Hardy martingales and Jensen's inequality

Abstract

Hardy martingales were introduced by Garling and used to study analytic functions on the N-dimensional torus TN, where analyticity is defined using a lexicographic order on the dual group ZN. We show how, by using basic properties of orders on ZN, we can apply Garling's method in the study of analytic functions on an arbitrary compact abelian group with an arbitrary order on its dual group. We illustrate our approach by giving a new and simple proof of a famous generalized Jensen's Inequality due to Helson and Lowdenslager.