Critical Points and Error Rank in Best $H_2$ Matrix Rational Approximation of Fixed McMillan Degree
Résumé
This paper deals with best rational approximation of prescribed McMillan degree to matrix-valued functions in the real Hardy space of the complement of the unit disk endowed with the Frobenius $L_2$-norm. We describe the topological structure of the set of approximants in terms of inner--unstable factorizations. This allows us to establish a two--sided tangential interpolation equation for the critical points of the criterion, and to prove that the rank of the error $F-H$ is at most $k-n$ when $F$ is rational of degree $k$ and $H$ is critical of degree $n$. In the particular case where $k=n$, it follows that $H=F$ is the unique critical point, and this entails a local uniqueness result when approximating near--rational functions.