Absolutely representative exponent systems of minimal type in function spaces with given growth near the boundary (Q1902780)

Let \(F= (f_n)\) be a decreasing (with respect to \(n\)) sequence of non-negative functions on a bounded convex domain \(D\subset \mathbb{C}\), \(0\in D\), and \(H_F(D)\) be the space of holomorphic functions \(f\) on \(D\), that \(\sup_{z\in D} |f(z)|\exp(- f_n(z))<+\infty\) for every \(n\in \mathbb{N}\). A sequence \(E(\Lambda):= (\exp \lambda_n z)\), where \(\Lambda= (\lambda_n)\) are pairwise distinct complex numbers, is called an absolutely representative system in \(H_F(D)\) if any function \(f\in H_F(D)\) is representable as a sum of series \(f(z)= \sum^\infty_{n= 1} c_n \exp \lambda_n z\) which absolutely converges in \(H_F(D)\). The authors study a minimal absolutely representative system in \(H_F(D)\), if \(f_n(z)= (1+ 1/n)\varphi(- \ln d(z))\), where \(d(z)\) is the distance between \(z\in D\) and \(\partial D\), and \(\varphi\) is a non-negative function that \(\varphi''(r)> 0\); \(\varphi'(r)\to + \infty\) for \(r\to + \infty\); \(\varphi'(r) \varphi(r)\to p\in (0, +\infty)\) for \(r\to +\infty\).