Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS

   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

Almost periodic factorization of certain
block triangular matrix functions


Authors: Ilya M. Spitkovsky and Darryl Yong
Journal: Math. Comp. 69 (2000), 1053-1070
MSC (1991): Primary 47A68, 47-04, 42A75
Published electronically: August 25, 1999
Supplement: Additional information related to this article.
MathSciNet review: 1659831
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let

\begin{displaymath}G(x)=\left[\begin{matrix}e^{i\lambda x}I_m & 0\\ c_{-1}e^{-i\nu x}+c_0+c_1 e^{i\alpha x} & e^{-i\lambda x}I_m \end{matrix}\right],\end{displaymath}

where $c_j\in\mathbb{C}^{m\times m}$, $\alpha,\nu>0$ and $\alpha+\nu=\lambda$. For rational $\alpha/\nu$ such matrices $G$ are periodic, and their Wiener-Hopf factorization with respect to the real line $\mathbb{R}$ always exists and can be constructed explicitly. For irrational $\alpha/\nu$, a certain modification (called an almost periodic factorization) can be considered instead. The case of invertible $c_0$ and commuting $c_1c_0^{-1}$, $c_{-1}c_0^{-1}$ was disposed of earlier-it was discovered that an almost periodic factorization of such matrices $G$ does not always exist, and a necessary and sufficient condition for its existence was found. This paper is devoted mostly to the situation when $c_0$ is not invertible but the $c_j$ commute pairwise ($j=0,\pm 1$). The complete description is obtained when $m\leq 3$; for an arbitrary $m$, certain conditions are imposed on the Jordan structure of $c_j$. Difficulties arising for $m=4$ are explained, and a classification of both solved and unsolved cases is given. The main result of the paper (existence criterion) is theoretical; however, a significant part of its proof is a constructive factorization of $G$ in numerous particular cases. These factorizations were obtained using Maple; the code is available from the authors upon request.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 47A68, 47-04, 42A75

Retrieve articles in all journals with MSC (1991): 47A68, 47-04, 42A75


Additional Information

Ilya M. Spitkovsky
Email: ilya@math.wm.edu

Darryl Yong
Email: dyong@u.washington.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-99-01161-8
PII: S 0025-5718(99)01161-8
Keywords: Almost periodic matrix functions, factorization, explicit computation
Received by editor(s): March 12, 1997
Received by editor(s) in revised form: September 18, 1998
Published electronically: August 25, 1999
Additional Notes: The first author’s research was partially supported by NSF Grant DMS-9800704
The second author’s research was started during a Research Experience for Undergraduates sponsored by the NSF at the College of William and Mary during the summer of 1995.
Article copyright: © Copyright 2000 American Mathematical Society