Trying to solve this problem :
Is $T$ invariant under Fourier transform ? Where : $T= \{f\in \mathcal{C}^{\infty} (\mathbb{R}), \forall n \in \mathbb{N}, |x|^nf(x) \to 0 \; \text{when}\; |x| \to \infty \}$
My answer is no, and here is what I've already done : Let $f : x \mapsto e^{-x^2} \sin (e^{x^2})$. We can easily check that $f \in T$. Now : $$\hat{f}(\lambda)= \int_{- \infty}^{\infty} e^{-x^2}\sin (e^{x^2})e^{-ix\lambda}\mathrm{d}x$$
Thus by Riemann-Lebesgue's theorem we have $\hat{f}(\lambda) \to 0$ when we let $\lambda \to \infty$, but I am pretty sure that $\hat f \notin T$. To show that I tried to show that : $$\lambda \hat f (\lambda)$$ is not wanishing to $0$, but I remain not able to prove this.
Can anyone help ? Or if anyone has a better example (more simple) of $f \in T$ such that $\hat f \notin T$ I am taking it !