Growth Behavior and Zero Distribution of Rational Approximants
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- Blatt, HP. & Kovacheva, R.K. Constr Approx (2011) 34: 393. doi:10.1007/s00365-010-9124-5
We investigate the growth and the distribution of zeros of rational uniform approximations with numerator degree ≤n and denominator degree ≤mn for meromorphic functions f on a compact set E of ℂ where mn=o(n/log n) as n→∞. We obtain a Jentzsch–Szegő type result, i.e., the zero distribution converges weakly to the equilibrium distribution of the maximal Green domain Eρ(f) of meromorphy of f if f has a singularity of multivalued character on the boundary of Eρ(f). 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.