= Nonintegrability of a radial triangular Fourier cutoff in three dimensions
{title2=$\mathcal F^{-1}[(1-|\xi|)_+]\notin L^1(\mathbb R^3)$}
The inverse <Fourier transform> of $(1-|\xi|)_+$ in three dimensions, with $a=2\pi|x|$, is $-4\pi\sin a/a^3+8\pi(1-\cos a)/a^4$. Its leading oscillatory $|x|^{-3}$ tail is not absolutely integrable against the radial <volume> element $4\pi|x|^2d|x|$. Multiplying by a Gaussian <Fourier transform> retains a nonzero boundary-<derivative> tail, so the radial cutoff need not map even <Schwartz functions> into $L^1$. A <tensor product> of one-dimensional triangular cutoffs is a valid <approximate identity>, and must not be confused with this Euclidean radial cutoff.
Back to article page
