Approximation and Interpolation of Singular Measures by Trigonometric Polynomials

Complex valued measures of finite total variation are a powerful signal model in many applications. Restricting to the d-dimensional torus, finitely supported measures can be exactly recovered from their trigonometric moments up to some order if this order is large enough. Here, we consider the approximation of general measures, e.g., supported on a curve, by trigonometric polynomials of fixed degree with respect to the 1-Wasserstein 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 indicator function on the support of the measure and to converge to zero outside.