The Riemann-Hilbert approach to strong asymptotics for orthogonal polynomials on [-1,1]

By Arno B. J. Kuijlaars, K. Mclaughlin, Walter Van Assche, Maarten Vanlessen. 2001

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Abstract We consider polynomials that are orthogonal on [−1,1] with respect to a modified Jacobi weight (1− x ) α (1+ x ) β h ( 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 Szegő 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.

Subfields: Algebra and Number Theory, Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/e679390eb77ce3991a45252563906e1ca8620523

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In Algebra and Number Theory

In Applied Mathematics

Citations in the corpus

Padé approximants for functions with branch points — strong asymptotics of Nuttall–Stahl polynomials (2011, methodology): “In this case the conclusion of Theorem 4 is contained in [38, 5, 32, 12].”
Asymptotics of Toeplitz, Hankel, and Toeplitz+Hankel determinants with Fisher-Hartwig singularities (2009, background): “ found by Basor and Ehrhardt in [6]. They use relations between Hankel and Toeplitz determinants which are less general than (1.40) but do not involve polynomials. For some other related results, see [26, 33]. Our ﬁnal task is to present asymptotics for the so-called Toeplitz+Hankel determinants. We”
Critical Measures, Quadratic Differentials, and Weak Limits of Zeros of Stieltjes Polynomials (2009, background): “[4], [24], [47], [49], [50], [64]; this is necessarily a very partial list).”
The Steepest Descent Method for Orthogonal Polynomials on the Real Line with Varying Weights (2008, methodology): “ use of Riemann-Hilbert techniques.) The Riemann-Hilbert method has been extended, and applied to various types of asymptotic questions in approximation theory (see, for example, [12], [15], [1], and [13]). With the exception of [12] and [15], all of these applications and extensions deal with weigh”
Asymptotics of Orthogonal Polynomials for a Weight with a Jump on [−1,1] (2009, methodology): “where α, β > −1 and h(x) is real analytic and strictly positive on [−1, 1], and Ξc is a step-like function, equal to 1 on [−1, 0) and c 2 > 0o n[0, 1]. Observe that w1 is the weight considered in [ 14 ].”