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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