## K. A. Penson ; Pawel Blasiak ; Gérard Duchamp ; A. Horzela ; A. I. Solomon - On certain non-unique solutions of the Stieltjes moment problem

dmtcs:507 - Discrete Mathematics & Theoretical Computer Science, September 15, 2010, Vol. 12 no. 2 - https://doi.org/10.46298/dmtcs.507
On certain non-unique solutions of the Stieltjes moment problem

Authors: K. A. Penson ; Pawel Blasiak ; Gérard Duchamp ; A. Horzela ; A. I. Solomon

We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ${\rho}_{1}^{(r)}(n)=(2rn)!$ and ${\rho}_{2}^{(r)}(n)=[(rn)!]^{2}$, $r=1,2,\dots$, $n=0,1,2,\dots$, \textit{i.e.} we find functions $W^{(r)}_{1,2}(x)>0$ satisfying $\int_{0}^{\infty}x^{n}W^{(r)}_{1,2}(x)dx = {\rho}_{1,2}^{(r)}(n)$. It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for $r>1$ both ${\rho}_{1,2}^{(r)}(n)$ 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 ${\rho}_{1,2}^{(r)}(n)$, such as the product ${\rho}_{1}^{(r)}(n)\cdot{\rho}_{2}^{(r)}(n)$ and $[(rn)!]^{p}$, $p=3,4,\dots$.

Volume: Vol. 12 no. 2
Published on: September 15, 2010
Imported on: March 26, 2015
Keywords: [MATH.MATH-FA] Mathematics [math]/Functional Analysis [math.FA],[MATH.MATH-CO] Mathematics [math]/Combinatorics [math.CO]