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

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    Martínez-Finkelshtein-Asymptotics of orthogonal.pdf (631.1Kb)
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    URI: http://hdl.handle.net/10835/1631
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    Author/s
    Martínez-Finkelshtein, Andrei; McLaughlin, K. T.-R.; Saff, E. B.
    Date
    2006
    Abstract
    Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form $$ W(z) = w(z) \prod_{k=1}^m |z-a_k|^{2\beta_k}, \quad |z|=1, \quad |a_k|=1, \quad \beta_k>-1/2, \quad k=1, ..., m, $$ where $w(z)>0$ for $|z|=1$ and can be extended as a holomorphic and non-vanishing 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, as well as 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.
    Palabra/s clave
    Asintótica
    Polinomios ortogonales
    Singularidades algebraicas
    Círculo
    Asymptotics
    Orthogonal polynomials
    Analytic weight
    Algebraic singularities
    Circle
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    • Artículos de revista Dpto. Matemáticas [135]
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