Almost periodic factorization of certain block triangular matrix functions

Ilya M. Spitkovsky, Darryl Yong

Research output: Contribution to journalArticle

Abstract

Let (equation presented) where cj ∈ ℂℳ × 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 c0 and commuting c1c0-1, c-1c0-1 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 c0 is not invertible but the Cj 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 cj. 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 languageEnglish (US)
Pages (from-to)1053-1070
Number of pages18
JournalMathematics of Computation
Volume69
Issue number231
StatePublished - 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

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