Some aspects of the finite moment problem

摘要

The approximate solution of the finite moment problem μk = ∫01xk−1ƒ(x) dx, k = 1, 2, 3, …, is considered. This problem is related to the problem of finding a best polynomial least squares approximation to a given function ƒ(x) in [0,1]. The connection with Laplace transform inversion is emphasized, and a set of special square matrices with integral elements is introduced, which has an intimate relation to the above two problems. These matrices are the well-known inverses of finite segments of the infinite Hilbert matrix.