TY - JOUR

T1 - Almost periodic factorization of block triangular matrix functions revisited

AU - Karlovich, Yuri I.

AU - Spitkovsky, Ilya M.

AU - Walker, Ronald A.

N1 - Funding Information:
*Corresponding author. E-mail: ilya@math.wm.edu 1 Research of these authors is partially supported by the NATO grant CRG 950332. 2Also partially supported by F.C.T. (Portugal), Grant PRAXIS XXI/BCC/4355/94. Depart-amento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1096 Lisboa, Portugal. E-mail: Yuri.Karlovic@math.ist.utl.pt 3Research of this author was partially supported by the Summer Research Grant from the College of William and Mary and by the NSF Grant DMS-98-00704. 4Research of this author was conducted during the summer of 1997 when he, then an undergraduate student of the University of Richmond, participated at the College of William and Mary's Research Experience for Undergraduates program. Sponsored by the NSF REU grant DMS-96-19577. E-mail: rawalker@math.lsa.umich.edu 0024-3795/99/$ – see front matter © 1999 Published by Elsevier Science Inc. All rights reserved. PII: S 0 0 2 4 - 3 7 9 5 ( 9 9 ) 0 0 0 4 1 - 5

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

AN - SCOPUS:0033412798

SN - 0024-3795

VL - 293

SP - 199

EP - 232

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 1-3

ER -