Let $C^\infty_c(\mathbb{R})$ denote the space of test functions(i.e. smooth and with compact support). Suppose $f\in C^\infty_c(\mathbb{R})$, let's consider decay of its Fourier transform $\hat{f}$. Of course $\hat{f}$ in Schwartz space. So $|\hat{f}(t)|=O(|t|^{-N})$ for each positive integer $N$.
Is this Optimal when we take all test functions into our consideration?
Something related: In this MO post, Joonas Ilmavirta proved for test functions, $\hat{f}$ can't have $e^{-k|x|^\gamma}$ decay rate, if $\gamma >1$. And the Standard Bump function $\Phi$, such that $|\hat{\Phi}(x)|=O(x^{-\frac{3}{4}}e^{-\sqrt{|x|}})$.
Let $\phi = e^{-1/(1-x^2)}1_{|x|< 1}$. Its Fourier transform is real and even.
For any $\hat{g}\ge 0$ continuous even decreasing on $[0,\infty)$ and such that $\forall n, \hat{g}=o(t^{-n})$
Then $$\hat{f}=\hat{g} \ast \hat{\phi}^2$$ doesn't decay faster than $\hat{g}$.
And its inverse Fourier transform $$f=g \cdot (\phi \ast \phi)$$ is $C^\infty_c$, since $g$ is $C^\infty$ and $\phi \ast \phi$ is $C^\infty_c$.