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Zbl 0944.42014
Kriecherbauer, T.; McLaughlin, K.T.-R.
Strong asymptotics of polynomials orthogonal with respect to Freud weights. (English)
[J] Int. Math. Res. Not. 1999, No.6, 299-333 (1999). ISSN 1073-7928; ISSN 1687-0247/e

In this important and interesting paper, the authors again illustrate the power of the Deift-Zhou steepest descent method associated with Riemann-Hilbert problems. Moreover, there is a new feature: they show how the technique may be applied, with suitable modifications, even in the presence of singularities. Let $\alpha>0$ and $$w(x):= \exp(-|x|^\alpha),\quad \alpha>0,\ x\in\bbfR.$$ Then we may define orthonormal polynomials $$p_n(x)= \gamma_n x^n+\cdots,\quad \gamma_n> 0,$$ satisfying $$\int p_n p_m w= \delta_{mn},\quad m,n\ge 0.$$ The authors obtain very precise (Plancherel-Rotach) asymptotics for the orthonormal polynomials in all regions of the plane, asymptotic relations for $\gamma_n$ with error terms, the zeros of $p_n$, the spacing between successive zeros, and so on.\par There are two remarkable features in this particular work: firstly, they can handle not just $\alpha$ an even positive integer, as was the case in their earlier papers, but also non-integer $\alpha$; secondly, the case $\alpha\le 1$ is particularly difficult and defied efforts at asymptotics using other methods, such as Bernstein-Szegö techniques. So this is one case, where Riemann-Hilbert techniques not only yield more precise results, but also work for weights where other methods failed to give any asymptotics at all.
[D.S.Lubinsky (Wits)]

MSC 2000:: *42C05 General theory of orthogonal functions and polynomials

Keywords: Freud weights; Plancherel-Rotach asymptotics; Deift-Zhou steepest descent method; orthonormal polynomials

Cited in: Zbl 1157.42007

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