= Fourier derivative identities for distributions
{c}
For $D_j=-i\partial_j$, the <Fourier transform of a tempered distribution> satisfies
$$
\widehat{D^\alpha u}=\xi^\alpha\widehat u,
\qquad \widehat{x^\beta u}=(-1)^{|\beta|}D^\beta\widehat u.
$$
The base cases follow from $\langle D_ju,\varphi\rangle=i\langle u,\partial_j\varphi\rangle$ and <integration by parts> in the Fourier integral. Iterating gives the <multi-index> formulas. The $D$ convention matters: for the unscaled derivative, $\widehat{\partial_j u}=i\xi_j\widehat u$.
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