The asymptotic behavior of orthogonal polynomials

V. M. Badkov

Sbornik. Mathematics, Tome 37 (1980) no. 1, pp. 39-51

Résumé

Let $\{\varphi_{\sigma,n}(z)\}_{n=0}^\infty$ be the system of polynomials orthonormal on the unit circumference with respect to the measure $\sigma$. By way of generalizing and strengthening a number of previous results, we show that if $\ln\sigma'(\theta)\in L^1[0,2\pi]$, $\sigma'(\theta)$ continuous and positive on $[a,b]\subset[0,2\pi]$, and $\omega(\sigma';\tau)_{[a,b]}\tau^{-1}\in L^1[0,b-a]$, then the polynomials $\varphi_{\sigma,n}^*(e^{i\theta})=e^{in\theta}\overline{\varphi_{\sigma,n}(e^{i\theta})}$ converge uniformly in $\theta$, inside $(a,b)$, to the Szegö function. The result so formulated is shown to be definitive. Bibligraphy: 16 titles.