the antiderivative of $x$. This degree, unknown by now, results from the decay rate of the singular values of the associated compact forward operator $A$, which is the composition of the compact simple integration operator mapping in $L^2(0,1)$ and the non-compact Hausdorff moment operator $B^{(H)}$ mapping from $L^2(0,1)$ to $\ell^2$. There is a seeming contradiction between (a) numerical computations, which show (even for large $n$) an exponential decay of the singular values for $n$-dimensional matrices obtained by discretizing the operator $A$, and \linebreak (b) a strongly limited smoothness of the well-known kernel $k$ of the Hilbert-Schmidt operator $A^*A$. Fact (a) suggests severe ill-posedness of the infinite dimensional Hausdorff moment problem, whereas fact (b) lets us expect the opposite, because exponential ill-posedness occurs in common just for $C^\infty$-kernels $k$. We recall arguments for the possible occurrence of a polynomial decay of the singular values of $A$, even if the numerics seems to be against it, and discuss some issues in the numerical approximation of non-compact operators.