Almost periodic factorization of block triangular matrix functions revisited Original Research Article

Let G be an n×n almost periodic (AP) matrix function defined on the real line image. 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], image. 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 matricesimageWe 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 (image) and the trinomial f ((Ω(f)={−ν,μ,α}) with −ν<μ<α, α+μ+νgreater-or-equal, slantedλ.