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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 problemArticle

Authors: K. A. Penson 1; Pawel Blasiak ORCID2; Gérard Duchamp 3; A. Horzela 2; A. I. Solomon 4

  • 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,.


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]
Funding:
    Source : HAL
  • Physique combinatoire; Funder: French National Research Agency (ANR); Code: ANR-08-BLAN-0243

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