Strong asymptotics of Jacobi-type kissing polynomials

Abstract

We investigate asymptotic behaviour of polynomials $p_n^{\omega }\left(z\right)$ satisfying varying non-Hermitian orthogonality relations ${\int }_{-1}^1{x}^k{p}_n^{\omega }\left(x\right)h\left(x\right){\mathrm{e}}^{\mathrm{i}\omega x}\mathrm{d}x=0,k\in \{0,\dots ,n-1\},$where $h\left(x\right)=h^{\ast }\left(x\right)(1-x{)}^{\alpha }(1+x{)}^{\text{β}},\omega =\lambda n,\lambda \ge 0$ and $h\left(x\right)$ is holomorphic and non-vanishing in a certain neighbourhood in the plane. These polynomials are an extension of so-called kissing polynomials ($\alpha =\text{β}=0$) introduced in Asheim et al. [A Gaussian quadrature rule for oscillatory integrals on a bounded interval. Preprint, 2012 Dec 6. arXiv:1212.1293] in connection with complex Gaussian quadrature rules with uniform good properties in ω. The analysis carried out here is an extension of what was done in Celsus and Silva [Supercritical regime for the kissing polynomials. J Approx Theory. 2020 Mar 18;225:Article ID: 105408]; Deaño [Large degree asymptotics of orthogonal polynomials with respect to an oscillatory weight on a bounded interval. J Approx Theory. 2014 Oct 1;186:33–63], and depends heavily on those works.

Keywords:

• Non-Hermitian orthogonality
• varying orthogonality
• Riemann–Hilbert analysis

2010 Mathematics Subject Classification:

• 42C05

Acknowledgments

The author is grateful to Maxim Yattselev for his guidance and the many useful discussions, suggestions, and comments. The author would also like to thank Alfredo Deaño and Guilherme Silva for their help, support and encouragement.

Disclosure statement

No potential conflict of interest was reported by the author(s).