Almost periodic factorization of block triangular matrix functions revisited

Yuri I. Karlovich, Ilya M. Spitkovsky, Ronald A. Walker

Research output: Contribution to journalArticle

Abstract

Let G be an n x n almost periodic (AP) matrix function defined on the real line ℝ. By the AP factorization of G we understand its representation in the form G = G+ΛG-, where G±1+ (G±1-) is an AP matrix function with all Fourier exponents of its entries being non-negative (respectively, non-positive) and Λ(x) = diag[eiλ1x, . . . , eiλnx], λ1, . . . , λn ∈ ℝ. This factorization plays an important role in the consideration of systems of convolution type equations on unions of intervals. In particular, systems of m equations on one interval of length λ lead to AP factorization of matrices G(x) = [eiλxIm 0f(x) e-iλxIm]. (0.1) We develop a factorization techniques for matrices of the form (0.1) under various additional conditions on the off-diagonal block f. The cases covered include f with the Fourier spectrum Ω(f) lying on a grid (Ω(f) ⊂ -ν + hℤ) and the trinomial f (Ω(f) = {-ν, μ, α}) with -ν < μ < α, α + |μ| + ν ≥ λ.

Original languageEnglish (US)
Pages (from-to)199-232
Number of pages34
JournalLinear Algebra and Its Applications
Volume293
Issue number1-3
DOIs
StatePublished - May 15 1999

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Keywords

  • Almost periodic matrix functions
  • Factorization

ASJC Scopus subject areas

  • Algebra and Number Theory
  • Numerical Analysis
  • Geometry and Topology
  • Discrete Mathematics and Combinatorics

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