Open questions asked to analysis and numerics concerning the Hausdorff moment problem
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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 ∈ L^2(0,1) from elements of the infinite dimensional sequence space ℓ^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 ℓ^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 (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^∞-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.
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