There are $3$ $\mathcal{S}$-type Spaces, namely $\mathcal{S}_\alpha\:,\: \mathcal{S}^\beta\:,\:\mathcal{S}_\alpha^\beta$. They are defined by:
$\mathcal{S}_\alpha: |x^k\varphi^{(q)}(x)|\le C_qA^kk^{k\alpha}\qquad (k,q=0,1,2,...)$
$\mathcal{S}^\beta: |x^k\varphi^{(q)}(x)|\le C_kB^qq^{q\beta}\qquad (k,q=0,1,2,...)$
$\mathcal{S}_\alpha^\beta: |x^k\varphi^{(q)}(x)|\le CA^kB^qk^{k\alpha}q^{q\beta}\qquad (k,q=0,1,2,...)$
where the constants $A,B,C_k,C_q$ depend on $\varphi \:\&\:\alpha+\beta\ge1$.
In the first space we have functions which are rapidly decaying as $|x|\to \infty$. The second space imposes conditions on the growth of derivatives of these functions as $|x|\to \infty$. The third space $$\mathcal{S}_\alpha^\beta\subset\mathcal{S}_\alpha\cap \mathcal{S}^\beta$$In general the converse also holds which is a classical result due to Kashpirovsky. I want to construct examples to get a better understanding of these definitions. From the definition it is clear that every function in these spaces is at least $C_c^\infty$. So intuitively the first thing that comes to my mind is Gaussian-like functions. Now say I consider $\mathcal{S}_1^2(\mathbb{R})$. Which type of functions belong to this space ? How to come up with suitable constants ?
Reference: Generalized Functions, Volume 2 by I.M.Gelfand and G.E. Shilov
