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We have $ f \in \mathcal{C}^{\infty}( \mathbb{R}) $ such that

$ \forall n \geq 0 \:, \exists C_n, \: \: |f^{(n)}(x)| \leq C_n (1 + |x|)^{2-n} $

  • Show that the distribution $ T_f $ is tempered.
  • Show that the distribution $ \widehat{T} \in \mathcal{C}^{\infty}( \mathbb{R}^{*}) $

I have noticed that $ |<T_f, \varphi > | \leq C_0 ( \lVert \varphi \rVert_1 + 2C_0 \lVert x \varphi \rVert_1 + C_0 \lVert x^2 \varphi \rVert_1 \leq C_0 ( \lVert \varphi \rVert_{\infty} + 2C_0 \lVert x \varphi \rVert_{\infty} + C_0 \lVert x^2 \varphi \rVert_{\infty} \leq KN_2( \varphi) $

And thus $ T_f \in \mathcal{S}' $

Now I know by Fubini that $ \widehat{T_f} = T_\widehat{f} \in \mathcal{S}' $

Then perhaps $ \widehat{f} $ is well defined after all even though f is a priori not in $ L^1 $ or $ \mathcal{S} $

So to show that $ \widehat{T} \in \mathcal{C}^{\infty}( \mathbb{R}^{*}) $ do we need to show that $ \widehat{f} \in \mathcal{C}^{\infty}( \mathbb{R}^{*}) $ ? How could we proceed? Thanks in advance.

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