An indeterminate rational moment problem and Carathéodory functions
An indeterminate rational moment problem and Carathéodory functions
Adhemar Bultheel
Pablo González-Vera
Erik Hendriksen
Olav Njåstad
Abstract:
Let $B_0 = 1$, and $B_n$ be the finite Blaschke products
with zeros $\alpha_1,..., \alpha_n$, for $n = 1,2,...$
and $\mathcal{L}$ is the span of $B_0,B_1,B_2,\ldots$
then we consider the following
moment problem:
Given a positive definite Hermitian inner product
$\langle\cdot,\cdot\rangle$ in $\mathcal{L}$
find a positive Borel measure $\mu$ on $[-\pi,\pi)$ such that
$$
\langle f,g \rangle = \int_{-\pi}^\pi f(e^{i\theta})\overline{g(e^{i\theta})} d\mu(\theta),\quad f,g\in\mathcal{L}.
$$
We assume that this moment problem is indeterminate.
Under some additional conditions on the $\alpha_n$, 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 $\infty$.
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://nalag.cs.kuleuven.be/papers/ade/IRMP/index.html",
DOI = "10.1016/j.cam.2007.05.002",
ZBL = "1149.30001",
MR = "2441231",
LIMO = "1128213",
}
File(s):
preprint.pdf (177K)