Nuttall's integral equation and Bernshtein's asymptotic formula for a complex weight

摘要

We obtain Nuttall's integral equation provided that the corresponding complex-valued function sigma(x) does not vanish and belongs to the Dini-Lipschitz class. Using this equation, we obtain a complex analogue of Bernshtein's classical asymptotic formulae for polynomials orthogonal on the closed unit interval Delta = [-1, 1] with respect to a complex-valued weight h(x) = sigma (x)/root 1-x(2).