Nonintegrability of a radial triangular Fourier cutoff in three dimensions (${\mathcal{F}}^{-1}[(1-|\xi |{)}_{+}]\notin L^1({\mathbb{R}}^3)$)

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 $4\pi |x{|}^{2}d|x|$. 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.