Some discrepancy theorems in the theory of weighted polynomial approximation (Q1269663)

The authors generalize discrepancy results in the theory of distribution of zeros of extremal polynomials appearing in the theory of weighted polynomial approximation on the real line. It is a very technical paper, making it practically impossible to give the main results explicitly in a review. They are concerned with bounds \[ \left| \nu(P)(I)-\lambda^{\ast}_n(I)\right| \leq c {\log C_n\over n}\log\left({n\over \log C_n}\right) \] and \[ \left| \nu({\mathcal T}_{n,p,Q})(I)-\lambda^{\ast}_n(I)\right| \leq c{(\log n)^2\over n}. \] Here \(\nu\) can be seen as the counting measure and \(\lambda^{\ast}\) plays the role of the equilibrium measure. The function \(Q\) satisfies several technical conditions and the polynomials \({\mathcal T}_{n,p,Q}\) (\(1<p\leq \infty)\) are a type of Chebyshev polynomials with respect to a weight function \(W_Q\).