Abstract
AbstractIn this paper, we investigate algebraic, differential and asymptotic properties of polynomials $p_n(x)$ that are orthogonal with respect to the complex oscillatory weight $w(x)=\mathrm {e}^{\mathrm {i}\omega x}$ on the interval $[-1,1]$, where $\omega>0$. We also investigate related quantities such as Hankel determinants and recurrence coefficients. We prove existence of the polynomials $p_{2n}(x)$ for all values of $\omega \in \mathbb {R}$, as well as degeneracy of $p_{2n+1}(x)$ at certain values of $\omega $ (called kissing points). We obtain detailed asymptotic information as $\omega \to \infty $, using recent theory of multivariate highly oscillatory integrals, and we complete the analysis with the study of complex zeros of Hankel determinants, using the large $\omega $ asymptotics obtained before.
Highlights
Introduction and motivationIn this paper we are concerned with orthogonal polynomials with respect to the complex weight function w(z) = eiωz on the interval [−1, 1]
An essential part of the problem is the location of the specific curve, among all possible smooth deformations of the original contour, that attracts the zeros of the orthogonal polynomials as the degree gets large; this leads to the crucial concept of S curve, which is distinguished by a precise symmetry property that involves both the logarithmic potential of equilibrium measures and the external field V (x)
The paper is organised as follows: in Section 2 we present identities for the Hankel determinants, the recurrence coefficients and the kissing polynomials that hold for finite n and ω; such results belong to the integrable systems approach to orthogonal polynomials, which is relevant since the weight function for the kissing polynomials is an exponential deformation of the classical Legendre weight
Summary
This framework was later expanded for complex-valued weight functions and complex OPs in the works of Stahl [43], Rakhmanov [42], Martınez–Finkelshtein and Rakhmanov [38], and Kuijlaars and Silva [34], among others In this scenario, an essential part of the problem is the location of the specific curve, among all possible smooth deformations of the original contour, that attracts the zeros of the orthogonal polynomials as the degree gets large; this leads to the crucial concept of S curve, which is distinguished by a precise symmetry property that involves both the logarithmic potential of equilibrium measures and the external field V (x). Recent extensions of this methodology include multiple orthogonal polynomials as well, that are connected to Hermite–Padeapproximation
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