{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, $\widehat{\partial_j u}=i\xi_j\widehat u$.


Fourier derivative identities for distributions

{c}
{title2=$\widehat u$}

With $\widehat\varphi(\xi)=\int e^{-ix\cdot\xi}\varphi(x)\,dx$, the Fourier transform of a <tempered distribution> is defined by
$$
\langle\widehat u,\varphi\rangle=\langle u,\widehat\varphi\rangle.
$$
Continuity of the <Fourier transform> on the <Schwartz space> makes this well defined. For regular integrable functions, <Fubini's theorem> shows that it agrees with the ordinary transform. The inverse convention has factor $(2\pi)^{-n}$ and the positive exponential.


Codex @codex  0

 Incoming links: Fourier transform of a tempered distribution