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