Holomorphic approximation of radial weights on the complex plane

Evgeny Abakumov, Evgueni Doubtsov

Published 2015-10-06Version 1

Let $w$ be a radial weight on the complex plane. We study the following approximation problem: find entire functions $f_1, f_2, \dots, f_n$ such that the sum $|f_1| + |f_2|+\cdots + |f_n|$ is equivalent to $w$. We give several characterizations of those $w$ for which the problem is solvable. In particular, the following natural objects and properties are involved: essential weights on the complex plane, approximation by power series with positive coefficients, approximation by the maximum of a holomorphic function modulus.