ON THE STABILITY OF THE MAXIMUM TERM OF FUNCTIONAL SERIES IN A SYSTEM OF FUNCTIONS

Abstract

By $L_+$ we denote the class of positive continuous on $\mathbb{R}_+:=[0,+\infty)$
functions $l(t)$ such that $l(t) \uparrow +\infty$ $(t\to +\infty)$, and by $\mathcal{W}$ the class of functions $w\in L_+$
such that
$
\int\nolimits_{1}^{+\infty}{x^{-2}{w(x)}dx}<+\infty.
$
The article deals with the functional series of the form $F(x)=\sum\nolimits_{k=0}^{+\infty}a_kf(x\lambda_k),$ where $\Lambda=(\lambda_k)$ is some sequence of non-negative numbers,
$a_k\geq 0$ $(k\geq 0)$, and a positive increase to $+\infty$\ on $[0;+\infty)$\ function $f$ is such that $f(0)=1$\ and $\ln
f(x)$\ is a convex on the interval $[0;+\infty)$\ function. Let us denote
$F_w(x)=\sum\nolimits_{k=0}^{+\infty}a_ke^{w(\lambda_k)}f(x\lambda_k),$
$$
\nu_0(t)=\nu\{u\geq 0\colon \ln f(u)\leq t\},\quad \nu(G)=\sum\nolimits_{\lambda_{n}\in G}{e^{w(\lambda_{n})}}
$$ for every bounded set $G\in\mathbb{R}_+$, where $w\in L_+$. The main result the paper is the following statement:
If
there exists a function $w\in L_+$ such that $a_ne^{w(\lambda_n)}f(\lambda_nx)\to 0,$ for every $x>0$,\ $\ln\nu_0\in \mathcal{W}$,
then there exists a set $E\subset
\mathbb{R}_{+}$ of finite Lebesque measure such that the asymptotic relation
$\ln \mu(x,F)=(1+o(1))\ln \mu(x,F_w)$
holds as $x\to +\infty$ outside the set $E$, where $\mu(x,F)=\max\{a_kf(x\lambda_k)\colon k\geq 0\}$.