# Different conventions between Fourier Transform and Characteristic Function

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.

• 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.
– uhoh
May 19, 2021 at 3:57
• 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.
– uhoh
May 19, 2021 at 4:02
• @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?
– Tom
May 19, 2021 at 14:22
• 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."
– uhoh
May 19, 2021 at 14:39

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