Weighted $L^{2}$ approximation of entire functions

Abstract:

Let $S$ be the space of entire functions $f(z)$ such that $||f(z)|{|^2} = \smallint \smallint |f(z){|^2}dm(z)$, where $m$ is a positive measure defined on the Borel sets of the complex plane. Write $dm(z) = K(z)d{A_z} = K(r,\theta )dAz$. Theorem 1. If $\ln {\inf _\theta }K(r,\theta )$ is asymptotic to $\ln {\sup _\theta } K(r,\theta )$ (together with other mild restrictions) then polynomials are dense in $S$. Theorem 2. Let $K(z) = {e^{ - \phi (z)}}$ where $\phi (z)$ is a convex function of $z$ such that all exponentials belong to $S$. Then polynomials are dense in $S$.