# Smooth tempered distribution

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 $$| | \leq C_0 ( \lVert \varphi \rVert_1 + 2C_0 \lVert x \varphi \rVert_1 + C_0 \lVert x^2 \varphi \rVert_1 \leq C_0 ( \lVert \varphi \rVert_{\infty} + 2C_0 \lVert x \varphi \rVert_{\infty} + C_0 \lVert x^2 \varphi \rVert_{\infty} \leq KN_2( \varphi)$$

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.