In 1998, Edwin Franks and Alan McIntosh published a fascinating paper [79] that provided various decompositions of bounded analytic functions defined on sectors or on more general domains. These decompositions have since found remarkable applications in the study of bounded \(H^\infty \) -functional calculus for either sectorial or Ritt operators. Before beginning the exposition of this topic in Chap. 7 , we dedicate the present chapter to the decompositions that will be used later on. More specifically, Sects. 6.1 and 6.2 will present discrete decompositions based on the work of Franks-McIntosh, while Sect. 6.3 will be devoted to an integral decomposition of a slightly different nature.