Analytic approximation of rational matrix functions

Abstract

For a rational matrix function $\Phi$ 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 $2\times 2$ matrix functions. It turns out that ``generically'' $°\AΦ\le\degΦ-2$. We prove that for an arbitrary $2\times 2$ rational function $\Phi$, $°\AΦ\le2\degΦ-3$ whenever $\mathrm{deg}\Phi \ge 2$. On the other hand, for $k\ge 2$, we construct a $2\times 2$ matrix function $\Phi$, for which $\mathrm{deg}\Phi =k$, 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.