Electronic Transactions on Numerical Analysis

This paper is devoted to the study of asymptotic zero distribution of Laurent-type approximants under certain extremality conditions analogous to the condition of Grothmann 1], which can be traced back t o W alsh's theory of exact harmonic majorants 8, 9]. We a l s o p r o ve results on the convergence of ray sequences of Laurent-type approximants to a function analytic on the closure of a nitely connected Jordan domain and on the zero distribution of optimal ray sequences. Some applications to the convergence and zero distribution of the best Lp approximants are given. The arising theory is similar to Walsh's theory of maximally convergent polynomials to a function in a simply connected domain 10]. 1. Majorization and zero distribution of Laurent-type rational functions. Let A be a bounded multiply connected domain whose boundary consists of a nite number of disjoint Jordan curves. We denote by C the extended complex plane, by fG l g n l=1 the set of bounded components of C n A and by the unbounded component. (It is clear that the G l and are Jordan domains and that CnA = (n l=1 G l).) Finally, for each l = 1 2 : : : n we associate an arbitrary but xed point a l 2 G l. We c o n tinue the study of the convergence and the limiting zero distribution of Laurent-type rationals of the form: R N (z) = k X j=0 t N j z j + n X l=1 ml X j=1 s N llj (z ; a l) ;j (1.1) where the multi-index N := (kkm 1 m 2 : : : m n) which w as started in 4]. A more detailed account on the subject can be found in 5]. In this paper, we shall consider diierent suucient conditions that yield the same type of zero distributions as in 4]. Note that we do not require that t N k 6 = 0 (i n c o n trast with 4]), but only that the highest positive p o wer d e (k) o f z with nonzero coeecient i n R N (z) satisses d e (k) k: Similarly, w e h a ve for the highest degree d l (m l) of the Laurent part of R N (z) associated with the pole a l that d l (m l) m …