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Arno B. J. Kuijlaars. 《Mathematics of Computation》1996,65(213):151-156
It is shown that the zeros of the Faber polynomials generated by a regular -star are located on the -star. This proves a recent conjecture of J. Bartolomeo and M. He. The proof uses the connection between zeros of Faber polynomials and Chebyshev quadrature formulas.
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We study asymptotics for orthogonal polynomials and other extremalpolynomials on infinite discrete sets, typical examples beingthe Meixner polynomials and the Charlier polynomials. Followingideas of Rakhmanov, Dragnev and Saff, weshow that the asymptoticbehaviour is governed by a constrained extremal energy problemfor logarithmic potentials, which can be solved explicitly.We give formulas for the contracted zero distributions, thenth root asymptotics and the asymptotics of the largest zeros.1991 Mathematics Subject Classification: 42C05, 33C25, 31A15. 相似文献
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A. B. J. Kuijlaars 《Constructive Approximation》1996,12(2):287-301
For a weight functionw: [a, b]→(0, ∞), we consider weighted polynomials of the formw n Pn where the degree ofP n is at mostn. The class of functions that can be approximated with such polynomials depends on the behavior of the densityv(t) of the extremal measure associated withw. We show that every approximable function must vanish at the endpointa ifv(t) behaves like (t?a) β ast→a with β>?1/2. We also present an analogous result for internal points. Our results solve some open problems posed by V. Totik and disprove a conjecture of G.G. Lorentz on incomplete polynomials. 相似文献
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We analyze a continuum limit of the finite non-periodic Toda lattice through an associated constrained maximization problem
over spectral density functions. The maximization problem was derived by Deift and McLaughlin using the Lax–Levermore approach,
initially developed for the zero dispersion limit of the KdV equation. It encodes the evolution of the continuum limit for
all times, including evolution through shocks. The formation of gaps in the support of the maximizer is indicative of oscillations
in the Toda lattice and the lack of strong convergence of the continuum limit. For large times, the maximizer tends to have
zero gaps, which is the continuum analogue of the sorting property of the finite lattice. Using methods from logarithmic potential
theory, we show that this behavior depends crucially on the initial data. We exhibit initial data for which the zero gap ansatz
holds uniformly in the spatial parameter (at large times), and other initial data for which this
uniformity fails at all times. We then construct an example of C
∞ smooth initial data generating, at a later time, infinitely many gaps in the support of the maximizer, while for even larger
times, all gaps have closed.
Received: 8 May 2000 / Accepted: 27 March 2001 相似文献
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We study unitary random matrix ensembles in the critical case where the limiting mean eigenvalue density vanishes quadratically at an interior point of the support. We establish universality of the limits of the eigenvalue correlation kernel at such a critical point in a double scaling limit. The limiting kernels are constructed out of functions associated with the second Painlevé equation. This extends a result of Bleher and Its for the special case of a critical quartic potential. The two main tools we use are equilibrium measures and Riemann‐Hilbert problems. In our treatment of equilibrium measures we allow a negative density near the critical point, which enables us to treat all cases simultaneously. The asymptotic analysis of the Riemann‐Hilbert problem is done with the Deift‐Zhou steepest‐descent analysis. For the construction of a local parametrix at the critical point we introduce a modification of the approach of Baik, Deift, and Johansson so that we are able to satisfy the required jump properties exactly. © 2005 Wiley Periodicals, Inc. 相似文献
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We consider n nonintersecting Brownian motion paths with p prescribed starting positions at time t=0 and q prescribed ending positions at time t=1. The positions of the paths at any intermediate time are a determinantal point process, which in the case p=1 is equivalent to the eigenvalue distribution of a random matrix from the Gaussian unitary ensemble with external source.
For general p and q, we show that if a temperature parameter is sufficiently small, then the distribution of the Brownian paths is characterized
in the large n limit by a vector equilibrium problem with an interaction matrix that is based on a bipartite planar graph. Our proof is
based on a steepest descent analysis of an associated (p+q)×(p+q) matrix-valued Riemann–Hilbert problem whose solution is built out of multiple orthogonal polynomials. A new feature of the
steepest descent analysis is a systematic opening of a large number of global lenses. 相似文献
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S. B. Damelin A. B. J. Kuijlaars 《Transactions of the American Mathematical Society》1999,351(11):4561-4584
The support of the equilibrium measure associated with an external field of the form , , with and a positive integer is investigated. It is shown that the support consists of at most two intervals. This resolves a question of Deift, Kriecherbauer and McLaughlin.
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A.B.J. Kuijlaars K.T.-R. McLaughlin W. Van Assche M. Vanlessen 《Advances in Mathematics》2004,188(2):337-398
We consider polynomials that are orthogonal on [−1,1] with respect to a modified Jacobi weight (1−x)α(1+x)βh(x), with α,β>−1 and h real analytic and strictly positive on [−1,1]. We obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval [−1,1], for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials. We also deduce asymptotic behavior for the Hankel determinants and for the monic orthogonal polynomials on the interval [−1,1]. For the asymptotic analysis we use the steepest descent technique for Riemann-Hilbert problems developed by Deift and Zhou, and applied to orthogonal polynomials on the real line by Deift, Kriecherbauer, McLaughlin, Venakides, and Zhou. In the steepest descent method we will use the Szeg? function associated with the weight and for the local analysis around the endpoints ±1 we use Bessel functions of appropriate order, whereas Deift et al. use Airy functions. 相似文献