= Large-distance tail of the Hilbert transform
{title2=$\mathcal Hf(x)=\widehat f(0)/(\pi x)+o(|x|^{-1})$}
For a <Schwartz function> $f$, the <Hilbert transform> satisfies $\mathcal Hf(x)=(\int f)/(\pi x)+o(|x|^{-1})$ at both infinities. Splitting the inverse <Fourier transform> at the sign multiplier's jump and integrating each half-line by parts extracts the zero-frequency value $\widehat f(0)$; the remainder vanishes by the <Riemann-Lebesgue lemma>. A nonzero zeroth moment therefore obstructs Schwartz decay.
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