Summary: Kolmogorov proposed the following problem: Given a system of integers \(0=k_0<k_1<\dots<k_j,\) find necessary and sufficient conditions on the system of positive numbers \({M_0,M_1,\dots,M_j}\) for the existence of a function \(f\in L_{\infty}(I), I=\mathbb R\) or \(I=\mathbb R_+\) such that \(\|f^{k_i}\|=M_i, i=0,1,\dots,j (\|f\| := \text{vrai sup} |f(t)|)\).
This problem was solved by Kolmogorov for \(I=\mathbb R\) and \(j=2\). Dzyadyk and Dubovik solved Kolmogorov’s problem for \(I=\mathbb R\) and the system of numbers \({M_0,M_k,M_{r-1},M_r}\). For \(j=2\) and \(I=\mathbb R_+\), necessary and sufficient conditions were found by Schoenberg and Cavaretta.
We solve Kolmogorov’s problem on the half-line for the system of numbers \({M_0,M_k,M_{r-1},M_r}\).