In this talk, I will discuss the recovery of an arbitrary measure on the $d$-dimensional torus, given trigonometric moments up to degree $n$. Considering the convolution of the measure with powers of the Fejér kernel, which can be computed efficiently from the truncated moment sequence, I will provide rates of convergence of the resulting polynomial density towards the measure in the $p$-Wasserstein distance, as the degree $n$ increases. In particular, I will show that the best possible rate for polynomial approximation is inversely proportional to the degree, and that it is achieved by adequately choosing the power to which the kernel is raised. Finally, I will discuss another class of polynomial approximations, similar although not based on convolution, that converge pointwise to the characteristic function of the support of the measure. This is joint work with Mathias Hockmann, Stefan Kunis and Markus Wageringel.