Spectral properties of Hankel matrices and numerical solutions of finite moment problems

After proving that any Hankel matrix generated by moments of positive functions is conditioned essentially the same as the Hilbert matrix of the same size, we show a preconditioning technique, i.e., a congruence transform of the original Hankel matrix that drastically reduces its ill-conditioning. Applications of this result to classical orthogonal polynomial sequences and to modified moment problems are given. Also, we outline an efficient algorithm for the computation of the function f(x) = w(x) exp (p(x)), where w(x) is positive and p(x) is a polynomial of degree n−1, from the knowledge of its first n moments.