Asymptotic Zero Distribution Asymptotic Zero Distribution of Laurent type Rational Functionsy

We study convergence and asymptotic zero distribution of sequences of rational functions with xed 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 A Edrei We obtain extensions of Edrei s results for more general se quences 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 do mains Faber Laurent polynomials Laurent Pad e approximants trigono metric polynomials etc Introduction The limiting behavior of zeros of sequences of polynomials is a classical subject that continues to receive much attention see e g because of its applications in function theory numerical analysis and approximation theory Two of the fundamental results of the subject are the theorems of Jentzsch and Szeg o on the zero distribution of partial sums of a power series It is rather surprising that although the study of zero distributions of power series sections has a long history analogous results for truncations of Laurent series Research supported in part by a University of Cyprus research grant zResearch done in partial ful llment of Ph D degree at the University of South Florida Research supported in part by NSF grant DMS Asymptotic Zero Distribution has been investigated only relatively recently by Edrei who in particular proved the following Theorem A Let A fz r jzj Rg be the exact annulus of convergence for the Laurent series f z X