On rational functions orthogonal to all powers of a given rational function on a curve

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Abstract

In this paper we study the generating function f(t) for the sequence of the moments, where ∫γPi(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, in the case where the functions ∫S1hi(z)gj(z)gl(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 ∫S1Li(z)m(z)dz=0, i≥i0.

Original languageEnglish
Pages (from-to)693-731
Number of pages39
JournalMoscow Mathematical Journal
Volume13
Issue number4
DOIs
StatePublished - 1 Jan 2013

Keywords

  • Abel equation
  • Cauchy type integrals
  • Center problem
  • Compositions
  • Moment problem
  • Periodic orbits

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