On certain non-unique solutions of the Stieltjes moment problemArticle
Authors: K. A. Penson 1; Pawel Blasiak 2; Gérard Duchamp 3; A. Horzela 2; A. I. Solomon 4
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K. A. Penson;Pawel Blasiak;Gérard Duchamp;A. Horzela;A. I. Solomon
1 Laboratoire de Physique Théorique de la Matière Condensée
2 H. Niewodniczanski Institute of Nuclear Physics
3 Laboratoire d'Informatique de Paris-Nord
4 Department of Physics and Astronomy [Milton Keynes]
We construct explicit solutions of a number of Stieltjes moment problems based on moments of the form ρ(r)1(n)=(2rn)! and ρ(r)2(n)=[(rn)!]2, r=1,2,…, n=0,1,2,…, \textit{i.e.} we find functions W(r)1,2(x)>0 satisfying ∫∞0xnW(r)1,2(x)dx=ρ(r)1,2(n). It is shown using criteria for uniqueness and non-uniqueness (Carleman, Krein, Berg, Pakes, Stoyanov) that for r>1 both ρ(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 ρ(r)1,2(n), such as the product ρ(r)1(n)⋅ρ(r)2(n) and [(rn)!]p, p=3,4,….
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