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Moscow Mathematical Journal

Volume 13, Issue 4, October–December 2013  pp. 693–731.

On Rational Functions Orthogonal to All Powers of a Given Rational Function on a Curve

Authors F. Pakovich (1)
Author institution: (1) Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, Israel

In this paper we study the generating function f(t) for the sequence of the moments ∫γ Pi(z)q(z)dz, i≥0, where P(z),q(z) are rational functions of one complex variable and γ is a curve in 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 γ. As an application, we prove a stronger form of the Wermer theorem, describing analytic functions satisfying the system of equations ∫S1 h>i(z)gj(z)g′(z) dz=0, i≥0, j≥0, 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 ∫S1 Li(z)m(z)dz=0, ii0.

2010 Mathematics Subject Classification. Primary: 30E99; Secondary: 34C99

Keywords:  Moment problem, center problem, Abel equation, periodic orbits, Cauchy type integrals, compositions

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