Analytic approximation of rational matrix functions

Abstract

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 x 2 matrix functions. It turns out that generically deg AΦ ≤ degΦ - 2. We prove that for an arbitrary 2 x 2 rational function Φ, degAΦ ≤ 2degΦ - 3 whenever degΦ ≥ 2. On the other hand, for k ≥ 2, we construct a 2 x 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.