Asymptotic Zero Distribution of Laurent-Type Rational Functions

Abstract

We study convergence and asymptotic zero distribution of sequences of rational functions with fixed location of poles that approximate an analytic function in a multiply connected domain. Although the study of zero distributions of polynomials has a long history, analogous results for truncations of Laurent series have been obtained only recently by Edrei (Michigan Math. J.29(1982), 43–57). We obtain extensions of Edrei's results for more general sequences of Laurent-type rational functions. It turns out that the limiting measure describing zero distributions is a linear convex combination of the harmonic measures at the poles of rational functions, which arises as the solution to a minimum weighted energy problem for a special weight. Applications of these results include the asymptotic zero distribution of the best approximants to analytic functions in multiply connected domains, Faber–Laurent polynomials, Laurent–Padé approximants, trigonometric polynomials, etc.