Riemann-Lebesgue lemma
= Riemann-Lebesgue lemma
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{wiki=Riemann–Lebesgue_lemma}
If $f\in L^1(\mathbb R^n)$, then its Fourier transform is continuous and tends to zero as $|\xi|\to\infty$. Approximate $f$ in $L^1$ by a compactly supported smooth function; integration by parts makes the approximant's transform decay, while $\|\widehat f-\widehat g\|_\infty\leq\|f-g\|_1$ transfers the conclusion.