 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 $\Vert \hat{f}-\hat{g}{\Vert }_{\infty }\le \Vert f-g{\Vert }_1$ transfers the conclusion.

The Riemann–Lebesgue lemma is a fundamental result in Fourier analysis, dealing with the behavior of the Fourier coefficients of integrable functions. It asserts that if \( f \) is an integrable function on the real line (or on a finite interval), then the Fourier coefficients of \( f \) tend to zero as the frequency goes to infinity.