Gowers norms have been studied extensively both in the direct sense, startingwith a function and understanding the associated norm, and in the inversesense, starting with the norm and deducing properties of the function. Insteadof focusing on the norms themselves, we study associated dual norms and dualfunctions. Combining this study with a variant of the Szemeredi RegularityLemma, we give a decomposition theorem for dual functions, linking the dualnorms to classical norms and indicating that the dual norm is easier tounderstand than the norm itself. Using the dual functions, we introduce higherorder algebras that are analogs of the classical Fourier algebra, which in turncan be used to further characterize the dual functions.
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