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While it is clear that there are several conventions for Fourier transforms of intragrable functions on $\mathbb{R}$, I don't think I have ever seen anything different from the three following conventions $$ \hat{f}(\xi) = \int f(x) e^{-i 2 \pi \xi x} dx \\ \hat{f}(\omega) = \int f(x) e^{-i \omega x} dx \\ \hat{f}(\omega) = \frac{1}{\sqrt{2 \pi}} \int f(x) e^{-i \omega x} dx \\ $$

On the other hand, it seems to me (please correct me if I am wrong) that the characteristic function of a random variable / probability measure is always (or very close to always) defined as $$ \hat{\mu}(\omega) = \int e^{i \omega x} d \mu(x) $$

All this seems to be confirmed by Wikipedia: https://en.wikipedia.org/wiki/Fourier_transform#Other_conventions

So the main difference between analysis / signal processing / electrical engineering on one side and probability on the other is mostly the mius sign.

Even though I do not think there is anything deep underneath this, i.e. just another convention (please confirm), I would be interested in understanding how we ended up with this confusion from a historical perspective.

Maybe there is a reason why the characteristic function was defined without a minus and nobody ever questioned that? Weren't the people who introduced the characteristic function in probability (who was that btw?) looking at analysis books/papers?

Note that my question is admittedly similar to, e.g., https://math.stackexchange.com/questions/2306738/fourier-transform-with-a-different-sign-convention but I am more insterested in the history.

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  • $\begingroup$ cross-posting is generally strongly discouraged, ideally you should close it there and migrate it here, or delete it there first. One reason cross-posting is bad is that it can result in answer fragmentation and/or duplicated efforts. An exception is sometimes acceptable after the original question is too old to migrate (60 days I think) but that's not a firm rule. Since 9 << 60 I think your duplicate post situation is considered not good. $\endgroup$
    – uhoh
    May 19, 2021 at 3:57
  • $\begingroup$ I had suboptimal results for a history question in Math SE It's been bumped a few times over the last four years, but there seems to be not so much interest, so it may be time to either bounty it or ask a similar question here. If I did post something similar here, I'd have to mention the older question in Math SE, and include some of the helpful material in comments as prior research. $\endgroup$
    – uhoh
    May 19, 2021 at 4:02
  • $\begingroup$ @uhoh I apologize. I will delete it from there. I did not know I could migrate posts.. actually I think I was once told i could not migrate (in that case I wanted to migrate from MSE to MO). Maybe the reason was that it s not possible to migrate old questions? $\endgroup$
    – Tom
    May 19, 2021 at 14:22
  • $\begingroup$ Ya,from meta FAQ find the lengthy What is migration and how does it work? which says "Only questions which are less than 60 days old can be migrated; this rule also applies to moderators." $\endgroup$
    – uhoh
    May 19, 2021 at 14:39

2 Answers 2

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There is a fourth convention $$\hat{f}(s)=\frac{1}{2\pi}\int e^{-its}f(t)dt,$$ which is used sometimes.

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  • $\begingroup$ Is f(x) missing on the right hand side? $\endgroup$
    – AChem
    May 19, 2021 at 14:49
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    $\begingroup$ @M. Farooq: thanks. I corrected. $\endgroup$
    – Alexandre Eremenko
    May 19, 2021 at 15:02
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I agree: the main difference is the minus sign in the exponent. The other differences are cosmetic.
For a probability measure, we normalize $$ \hat{\mu}(\omega) = \int e^{i \omega x} d \mu(x) $$ because $\mu$ has total mass $1$.

For a Fourier series we normalize $$c_n = \frac{1}{L}\int_0^L e^{-i(2\pi/L)nx}\;dx$$ to make the exponentials into an orthonormal system.

For the Fourier transform, we have neither of these reasons for the normalization. Which leads to the various conventions.

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