  We present a short and elementary proof of the Ajtai-Koml\'os-Tusn\'ady (AKT)
optimal matching theorem in dimension 2 via Fourier analysis and a smoothing
argument. The upper bound applies to more general families of samples,
including dependent variables, of interest in the study of rates of convergence
for empirical measures. Following the recent pde approach by L. Ambrosio, F.
Stra and D. Trevisan, we also adapt a simple proof of the lower bound.
