A general discrepancy theorem (Q1320529)

The discrepancy \(D [\sigma]\) of a signed measure \(\sigma\) on a Jordan arc (curve) \(E\) is defined by \(D [\sigma] : = \sup | \sigma (J) |\), the supremum being taken over all subarcs \(J\) of \(E\). If \(\{\nu_ n\}\) is a sequence of Borel measures on \(E\) the discrepancy \(D [\nu_ n - \mu]\) serves as a measurement on the weak-star convergence of \(\{\nu_ n\}\) to a Borel measure \(\mu\). Main result: Let \(E\) be a Jordan arc (curve) of class \(C^{1+}\). Let \(p_ n\) be a monic polynomial of degree \(n\) with zeros \(z_ i\) in \(E\), \(1 \leq i \leq n\), such that (a) \(\max_{z \in E} | p_ n (z) | \leq A_ n (\text{cap} E)^ n\), (b) \(| p_ n' (z_ i) | \geq B_ n^{-1} (\text{cap} E)^ n\), (c) \(C_ n : = \max (A_ n, B_ n,n) \leq e^{n/e}\). Finally let \(\nu_ n\) denote the measure which associates the mass \(1/n\) with each of the zeros \(z_ i\). Then \[ D [\nu_ n - \mu_ E] \leq c {\log C_ n \over n} \log {n \over \log C_ n}, \] where \(c\) is a positive constant depending only on \(E\) and \(\mu_ E\) is the equilibrium measure of \(E\). The main result is applied to get estimates of the distribution of Fekete points, extreme points of polynomials of best approximation and zeros of orthogonal polynomials on the unit circle and on compact intervals.