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

Philippe Rambour

Published 2013-10-17, updated 2014-06-13Version 2

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.