On equivalent slowly varying functions

Let $0<t_0<t_1<\dots, \lim\limits_{n\rightarrow +\infty}t_n = +\infty, \sup_n(t{n+1}-t_n )<+\infty.$ For an upward convex slowly varying function $L(t)>0$ an equivalent slowly varying function $L_1(t)$ has been constructed that is convex, infinitely differentiable, and that coincides with $L(t)>0$ on a beforehand given numerical sequence $\{t_n\}$.