The asymptotic representation at a point of the derivative of orthonormal polynomials

B. L. Golinskii

Matematičeskie zametki, Tome 19 (1976) no. 5, pp. 659-672

Résumé

A theorem is proved on the asymptotic representation at the pointe $e^{i\theta_0}$ of the first derivative of polynomials, orthonormal on the unit circumference, under the following conditions: the weight $\varphi(\theta)$ is bounded from above, the function $\varphi^{-2}(\theta)$ is summable on the segment $[-\pi,\pi]$; at the $\eta_0$ neighborhood of the point $\theta=\theta_0$ the weight is bounded from below by a positive constant and has a bounded variation; the trigonometric conjugate $\widetilde{\ln\varphi(\theta_0)}$ exists. These restrictions are less restrictive than those in Ch. Hörup's similar theorem.