ON THE STABILITY OF THE MAXIMUM TERM OF FUNCTIONAL SERIES IN A SYSTEM OF FUNCTIONS - Bodnarchuk, A. Yu.; Skaskiv, O. B. (PRECARPATHIAN BULLETIN OF THE SHEVCHENKO SCIENTIFIC SOCIETY. Number, 2025)
By L_+ we denote the class of positive continuous on R_+:=[0,+∞) functions l(t) such that l(t) +∞ (t→ +∞), and by W the class of functions wL_+ such that olimits_1^+∞x^-2w(x)dx<+∞. The article deals with the functional series of the form F(x)=olimits_k=0^+∞a_kf(xλ_k), where Λ=(λ_k) is some sequence of non-negative numbers, a_k≥ 0 (k≥ 0), and a positive increase to +∞ on [0;+∞) function f is such that f(0)=1 and f(x) is a convex on the interval [0;+∞) function. Let us denote F_w(x)=olimits_k=0^+∞a_ke^w(λ_k)f(xλ_k), u_0(t)= u{u≥ 0f(u)≤ t},u(G)=olimits_λ_nGe^w(λ_n) for every bounded set GR_+, where wL_+. The main result the paper is the following statement: If there exists a function wL_+ such that a_ne^w(λ_n)f(λ_nx)→ 0, for every x>0, u_0W, then there exists a set E R_+ of finite Lebesque measure such that the asymptotic relation μ(x,F)=(1+o(1))μ(x,F_w) holds as x→ +∞ outside the set E, where μ(x,F)={a_kf(xλ_k)k≥ 0}.