Asymptotic zero distribution of Laurent-type rational functions

[摘要] 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 ui polynomials has a long history, analogous results for truncations of Laurent series have bren 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. II 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-Pade approximants, trigonometric polynomials, etc. (C) 1997 Academic Press.

[发布日期] 1997-04-01