Small entire functions with extremely fast growth
Generalizing his earlier results [ibid. 201, No. 3, 1002-1009 (1996; Zbl 0855.30030)] the author produces a linear space \(M\) of entire functions dense in the space of all entire functions such that for every \(f\in M\), \(\exp(|z|^\alpha)f^{(j)} (z)\to 0\) for \(z\to\infty\) in \(S\) and all \(j\in\mathbb{N}_0\), \(\int_\ell f^{(j)}ds=0\) for all \(j\in\mathbb{N}_0\) and all straight lines \(\ell\), while \(f\) is `very large' (infinite growth index) globally. Here \(S\) is any sector with tip at 0, or any parallel strip, and \(\alpha\) is any number in \((0,\frac{1}{2})\). The main tool is a theorem by Arakelian on approximation with speed of a function \(g\in A(F)\) by entire functions, if \(F\) is an Arakelian set.
scientific article; zbMATH DE number 1005695