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

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 $s_{\alpha }=L(x^{\alpha })$.