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Growth behavior and zero distribution of rational approximants. (English) Zbl 1236.41014
From authors’ abstract: The authors investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree $\le n$ and denominator degree $\le {m}_{n}$ for meromorphic functions $f$ on a compact set $E$ of $ℂ$ where ${m}_{n}=o\left(n/logn\right)$ as $n\to \infty$. They obtain a Jentzsch-Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain ${E}_{\rho \left(f\right)}$ of meromorphy of $f$ if $f$ has a singularity of multivalued character on the boundary of ${E}_{\rho \left(f\right)}$. The paper extends results for polynomial approximation and rational approximation with fixed degree of the denominator. As applications, Padé approximation and real rational best approximants are considered.
##### MSC:
 41A20 Approximation by rational functions 26C15 Rational functions (real variables) 30E10 Approximation in the complex domain 41A21 Padé approximation 41A25 Rate of convergence, degree of approximation
##### References:
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