# Fourier transform inconsistency

Bug introduced in 5 or earlier and persisting through 11.3.0

On Mathematica 11.0.0 the Fourier transform of $x\theta(x)$ gives the expected result:

In: FourierTransform[x*UnitStep[x], x, t]
Out: -(1/(Sqrt[2*Pi]*t^2)) - I*Sqrt[Pi/2]*Derivative[1][DiracDelta][t]


But the Fourier transform of $(x-a)\theta(x)$ misses the derivative of a delta function, and does not reduce to the previous answer when $a\rightarrow 0$:

In: FourierTransform[(x - a)*UnitStep[x], x, t]
Out: -(1/(Sqrt[2*Pi]*t^2)) - (I*a)/(Sqrt[2*Pi]*t) - a*Sqrt[Pi/2]*DiracDelta[t]

• Related? Sep 9, 2016 at 16:08
• @march -- the difference between the first and second integrands is $a\theta(x)$, which is a smooth function of $a$, without any discontinuity at $a=0$, so taking the limit $a\rightarrow 0$ of the integral over $x$ should be fine. Sep 9, 2016 at 17:54
• @march There is clearly an inconsistency if you expect the Fourier transform to be invertible (a fundamental property): InverseFourierTransform[FourierTransform[(x-a) Sign[x],x,t],t,x] doesn't produce the original function back again. So I think it has to be called a bug.
– Jens
Sep 9, 2016 at 18:13
• Distribution over + should clearly be valid in this case e.g. (FourierTransform[x*UnitStep[x], x, t] - FourierTransform[a*UnitStep[x], x, t]) - FourierTransform[(x - a)*UnitStep[x], x, t] // FullSimplify should be zero, and it is not. This is a bug Sep 11, 2016 at 9:22
• FWIW, the bug persists even if you replace UnitStep[] with HeavisideTheta[], even if the latter function is supposedly the function intended for use with the built-in integral transforms. Oct 4, 2016 at 2:06

## 1 Answer

I've reported the issue to Wolfram and received this response:

Thank you for taking the time to send in this report. It does appear that the inverse Fourier transform of the Fourier transform of a particular function is not behaving properly. In particular it is not giving the original function back, as it should. I will forward an incident report to our developers regarding this issue.

So this is a bug. The case number is CASE:3710457.

• I decided to put the bug report case number in the answer instead of a comment, because this will make it easier to find in the future, in case I get additional feedback...
– Jens
Sep 12, 2016 at 23:07
• thank you, Jens, nice to catch up with you here! Sep 13, 2016 at 9:41