Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle
| dc.creator | Martinez-Finkelshtein, A. | |
| dc.creator | McLaughlin, K. T. -R. | |
| dc.creator | Saff, E. B. | |
| dc.date | 2006-05-29 | |
| dc.date.accessioned | 2026-07-07T07:14:35Z | |
| dc.date.available | 2026-07-07T07:14:35Z | |
| dc.description | Strong asymptotics of polynomials orthogonal on the unit circle with respect to a weight of the form $$ W(z) = w(z) \prod_{k=1}^m |z-a_k|^{2β_k}, \quad |z|=1, \quad |a_k|=1, \quad β_k>-1/2, \quad k=1, ..., m, $$ 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. | |
| dc.description | 36 pages, 9 figures | |
| dc.identifier | https://arxiv.org/abs/math/0605715 | |
| dc.identifier | http://arxiv.org/abs/math/0605715 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/112936 | |
| dc.subject | Classical Analysis and ODEs | |
| dc.subject | Complex Variables | |
| dc.subject | 42C05; 41A60 | |
| dc.title | Asymptotics of orthogonal polynomials with respect to an analytic weight with algebraic singularities on the circle | |
| dc.type | text |