Inverse Fourier transform of a function multiplied by $x$

Question

Is there a way to write a general analytic expression for the inverse Fourier transform of a function multiplied by $x$?

I.e.

$$\mathcal F[g](x)=\int \text{d}t\, g(t)e^{-itx}$$

And we would like to find the inverse Fourier transform of

$$x\mathcal F[g](x)$$

i.e. $$\int \text{d}x \mathcal F[g](x) x e^{irx}$$

Answer 1

That's actually quite easy: $$ x\mathcal{F}[g](x) = x \int \mathrm{d}t \, g(t) e^{-itx} \, dt = \int \mathrm{d}t \, g(t) x e^{-itx} \, dt = \int \mathrm{d}t \, g(t) \left(i\frac{d}{dt} e^{-itx}\right) \, dt \\ = i \int \mathrm{d}t \, g(t) \left(\frac{d}{dt} e^{-itx}\right) \, dt = -i \int \mathrm{d}t \, g'(t) e^{-itx} \, dt = -i \mathcal{F}[g'](x). $$ Thus, $$ x\mathcal{F}[g] = -i \mathcal{F}[g']. $$