An indeterminate rational moment problem and Carathéodory functions


Adhemar Bultheel       Pablo González-Vera       Erik Hendriksen       Olav Njåstad


Abstract: Let B0 = 1, and Bn be the finite Blaschke products with zeros α1,..., αn, for n = 1,2,... and £ is the span of B0, B1, B2, ... then we consider the following moment problem:
Given a positive definite Hermitian inner product ⟨ . , . ⟩ in £, find a positive Borel measure μ on [-π,π) such that

⟨ f,g ⟩ = ∫-ππ f(eiθ)[g(eiθ)]* dμ(&theta),   f,g ∈ £.

We assume that this moment problem is indeterminate. Under some additional conditions on the &alphan, we will describe a one-to-one correspondence between the collection of all solutions to this moment problem and the collection of all Carathéodory functions augmented by the constant ∞.

Status:
Published on line July 11, 2008

BiBTeX entry: 

   @article{ArtBGHN06a,
      author = "A. Bultheel and P. Gonz{\'a}lez-Vera and E. Hendriksen and O. Nj{\aa}stad",
      journal = "Journal of Computational and Applied Mathematics",
      pages = "359-369",
      title = "An indeterminate rational moment problem and {C}arath{\'e}odory functions",
      volume = "219",
      number = "2",
      year = "2008",
      url = "http://www.cs.kuleuven.be/~nalag/papers/ade/IRMP/index.html",
      note = "Published on line 11-July-2008",
      DOI = "10.1016/j.cam.2007.05.002",
      ZBL = "05309301",
      MR = "2441231",
      LIRIAS = "201862",
   }
File(s): preprint.pdf (177K)

Adhemar Bultheel <Adhemar.Bultheel at cs.kuleuven.be>