Kolmogorov problem on the class of multiply monotone functions

TL;DR

This paper establishes necessary and sufficient conditions for the existence of an (r-1)-monotone function on the negative half-line with prescribed norms of its derivatives at specific orders, advancing the understanding of the Kolmogorov problem.

Contribution

It provides the first complete characterization of the Kolmogorov problem for multiply monotone functions with given derivative norms at four specific orders.

Findings

Derived necessary and sufficient conditions for the problem

Solved the Kolmogorov problem for a new class of functions

Enhanced the theoretical framework for function approximation

Abstract

Necessary and sufficient conditions for positive numbers $M_{k_1}, M_{k_2}, M_{k_3}, M_{k_4}$, $0 = k_1 < k_2<k_2\leq r-2$, $k_4=r$, to guarantee the existence of an $r-1$-monotone function defined on the negative half-line and such that $\|x^{(k_i)}\| = M_{k_i}$, $i=1,2,3,4$ were found.