Constructive factorization of some almost periodic triangular matrix functions with a quadrinomial off diagonal entry

Using an abbreviation eμ to denote the function eiμx on the real line R, let G=[eλ0fe−λ], where f is a linear combination of the functions eα, eβ, eα−λ, eβ−λ with some (0<)α,β<λ. The criterion for G to admit a canonical factorization was established recently by Avdonin, Bulanova and Moran (2007) [1]. We give an alternative approach to the matter, proving the existence (when it does take place) via deriving explicit factorization formulas. The non-existence of the canonical factorization in the remaining cases then follows from the continuity property of the geometric mean.