### Abstract

Let (equation presented) where c_{j} ∈ ℂ^{ℳ × ℳ}, α, v > 0 and α + v = λ. For rational α/v such matrices G are periodic, and their Wiener-Hopf factorization with respect to the real line ℝ always exists and can be constructed explicitly. For irrational α/v, a certain modification (called an almost periodic factorization) can be considered instead. The case of invertible c_{0} and commuting c_{1}c_{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, ±1). The complete description is obtained when ℳ < 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.

Original language | English (US) |
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Pages (from-to) | 1053-1070 |

Number of pages | 18 |

Journal | Mathematics of Computation |

Volume | 69 |

Issue number | 231 |

State | Published - Jul 1 2000 |

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### Keywords

- Almost periodic matrix functions
- Explicit computation
- Factorization

### ASJC Scopus subject areas

- Algebra and Number Theory
- Computational Mathematics
- Applied Mathematics

### Cite this

*Mathematics of Computation*,

*69*(231), 1053-1070.