Splitting and factorization with respect to a contour

Abstract

This chapter contains mostly well-known material presented in a form needed for some of the further chapters. This material can not always be found concentrated in one place with complete proofs. The main theme in this chapter is to study continuous functions on a closed contour which admit an additive splitting as a sum or a product (with additional properties) of two functions; one continuous and analytic inside relative to the contour and the second outside. Not all continuous functions admit a splitting. We give here complete descriptions when continuous functions admit such representations and an example when this does not happen. We prove that functions from the algebras of Hölder, differentiable, and Wiener functions admit additive and multiplicative splittings inside these algebras under natural conditions. A local principle is also deduced.