Is there a proper description of the space $$\{\hat f\ | \ f\in C^\infty \ s.t. \forall \alpha\in\mathbb{N}^n,\forall \epsilon>0\exists K\subseteq \mathbb{R}^n\ K\ \text{compact};\ \sup_{x\in \mathbb{R}^n\backslash K}|\partial^\alpha_x f(x)|< \epsilon\}$$Especially I'm interested in the order of these distributions and their decay properties.

To be precise are there constants $C_{K,m}$, for every $K\subseteq\mathbb{R}^n$ compact and $m\in\mathbb{R}$, s.t. $$\left|\int \hat f(\xi)\phi(\xi-b)\ d \xi\right|\leq C_{K,m}(1+b^2)^{-m}\|\phi\|_{\infty}\quad \forall\phi\in C_c(K)$$ or anything slightly worse?

Edit: The assertion automatically holds for all $m\in\mathbb{R}$ as long as it holds for any $m_0\in\mathbb{R}$, since all derivatives are again of the same form. This maximally exploits smoothness, so to establish this starting estimate or to give a counterexample, it is no loss assuming only $f\in C_0(\mathbb{R}^n)$.

Further the condition vanishing at infinity must be exploited, since on $C^\infty_b(\mathbb{R}^n)$ the assertion does not hold! (e.g. for $\Theta$ the Heaviside step function and $\phi$ some Schwartz-function $\widehat{\phi\ast\Theta}(\xi)=\hat\phi\cdot 1/2(\delta(\xi)-\text{p.v.}\frac{i}{\pi \xi})$ is of order 1)

Theorie des Distributions(page 199). $\endgroup$ – Jochen Wengenroth Sep 15 '16 at 12:41