Abstract
Let \(w\) be an unbounded radial weight on the complex plane. We study the following approximation problem: find a proper holomorphic map \(f: \mathbb {C}\rightarrow \mathbb {C}^n\) such that |f| is equivalent to \(w\). We give several characterizations of those \(w\) for which the problem is solvable. In particular, a constructive characterization is given in terms of tropical power series. Moreover, the following natural objects and properties are involved: essential weights on the complex plane, approximation by power series with positive coefficients, and approximation by the maximum of a holomorphic function modulus. Extensions to several complex variables and approximation by harmonic maps are also considered.
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The authors are grateful to José Bonet and Jari Taskinen for useful discussions.
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Communicated by Doron S. Lubinsky.
Evgueni Doubtsov was supported by the Russian Science Foundation (Grant No. 14-41-00010).
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Abakumov, E., Doubtsov, E. Approximation by Proper Holomorphic Maps and Tropical Power Series. Constr Approx 47, 321–338 (2018). https://doi.org/10.1007/s00365-017-9375-5
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DOI: https://doi.org/10.1007/s00365-017-9375-5
Keywords
- Entire function
- Radial weight
- Proper holomorphic map
- Tropical power series
- Essential weight
- Log-convex function