  Complex signed measures of finite total variation are a powerful signal model
in many applications. Restricting to the $d$-dimensional torus, finitely
supported measures allow for exact recovery if the trigonometric moments up to
some order are known. Here, we consider the approximation of general measures,
e.g., supported on a curve, by trigonometric polynomials of fixed degree with
respect to the Wasserstein-1 distance. We prove sharp lower bounds for their
best approximation and (almost) matching upper bounds for effectively
computable approximations when the trigonometric moments of the measure are
known. A second class of sum of squares polynomials is shown to interpolate the
characteristic function on the support of the measure and to converge to zero
outside.
