Discrepancy Theorems for the Zero Distribution of Rational Functions

Abstract

In the theory of zero distribution of polynomials, there are two branches that are closely related to each other. One branch deals with the conditions under which the normalized zero counting measure of a sequence of polynomials converges in the weak-star sense to the equilibrium measure of a Jordan curve. The other consists of discrepancy theorems, where the maximum deviation of these two measures is estimated. There are also theorems on weak-star convergence of the zero distribution of special sequences of rational functions to a measure that depends on the poles of these functions. In this paper, we give appropriate discrepancy theorems for rational functions. These new results also imply theorems about the weak-star convergence of the zero distribution of more general sequences of rational functions.