Bourbaki

Remark

Tag 031J, permanent. Cite it as [Bourbaki, Tag 031J].

Théories spectrales, chapter IV, § 3. Distributions et distributions tempérées

The elementary formulas concerning the Fourier transform of measures remain valid for the Fourier transform of tempered distributions. For example, for f∈S′(Rn)f\in \mathscr{S}'(\mathbf{R}^n) and α∈Nn\alpha \in \mathbf{N}^n, one has

F(∂αf)=(2iπ)∣α∣XαF(f) \mathscr{F}(\partial^{\alpha}f) = (2i\pi )^{|\alpha|}X^{\alpha}\mathscr{F}(f)
F(Xαf)=(−2iπ)−∣α∣∂α(F(f)) \mathscr{F}(X^{\alpha}f) = (-2i\pi )^{-|\alpha|}\partial^{\alpha}(\mathscr{F}(f))

by lemma 11.

Bourbaki sets the proof directly after the statement, and the corpus does not separate them, so what is above is the statement and its proof together.

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Cites

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Machine translation of the English, by gpt-5-mini, gpt-5.4, against glossary version 34. Not checked by a person. That model is a cut down version of the one the rest of this language was written by.