### 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[e^{iλ1x}, . . . , e^{iλnx}], λ_{1}, . . . , λ_{n} ∈ 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 0}f(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 language | English (US) |
---|---|

Pages (from-to) | 199-232 |

Number of pages | 34 |

Journal | Linear Algebra and Its Applications |

Volume | 293 |

Issue number | 1-3 |

DOIs | |

State | Published - May 15 1999 |

### Fingerprint

### 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

*Linear Algebra and Its Applications*,

*293*(1-3), 199-232. https://doi.org/10.1016/S0024-3795(99)00041-5

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

Research output: Contribution to journal › Article

*Linear Algebra and Its Applications*, vol. 293, no. 1-3, pp. 199-232. https://doi.org/10.1016/S0024-3795(99)00041-5

}

TY - JOUR

T1 - Almost periodic factorization of block triangular matrix functions revisited

AU - Karlovich, Yuri I.

AU - Spitkovsky, Ilya

AU - Walker, Ronald A.

PY - 1999/5/15

Y1 - 1999/5/15

N2 - 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.

AB - 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.

KW - Almost periodic matrix functions

KW - Factorization

UR - http://www.scopus.com/inward/record.url?scp=0033412798&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0033412798&partnerID=8YFLogxK

U2 - 10.1016/S0024-3795(99)00041-5

DO - 10.1016/S0024-3795(99)00041-5

M3 - Article

VL - 293

SP - 199

EP - 232

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

SN - 0024-3795

IS - 1-3

ER -