On Bornological Spaces of Series in Systems of Functions

Abstract

Let $f$ be an entire transcendental function, $M_f(r) = \max\{| f(z)| : |z| = r\}$, $(\lambda_n)$ be a sequence of positive numbers increasing to $+\infty$ and suppose that the series \[ A(z) = \sum_{n=1}^{\infty} a_n f(\lambda_n z) \] regularly converges in $\mathbb{C}$, i.e., \[ \sum_{n=1}^{\infty} |a_n| M_f(r\lambda_n) < +\infty \quad \text{for all } r \in [0,+\infty). \] Bornology is introduced on a set of such series as a system of functions $f(\lambda_n z)$, and its connection with Fréchet spaces is studied.