The Fourier transform of x·f(x) is related to the derivative of the transform of f(x) with respect to angular frequency, with a sign/factor depending on the transform convention (here using F(ω)=∫f(x) e^{iωx} dx).
Differentiating F(ω)=∫f(x)e^{iωx}dx with respect to ω gives dF/dω = i∫x f(x) e^{iωx}dx = i·FT[xf(x)](ω), so FT[xf(x)](ω) = (1/i)·dF/dω = -i·dF/dω. Given F(ω)=ω², dF/dω = 2ω,...
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