Almost periodic factorization of block triangular matrix functions revisited

Yuri I. Karlovich, Ilya 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

    Fingerprint

    Triangular matrix
    Block Matrix
    Matrix Function
    Almost Periodic
    Factorization
    Periodic Functions
    Factorization of Matrices
    Fourier Spectrum
    Interval
    Convolution
    Real Line
    Union
    Non-negative
    Exponent
    Grid
    Form

    Keywords

    • Almost periodic matrix functions
    • Factorization

    ASJC Scopus subject areas

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

    Cite this

    Almost periodic factorization of block triangular matrix functions revisited. / Karlovich, Yuri I.; Spitkovsky, Ilya; Walker, Ronald A.

    In: Linear Algebra and Its Applications, Vol. 293, No. 1-3, 15.05.1999, p. 199-232.

    Research output: Contribution to journalArticle

    Karlovich, Yuri I. ; Spitkovsky, Ilya ; Walker, Ronald A. / Almost periodic factorization of block triangular matrix functions revisited. In: Linear Algebra and Its Applications. 1999 ; Vol. 293, No. 1-3. pp. 199-232.
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