Constructive almost periodic factorization of some triangular matrix functions

A factorability criterion is obtained constructively, and the respective factorization obtained explicitly, for 2 × 2 triangular almost periodic matrix functions of the form [(e_λ, 0; f, e_{- λ})]. Here f = c_{- 1} e_{- α} - c₀ + c₁ e_β, e_μ (x) : = e^{i μ x}, c_j are non-zero constants and 0 < α, β, α + β < λ ≤ α + β + max {α, β} with α / β being irrational. Note that the factorization problem, even for triangular matrix functions as above with an arbitrary trinomial f, is open. The result obtained is yet another step towards its solution.