Absolutely Continuous Representing Measures of Moment Sequences with an Emphasis on the Regularity of the Densities

TL;DR

This paper studies the conditions under which linear functionals have absolutely continuous measures with densities, focusing on how the moments influence the regularity of these densities.

Contribution

It characterizes the relationship between moments and the regularity of densities of absolutely continuous representing measures.

Findings

Established criteria linking moments to density regularity.

Provided conditions ensuring the existence of absolutely continuous measures.

Analyzed the influence of moments on the smoothness of densities.

Abstract

In this work we investigate and characterize linear functionals $L:\mathbb{R}[x_1,\dots,x_n]\to\mathbb{R}$ with absolutely continuous representing measures $\mu$, i.e., $\mathrm{d}\mu(x) = g(x)\,\mathrm{d} x$ for some density $g$. We focus on the regularity of $g$ and how it is influenced/determined by the moments $s_\alpha = L(x^\alpha)$.