Advanced Search

Query:

Fill in the form and click »Search«...

Format:

Display: entries per page entries

Zbl 1149.30001
Bultheel, Adhemar; González-Vera, Pablo; Hendriksen, Erik; Njåstad, Olav
An indeterminate rational moment problem and Carathéodory functions. (English)
[J] J. Comput. Appl. Math. 219, No. 2, 359-369 (2008). ISSN 0377-0427

A moment problem is called indeterminate if it has more than one solution. It is known that if the Hamburger moment problem is indeterminate, then there is a one-to-one correspondence between the collection of all the solutions to this moment problem and the collection of all Nevanlinna functions augmented by the constant $\infty$. The purpose of the present paper is to prove a similar statement for a rational moment problem that arises in the study of certain rational functions with poles outside the closed unit disk in the extended complex plane. Let $\{\alpha_n\}^\infty_{n=1}$ be a sequence of points in the open unit disk in the complex plane and let $$\Bbb B_0=1\quad \text{and}\quad \Bbb B_n(z)=\prod_{k=0}^n\frac{\overline{\alpha_k}}{\vert \alpha_k\vert } \frac{\alpha_k-z}{1-\overline{\alpha_k}z},\quad n=1,2,\dots,$$ ($\overline{\alpha_k}/\vert \alpha_k\vert =-1$ when $\alpha_k = 0$). Put $\Cal L = \text{span}\{\Bbb B_n: n = 0, 1,2, \dots\}$ and consider the following ``moment'' problem: Given a positive-definite Hermitian inner product $\langle\cdot,\cdot\rangle$ in $\Cal L$, find all positive Borel measures $\nu$ on $[-\pi,\pi)$ such that $$\langle f,g\rangle = \int_{-\pi}^\pi f\left(e^{i\theta}\right) \overline{g\left(e^{i\theta}\right)}\, d\nu(\theta)\quad \text{for}\quad f,g\in\Cal L.$$ Assuming that this moment problem is indeterminate, that is $\sum_{n=0}^\infty(1-\vert \alpha_n\vert )<\infty$, under some additional condition on the $\alpha_n$, the authors establish 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 $\infty$.
[Vasily A. Chernecky (Odessa)]

MSC 2000:: *30-06 Proceedings of conferences (functions of a complex variable)
30E05 Moment problems, etc.

Keywords: orthogonal rational functions; positive Borel measures; Riesz-Herglotz transform

Comment on this Item PDF MathML XML ASCII DVI PS BibTeX Online Ordering Article Journal

	Overhang
	Lie groups, physics and geometry. An introduction for physicists, engineers and chemists.
	Elementary number theory. Primes, congruences, and secrets.

Advanced Search

Zentralblatt MATH Berlin [Germany]