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

A. Martínez-Finkelshtein, Kenneth D T Mclaughlin, E. B. Saff

Research output: Contribution to journalArticle

18 Citations (Scopus)

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 languageEnglish (US)
Article number91426
JournalInternational Mathematics Research Notices
Volume2006
DOIs
StatePublished - 2006

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Unit circle
Orthogonal Polynomials
Circle
Toeplitz Determinant
Singularity
Distribution of Zeros
Zeros of Polynomials
Steepest Descent
Ring or annulus
Argand diagram
Hilbert
Valid
Alternatives
Coefficient
Form

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

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.

In: International Mathematics Research Notices, Vol. 2006, 91426, 2006.

Research output: Contribution to journalArticle

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