= Fourier transform of a derivative
{c}
{title2=$\widehat{f'}(\omega)=i\omega\widehat f(\omega)$}
For a <Schwartz function>, with the angular-frequency convention,
$$
\widehat{f'}(k)=ik\widehat f(k).
$$
More generally, the identity holds when <integration by parts> is justified and the boundary term vanishes. Repeated <differentiation> gives $\widehat{f^{(m)}}(k)=(ik)^m\widehat f(k)$. The short proof transfers the <derivative> from $f$ to $e^{-ikx}$ by <integration by parts>. High <frequencies> receive larger multipliers, so rapid ripples contribute more strongly to a <derivative>. For the <heat equation>, the second <derivative> becomes multiplication by $-k^2$, directly explaining the faster decay of fine spatial structure.
Back to article page
