Asymptotic of the terms of the Gegenbauer polynomial on the unit circle and applications to the inverse of Toeplitz matrices

TL;DR

This paper studies orthogonal polynomials on the circle with specific weights and derives asymptotics for the inverse of associated Toeplitz matrices, offering insights into their eigenvalues.

Contribution

It provides new asymptotic formulas for the entries of the inverse Toeplitz matrices related to a class of weights, advancing understanding of their spectral properties.

Findings

Asymptotic behavior of orthogonal polynomials on the circle with given weights

Asymptotic formulas for entries of inverse Toeplitz matrices

Lower bounds on eigenvalues of these matrices

Abstract

The first part of this paper is devoted to the study of the orthogonal polynomial on the circle, with respect of a weight of type $f_\alpha (\theta) = (2\cos \theta- 2\cos \theta_0)^{2\alpha} c_1$ with $\theta_0 \in ]0,\pi[$, -1/2 <\alpha<1/2 and c_1 a sufficiently smooth function. In a second part of the paper we obtain an asymptotic of the entries $(T_N f_\alpha)^{-1}_{k+1,l+1}$ for sufficiently large values of $k,l$, that provides a lower bound on the eigenvalues of this matrix.