Almost periodic factorization of block triangular matrix functions revisited

Abstract

Let G be an n×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 i λ 1 x ,…,e i λ n x ], 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 (0.1) G(x)= e iλx I m 0 f(x) e − iλx I m . 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+h Z ) and the trinomial f (( Ω(f)={−ν,μ,α}) with − ν< μ< α, α+| μ|+ ν⩾ λ.