The representation of a meromorphic function as the quotient of entire functions and Paley problem in ℂn: survey of some results

Анотація

The classical representation problem for a meromorphic function $f$ in $\mathbb{C}^n$, $n\ge 1$,consists in representing $f$ as the quotient $f=g/h$ of two entire functions $g$ and $h$, each with logarithm of modulus majorized by a function as close as possible to the Nevanlinna characteristic. Here we introduce generalizations of the Nevanlinna characteristic and give a short survey of classical and recent results on the representation of a meromorphic function in terms such characteristics. When $f$ has a finite lower order, the Paley problem on best possible estimates of the growth of entire functions $g$ and $h$ in the representations $f = g/h$ will be considered. Also we point out to some unsolved problems in this area.