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

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 $A^{\ast }A$. 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.