English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

As in the paper [G. Groenewald, M.A. Kaashoek, A.C.M. Ran, Wiener-Hopfindices of unitary functions on the unit circle in terms of realizations andrelated results on Toeplitz operators. \emph{Indag. Math.} 28 (2017) 694--710]our aim is to obtain explicitly the Wiener-Hopf indices of a rational $m\timesm$ matrix function $R(z)$ that has no poles and no zeros on the unit circle$\mathbb{T}$ but, in contrast with that paper, the function $R(z)$ is notrequired to be unitary on the unit circle. On the other hand, using aDouglas-Shapiro-Shields type of factorization, we show that $R(z)$ factors as$R(z)=\Xi(z)\Psi(z)$, where $\Xi(z)$ and $\Psi(z)$ are rational $m\times m$matrix functions, $\Xi(z)$ is unitary on the unit circle and $\Psi(z)$ is aninvertible outer function. Furthermore, the fact that $\Xi(z)$ is unitary onthe unit circle allows us to factor as $\Xi(z) =V(z)W^*(z)$ where $V(z)$ and$W(z)$ are rational bi-inner $m\times m$ matrix functions. The latter allows usto solve the Wiener-Hopf indices problem. To derive explicit formulas for thefunctions $V(z)$ and $W(z)$ requires additional realization properties of thefunction $\Xi(z)$ which are given in the last two sections.

Wiener-Hopf factorization indices of rational matrix functions with respect to the unit circle in terms of realization