Universal distribution of limit points. (English) Zbl 1299.30011
Summary: We consider sequences of functions that have in some sense a universal distribution of limit points of zeros in the complex plane. In particular, we prove that functions having universal approximation properties on compact sets with connected complement automatically have such a universal distribution of limit points. Moreover, in the case of sequences of derivatives, we show connections between this kind of universality and some rather old results of Edrei/MacLane and Pólya. Finally, we show the lineability of the set what we call Jentzsch-universal power series.
MSC:
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
| 30K99 | Universal holomorphic functions of one complex variable |
| 30B10 | Power series (including lacunary series) in one complex variable |
| 30K05 | Universal Taylor series in one complex variable |
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