Generators and interpolation in algebras of entire functions (Q5947109)

Let \(R_p\) be the algebra of functions of infinite type with respect to \(p\): NEWLINE\[NEWLINE R_p:= \Biggl\{f\in H(\mathbb{C}) \biggl|\limsup\limits_{r\to\infty}\frac{\log\log M(f,r)}{\log p(r)} \leq 1\Biggr\} . NEWLINE\]NEWLINE Here \(p\) is a subharmonic function satisfying certain technical conditions, but which allows \(f\) to be an entire function of infinite order. Let \(V=\{z_k,m_k\}\) be a multiplicity variety, where the \(z_k\) are distinct points of \(\mathbb{C}\) with \(|z_k|\to\infty\), and \(m_k\) are positive integers. NEWLINENEWLINENEWLINETranslating a theorem of Berenstein and Taylor to one concerning the algebras \(R_p\) and using an auxiliary theorem, the author obtains:NEWLINENEWLINENEWLINETheorem. Let \(f\in R_p\), \(V=V(f)\). If NEWLINE\[NEWLINE \limsup\limits_{k\to\infty}\frac{\log\log\left(\frac{|f^{(m_k)}(z_k)|}{m_k!} \right)^{-1}} {\log p(z_k)}\leq 1 NEWLINE\]NEWLINE then for every sequence \(a_{k,j}\) satisfying NEWLINE\[NEWLINE \limsup\limits_{k\to\infty}\frac{\log\log\max_j|a_{k,j}|} {\log p(z_k)}\leq 1 NEWLINE\]NEWLINE there is a function \(g\in R_p\) such that NEWLINE\[NEWLINE \frac{g^{(j)}(z_k)}{j!}= a_{k,j},\quad 0\leq j\leq m_k,\quad k=1,2,\dotsNEWLINE\]