We define a Fourier transform \(\mathcal{F}\) for a tempered distribution \(u\in \mathcal{S}'(\mathbb{R}^d)\) as follows,

\[ \mathcal{F}(u)(\varphi)=\hat{u}(\varphi)=u(\hat{\varphi}). \]

It is not difficult to check, that \(\hat{u}\) is again a tempered distribution.

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• Fourier transform

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• The Fourier transform for Distributions coincide with the usual one.
• \(\mathcal{F}\) is a homeomorphism.

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• Dirac distribution