This article is cited in
8 papers
On rational functions orthogonal to all powers of a given rational function on a curve
F. Pakovich Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, Israel
Abstract:
In this paper we study the generating function
$f(t)$ for the sequence of the moments
$\int_\gamma P^i(z)q(z)\,dz$,
$i\geq0$, where
$P(z),q(z)$ are rational functions of one complex variable and
$\gamma$ is a curve in
$\mathbb C$. We calculate an analytical expression for
$f(t)$ and provide conditions implying that
$f(t)$ is rational or vanishes identically. In particular, for
$P(z)$ in generic position we give an explicit criterion for a function
$q(z)$ to be orthogonal to all powers of
$P(z)$ on
$\gamma$. As an application, we prove a stronger form of the Wermer theorem, describing analytic functions satisfying the system of equations
$\int_{S^1}h^i(z)g^j(z)g'(z)\,dz=0$,
$i\geq0$,
$j\geq0$, in the case where the functions
$h(z),g(z)$ are rational. We also generalize the theorem of Duistermaat and van der Kallen about Laurent polynomials
$L(z)$ whose integer positive powers have no constant term, and prove other results about Laurent polynomials
$L(z),m(z)$ satisfying
$\int_{S^1}L^i(z)m(z)\,dz=0$,
$i\geq i_0$.
Key words and phrases:
moment problem, center problem, Abel equation, periodic orbits, Cauchy type integrals, compositions.
MSC: Primary
30E99; Secondary
34C99 Received: June 15, 2012; in revised form
March 7, 2013
Language: English
DOI:
10.17323/1609-4514-2013-13-4-693-731
Fulltext:
References