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Abstract:
Abstract. 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 to Walsh's theory of exact harmonic majorants [8, 9]. We also prove results on the convergence of ray sequences of Laurent-type approximants to a function analytic on the closure of a finitely 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]. Key words. Laurent-type rational functions, zero distributions, convergence, optimal ray sequences, best Lp approximants. AMS subject classifications. 30E10, 30C15, 41A20, 31A15. 1. Majorization and zero distribution of Laurent-type rational functions. Let A be a bounded multiply connected domain whose boundary consists of a finite number of disjoint Jordan curves. We denote by C the extended complex plane, by {G
Citations
| 101 | Interpolation and Approximation by Rational Functions – Walsh - 1969 |
| 57 | Potential Theory in Modern Function Theory", Chelsea Publ – Tsuji - 1950 |
| 24 | Analytic Functions – Nevanlinna - 1970 |
| 2 | Ostrowski gaps, overconvergence and zeros of polynomials, in Approximation Theory – Grothmann - 1989 |
| 2 | On representation of analytical functions by power series – Ostrowski - 1926 |
| 2 | Asymptotic zero distribution of Laurent-type rational functions – Papamichael, Pritsker, et al. - 1995 |
| 2 | The analogue for maximally convergent polynomials of Jentzsch's theorem – Walsh - 1959 |
| 1 | Overconvergence, degree of convergence, and zeros of sequences of analytic functions – Walsh - 1946 |
