ERDOS-TURAN-TYPE THEOREMS ON PIECEWISE-SMOOTH CURVES AND ARCS

If L is a Jordan curve of a Jordan arc and pn is a monic polynomial of degree n we obtain estimates for the discrepancy between the equilibr ium measure mu(L) of L and the distribution nu(pn) of the zeros of p(n ) based on one-sided bounds for the difference U(mu(L)-nu(pn), z) of t heir logarithmic potentials. These new estimates generalize known resu lts to the case that L is not smooth, i.e., corners of L are allowed, but cusps are not. Moreover, the results are independent of the angles at the corners. The method of proof shows that both situations--upper or lower bounds of U(mu(L)-nu(pn), z)--can be treated simultaneously. As an application, the distribution of Fekete points and extremal poi nts of best uniform approximants can be investigated generalizing resu lts of Kleiner [14] and Blatt and Grothmann [6] to Jordan curves and a rcs with corners. (C) 1997 Academic Press