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]
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. $\endgroup$+
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 $\endgroup$UnitStep[]
withHeavisideTheta[]
, even if the latter function is supposedly the function intended for use with the built-in integral transforms. $\endgroup$