Complete monotonicity and bernstein properties of functions are characterized by their restriction on N

TL;DR

This paper characterizes completely monotone and Bernstein functions through their behavior on non-negative integers, providing new algebraic and restriction-based criteria that extend classical moment theorems.

Contribution

It introduces novel characterizations of these functions using elementary algebraic mappings and integer restrictions, addressing a key question about their properties from discrete data.

Findings

Complete characterization of monotonicity via integer restrictions

Extension of Hausdorff's moment theorem to functions

New algebraic criteria for Bernstein functions

Abstract

We give several new characterizations of completely monotone functions and Bernstein functions via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the behavior of their restriction on the set of non-negative integers. We give a complete answer to the following question: Can we affirm that a function is completely monotone (resp. a Bernstein function) if we know that the sequence formed by its restriction on the integers is completely monotone (resp. alternating)? This approach constitutes a kind of converse of Hausdorff's moment characterization theorem in the context of completely monotone sequences.