  In this paper, we study the approximate minimization problem of weighted
finite automata (WFAs): to compute the best possible approximation of a WFA
given a bound on the number of states. By reformulating the problem in terms of
Hankel matrices, we leverage classical results on the approximation of Hankel
operators, namely the celebrated Adamyan-Arov-Krein (AAK) theory.
  We solve the optimal spectral-norm approximate minimization problem for
irredundant WFAs with real weights, defined over a one-letter alphabet. We
present a theoretical analysis based on AAK theory, and bounds on the quality
of the approximation in the spectral norm and $\ell^2$ norm. Moreover, we
provide a closed-form solution, and an algorithm, to compute the optimal
approximation of a given size in polynomial time.
