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Let $f : \mathbb{Z}/N\mathbb{Z} \rightarrow \mathbb{C}$ be a function. Is its Fourier transform typically defined by

$$\widehat{f}(\xi) = \sum_{x \in \mathbb{Z}/N \mathbb{Z}} f(x) e^{- 2 \pi i x \xi /N},$$

or the same expression with $\frac{1}{N}$ in front?

I recently encountered both in material on additive combinatorics. This is just a choice of normalization; if the factor of $1/N$ is not included, then it appears in the Fourier inversion formula instead.

Is one of these substantially more common than the other (perhaps just in additive combinatorics), or does the choice of normalization tell me something implicit about the author's approach? (The source including $1/N$ in front is using the language of probability theory; perhaps this has something to do with it.)

Thank you!

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# Choice of normalization of the finite Fourier transform

Let $f : \mathbb{Z}/N\mathbb{Z} \rightarrow \mathbb{C}$ be a function. Is its Fourier transform typically defined by

$$\widehat{f}(\xi) = \sum_{x \in \mathbb{Z}/N \mathbb{Z}} e^{- 2 \pi i x \xi /N},$$

or the same expression with $\frac{1}{N}$ in front?

I recently encountered both in material on additive combinatorics. This is just a choice of normalization; if the factor of $1/N$ is not included, then it appears in the Fourier inversion formula instead.

Is one of these substantially more common than the other (perhaps just in additive combinatorics), or does the choice of normalization tell me something implicit about the author's approach? (The source including $1/N$ in front is using the language of probability theory; perhaps this has something to do with it.)

Thank you!