### Abstract

Strong asymptotics of orthogonal polynomials on the unit circle with respect to a weight of the form W(z)=w(z) Π k=1 m | z-ak | 2βk, |z|=1, | ak |=1, βk >-1/2, k=1,...,m, where w(z)>0 for |z|=1 can be extended as a holomorphic and nonvanishing function to an annulus containing the unit circle. The formulas obtained are valid uniformly in the whole complex plane. As a consequence, we obtain some results about the distribution of zeros of these polynomials, the behavior of their leading and Verblunsky coefficients, and we give an alternative proof of the Fisher-Hartwig conjecture about the asymptotics of Toeplitz determinants for such type of weights. 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) |
---|---|

Article number | 91426 |

Journal | International Mathematics Research Notices |

Volume | 2006 |

DOIs | |

State | Published - 2006 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*International Mathematics Research Notices*,

*2006*, [91426]. https://doi.org/10.1155/IMRN/2006/91426

**Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle.** / Martínez-Finkelshtein, A.; Mclaughlin, Kenneth D T; Saff, E. B.

Research output: Contribution to journal › Article

*International Mathematics Research Notices*, vol. 2006, 91426. https://doi.org/10.1155/IMRN/2006/91426

}

TY - JOUR

T1 - Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle

AU - Martínez-Finkelshtein, A.

AU - Mclaughlin, Kenneth D T

AU - Saff, E. B.

PY - 2006

Y1 - 2006

N2 - Strong asymptotics of orthogonal polynomials on the unit circle with respect to a weight of the form W(z)=w(z) Π k=1 m | z-ak | 2βk, |z|=1, | ak |=1, βk >-1/2, k=1,...,m, where w(z)>0 for |z|=1 can be extended as a holomorphic and nonvanishing function to an annulus containing the unit circle. The formulas obtained are valid uniformly in the whole complex plane. As a consequence, we obtain some results about the distribution of zeros of these polynomials, the behavior of their leading and Verblunsky coefficients, and we give an alternative proof of the Fisher-Hartwig conjecture about the asymptotics of Toeplitz determinants for such type of weights. 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 - Strong asymptotics of orthogonal polynomials on the unit circle with respect to a weight of the form W(z)=w(z) Π k=1 m | z-ak | 2βk, |z|=1, | ak |=1, βk >-1/2, k=1,...,m, where w(z)>0 for |z|=1 can be extended as a holomorphic and nonvanishing function to an annulus containing the unit circle. The formulas obtained are valid uniformly in the whole complex plane. As a consequence, we obtain some results about the distribution of zeros of these polynomials, the behavior of their leading and Verblunsky coefficients, and we give an alternative proof of the Fisher-Hartwig conjecture about the asymptotics of Toeplitz determinants for such type of weights. 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.

UR - http://www.scopus.com/inward/record.url?scp=33749525489&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749525489&partnerID=8YFLogxK

U2 - 10.1155/IMRN/2006/91426

DO - 10.1155/IMRN/2006/91426

M3 - Article

AN - SCOPUS:33749525489

VL - 2006

JO - International Mathematics Research Notices

JF - International Mathematics Research Notices

SN - 1073-7928

M1 - 91426

ER -