Spaces of series in system of functions

Abstract

The Banach and Fr\'{e}chet spaces of series $A(z)=\sum_{n=1}^{\infty}a_nf(\lambda_nz)$ regularly converging in ${\mathbb C}$,where $f$ is an entire transcendental function and $(\lambda_n)$ is a sequence of positive numbers increasing to $+\infty$, are studied.Let $M_f(r)=\max\{|f(z)|:\,|z|=r\}$, $\Gamma_f(r)=\frac{d\ln\,M_f(r)}{d\ln\,r}$, $h$ be positive continuous function on $[0,+\infty)$increasing to $+\infty$ and ${\bf S}_h(f,\Lambda)$ be a class of the function $A$ such that $|a_n|M_f(\lambda_nh(\lambda_n))$ $\to 0$ as$n\to+\infty$. Define $\|A\|_h=\max\{|a_n|M_f(\lambda_nh(\lambda_n)):n\ge 1\}$. It is proved that if$\ln\,n=o(\Gamma_f(\lambda_n))$ as $n\to\infty$ then $({\bf S}_h(f,\Lambda),\|\cdot\|_h)$ is a non-uniformly convexBanach space which is also separable.In terms of generalized orders, the relationship between the growth of $\mathfrak{M}(r,A)=\break=\sum_{n=1}^{\infty} |a_n|M_f(r\lambda_n)$,the maximal term $\mu(r,A)= \max\{|a_n|M_f(r\lambda_n)\colon n\ge 1\}$ and the central index$\nu(r,A)= \max\{n\ge 1\colon |a_n|M_f(r\lambda_n)=\mu(r,A)\}$ and the decrease of the coefficients $a_n$.The results obtained are used to construct Fr\'{e}chet spaces of series in systems of functions.