It is known that the spectral factorization mapping is bounded, but not continuous, on L-infinity, the space of essentially bounded measurable functions on the unit circle. In this article we study the spectral factorization mapping on decomposing Banach algebras. The most important example of a decomposing Banach algebra is the Wiener algebra, the space of all absolutely convergent Fourier series. It is shown that the spectral factorization mapping is locally Lipschitz continuous, but not bounded, on all decomposing Banach algebras in consideration. An application is given to the construction of approximate normalized coprime factorization.