Orthogonal polynomials with respect to a class of Fisher-Hartwig symbols and inverse of Toeplitz matrices

Abstract

Using previous results we propose an asymptotic expression of the coefficients of the orthogonal polynomials on the unit circle with respect to a weight of type \(f_{\alpha _{1},\ldots ,\alpha _{M},\, \theta _{1},\ldots ,\theta _{M}}\) defined by \({ \theta \mapsto \prod \nolimits _{1\le j \le M} \vert 1 - e^{i(\theta -\theta _{j})}\vert ^{2\alpha _{j}} c}\) with \(\theta _{j}\in ]-\pi ,\pi ]\), \(\theta _{i}\ne \theta _{j}\), and \(-\frac{1}{2} < \alpha _{j}<\frac{1}{2}\) and c a sufficiently smooth regular function. As a corollary we give an asymptotic expansion of the entries of \(T_{N}^{-1}\big ( f_{\alpha _{1},\ldots ,\alpha _{M},\theta _{1},\ldots ,\theta _{M},\, \theta _{1},\ldots ,\theta _{M}}\big )\) for \(\max ( \alpha _{1},\ldots ,\alpha _{M})\in ]0, \frac{1}{2}[\).