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 $2\times2$ matrix functions. It turns out that ``generically'' $°\AΦ\le\degΦ-2$. We prove that for an arbitrary $2\times2$ rational function $Φ$, $°\AΦ\le2\degΦ-3$ whenever $\degΦ\ge2$. On the other hand, for $k\ge2$, we construct a $2\times2$ matrix function $Φ$, for which $\degΦ=k$, while $°\AΦ=2k-3$. Moreover, we conduct a detailed analysis of the situation when the inequality $°\AΦ\le\degΦ-2$ can violate and obtain best possible results.