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Although I am not completely familiar with the subject but I have met two 'dual' definitions of the Discrete Fourier Transform of a function $ f: \mathbb{Z} / N \mathbb{Z} \rightarrow \mathbb{C} $ , one is $ \widehat{f}(r) = \sum_{ s \in \mathbb{Z}_N } f(s) \omega^{-rs} $ where $ \omega = exp (2 \pi i /N ) $ (for example see Timothy Gowers's paper concerning Szemeredis Theorem) and the other one is the same without the minus $ \widehat{f}(r)= \sum_{x \in \mathbb{Z}_N} \left( f(x) e^{ \dfrac{2 \pi r ix}{N} } \right) $ (for example see here http://www.dms.umontreal.ca/~andrew/PDF/ProcAddPap.pdf page 12). Am I am missing something concerning duality of bilinear forms over additive groups for example?

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    thnx Thomas. @joriki This transform is used in Additive Combinatorics. It is just a tag. See Terence Tao's and Van Vu's book or Gowers's paper mentioned above. Thnx anyway.2012-10-10

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