Nakhlé Asmar and Stephen Montgomery-Smith, Hardy martingales and Jensen's Inequality.
Bull. Australian Math. Soc., 55, (1997), 185-195.
Hardy martingales were introduced by Garling and used to study analytic functions on the \(N\)-dimensional torus \(\mathbb T^N\), where analyticity is defined using a lexicographic order on the dual group \(\mathbb Z^N\). We show how, by using basic properties of orders on \(\mathbb Z^N\), 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.
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