Almost periodic factorization of 2 × 2 triangular matrix functions: New cases of off diagonal spectrum

Abstract

Many known results on almost periodic factorization of almost periodic 2 × 2 triangular matrix functions of the form (Formula presented.) are reviewed from a unified point of view, with particular attention to the case when the off diagonal entry is at most a quadrinomial almost periodic function. New results are obtained on almost periodic factorization for off diagonal entry having its Bohr-Fourier spectrum in a union of two shifted grids, i.e., arithmetic progressions, with the same difference, and perhaps an additional point. When specializing these results to the case of off diagonal almost periodic trinomials, new cases of factorability are obtained. The main technical tool is the Portuguese transformation, a known algorithm.