### 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 2006 |

### Fingerprint

### Keywords

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

### ASJC Scopus subject areas

- Mathematics(all)
- Analysis

### Cite this

*Constructive Approximation*,

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

**Szego orthogonal polynomials with respect to an analytic weight : Canonical representation and strong asymptotics.** / Martínez-Finkelshtein, A.; Mclaughlin, Kenneth D T; Saff, E. B.

Research output: Contribution to journal › Article

*Constructive Approximation*, vol. 24, no. 3, pp. 319-363. https://doi.org/10.1007/s00365-005-0617-6

}

TY - JOUR

T1 - Szego orthogonal polynomials with respect to an analytic weight

T2 - Canonical representation and strong asymptotics

AU - Martínez-Finkelshtein, A.

AU - Mclaughlin, Kenneth D T

AU - Saff, E. B.

PY - 2006/11

Y1 - 2006/11

N2 - 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.

AB - 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.

KW - Cauchy transform

KW - Orthogonal polynomials

KW - Scattering function

KW - Uniform asymptotics

KW - Unit circle

KW - Verblunsky coefficients

UR - http://www.scopus.com/inward/record.url?scp=33747637122&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33747637122&partnerID=8YFLogxK

U2 - 10.1007/s00365-005-0617-6

DO - 10.1007/s00365-005-0617-6

M3 - Article

VL - 24

SP - 319

EP - 363

JO - Constructive Approximation

JF - Constructive Approximation

SN - 0176-4276

IS - 3

ER -