A tempered distribution $u\in \mathcal{S}'(\mathbb{R})$ is said to be rapidly decreasing if for every $f \in \mathcal{S}(\mathbb{R})$, $u*f \in \mathcal{S}(\mathbb{R})$.
One rough way to motivate this definition is that $u*f$ has the best regularity between $f$ and $u$ (and thus, for $f$ smooth, it is smooth) and the worst decay between $f$ and $u$ (for $f$ rapidly decreasing, it has consequently the decay of $u$).
I am wondering if we could define similarly the space of distributions with algebraic (or slow) decay in the following way. First, a smooth function $f$ has an algebraic decay if we have for every integer $n$ a bound $$|f^{(n)}(t)| \leq \frac{C}{1 + |t|^\alpha}$$ for some $C$ and $\alpha >0$. Then, $u \in \mathcal{S}'(\mathbb{R})$ is said to have a algebraic decay if for every $f\in \mathcal{S}(\mathbb{R})$, the smooth function $u*f$ has an algebraic decay.
Questions: Is the space of distributions with algebraic decay already introduced and studied somewhere? Is the previous definition the correct one for that concept? What are the properties of such distributions? Especially, what can we say about the Fourier transform of a distribution with algebraic decay?
