Analytic Approximation of Rational Matrix Functions

For a rational matrix function Φ with poles outside the unit circle, we estimate the degree of the unique superoptimal approximation AΦ by matrix functions analytic in the unit disk. We obtain sharp estimates in the case of 2 × 2 matrix functions. It turns out that "generically" deg AΦ ≤ degΦ – 2. We prove that for an arbitrary 2 × 2 rational function Φ, deg AΦ ≤ 2degΦ – 3 whenever degΦ ≥ 2. On the other hand, for k ≥ 2, we construct a 2 × 2 matrix function Φ, for which degΦ = k, while deg AΦ = 2k–3. Moreover, we conduct a detailed analysis of the situation when the inequality deg AΦ ≤ degΦ–2 can violate and obtain best possible results.