where , and . For rational such matrices are periodic, and their Wiener-Hopf factorization with respect to the real line always exists and can be constructed explicitly. For irrational , a certain modification (called an almost periodic factorization) can be considered instead. The case of invertible and commuting , was disposed of earlier-it was discovered that an almost periodic factorization of such matrices does not always exist, and a necessary and sufficient condition for its existence was found. This paper is devoted mostly to the situation when is not invertible but the commute pairwise (). The complete description is obtained when ; for an arbitrary , certain conditions are imposed on the Jordan structure of . Difficulties arising for 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 in numerous particular cases. These factorizations were obtained using Maple; the code is available from the authors upon request.
©1999 American Mathematical Society
Spitkovsky, IM, Yong, DH. Almost periodic factorization of certain block triangular matrix functions. Math Comp. 1999;69(231): 1053-1070.