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

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

2006-05-29

arxiv.org/abs/math/0605715v1

Mathematics

Classical Analysis and ODEs

36 pages, 9 figures

Scientific paper

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.

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