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

Using an abbreviation e μ to denote the function e i μ x on the real line R , let G = [ e λ 0 f e − λ ] , where f is a linear combination of the functions e α , e β , e α − λ , e β − 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.