# On multiply monotone functions

## TL;DR

This paper investigates the algebraic properties of multiply monotone functions, providing conditions under which certain functions can be represented as Fourier transforms, with implications for analysis on $
_+$.

## Contribution

It introduces new algebraic insights into multiply monotone functions and establishes criteria for Fourier transform representation of specific functions.

## Key findings

- Algebraic structure of differences of multiply monotone functions analyzed.
- Sufficient conditions for Fourier transform representation of certain functions derived.
- Criteria for functions of the form $f_0(|x|_{p,d})$ to be Fourier transformable established.

## Abstract

In this paper, the algebra of the differences of two multiply monotone functions on $\mathbb{R}_+=(0,+\infty)$ is studied. A sufficient condition for the function $f_0\big(|x|_{p,d}\big)$, where $|x|_{p,d}=\Big(\sum\limits_{j=1}^d|x_j|^p\Big)^{\frac{1}{p}}$, $p\in(0,+\infty]$, to be represented as the Fourier transform is given.