### Abstract

We provide a representation in terms of certain canonical functions for a sequence of polynomials orthogonal with respect to a weight that is strictly positive and analytic on the unit circle. These formulas yield a complete asymptotic expansion for these polynomials, valid uniformly in the whole complex plane. As a consequence, we obtain some results about the distribution of zeros of these polynomials. The main technique is the steepest descent analysis of Deift and Zhou, based on the matrix Riemann-Hilbert characterization proposed by Fokas, Its, and Kitaev.

Original language | English (US) |
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Pages (from-to) | 319-363 |

Number of pages | 45 |

Journal | Constructive Approximation |

Volume | 24 |

Issue number | 3 |

DOIs | |

State | Published - Nov 1 2006 |

### Keywords

- Cauchy transform
- Orthogonal polynomials
- Scattering function
- Uniform asymptotics
- Unit circle
- Verblunsky coefficients

### ASJC Scopus subject areas

- Analysis
- Mathematics(all)
- Computational Mathematics

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## Cite this

Martínez-Finkelshtein, A., McLaughlin, K. T. R., & Saff, E. B. (2006). Szego orthogonal polynomials with respect to an analytic weight: Canonical representation and strong asymptotics.

*Constructive Approximation*,*24*(3), 319-363. https://doi.org/10.1007/s00365-005-0617-6