Finite-term relations for planar orthogonal polynomials
Date
2007ISSN
1661-8254Source
Complex Analysis and Operator TheoryVolume
1Issue
3Pages
447-456Google Scholar check
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We prove by elementary means that, if the Bergman orthogonal polynomials of a bounded simply-connected planar domain, with sufficiently regular boundary, satisfy a finite-term relation, then the domain is algebraic and characterized by the fact that Dirichlet's problem with boundary polynomial data has a polynomial solution. This, and an additional compactness assumption, is known to imply that the domain is an ellipse. In particular, we show that if the Bergman orthogonal polynomials satisfy a three-term relation then the domain is an ellipse. This completes an inquiry started forty years ago by Peter Duren. © 2007 Birkhäuser Verlag Basel/Switzerland.