On certain non-unique solutions of the Stieltjes moment problem
We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form rho((r))(1)(n) = (2rn)! and rho((r))(2)(n) = [(rn)!](2), r = 1, 2, ... , n = 0, 1, 2, ... , i.e. we find functions W-1,2((r))(x) > 0 satisfying integral(infinity)(0) x(n)W(1,2)((r))(x)dx = rho((r))(1,2)(n). It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Gut, Pakes, Stoyanov) that for r > 1 both rho((r))(1,2)(n) give rise to non-unique solutions. Examples of such solutions are constructed using the technique of the inverse Mellin transform supplemented by a Mellin convolution. We outline a general method of generating non-unique solutions for moment problems generalizing rho((r))(1,2)(n), such as the product rho((r))(1)(n) . rho((r))(2)(n) and [(rn)!](p), p = 3, 4, ....
stieltjes moment problem mellin transform classical moment problem