Finite-Term Relations for Planar Orthogonal Polynomials

Abstract.

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.

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Correspondence to Mihai Putinar.

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To Peter Duren on the occasion of his seventieth birthday

The first author was partially supported by the National Science Foundation Grant DMS- 0350911.

Received: October 15, 2006. Revised: January 22, 2007.

Communicated by Daniel Alpay.

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Putinar, M., Stylianopoulos, N.S. Finite-Term Relations for Planar Orthogonal Polynomials. Complex anal.oper. theory 1, 447–456 (2007). https://doi.org/10.1007/s11785-007-0013-2

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Mathematics Subject Classification (2000).

  • Primary 30C10
  • Secondary 47B32, 30C40, 31A25

Keywords.

  • Orthogonal polynomials
  • algebraic curves
  • Dirichlet’s problem
  • Bergman space