Let $S$ denote the space of Schwartz functions and for $d , n=1,2,3,\ldots $ define $$ S_{\emptyset ,n}\left(\mathbb{R}^{d}\right) =\left\{ \phi \in S\left( \mathbb{R}^{d}\right) :D^{\alpha }\phi \left( 0\right) =0,\quad \left\vert \alpha \right\vert <n\right\}, $$ where I have used standard multi-index notation and $n$ can be referred to as the order.
My first question is: Is the pointwise function product space $S_{\emptyset ,n}S_{\emptyset ,n}$ a vector space?
It is not difficult to show that $S_{\emptyset ,n}S_{\emptyset ,n}\subset S_{\emptyset ,2n}$ and a related question is: Under what conditions is a function in $S_{\emptyset ,2n}$ a member of $S_{\emptyset ,n}S_{\emptyset ,n}$?
I think a good first step would be to consider the case $n=d=1$. In this case $$ S_{\emptyset ,1}\left( \mathbb{R}^{1}\right) =\left\{ \phi \in S\left( \mathbb{R}^{1}\right) :\phi \left( 0\right) =0\right\}, $$
and I now ask: Is the pointwise function product space $$ S_{\emptyset ,1}\left( \mathbb{R}^{1}\right) S_{\emptyset ,1}\left( \mathbb{R% }^{1}\right) :=\left\{ \sigma \in S\left( \mathbb{R}^{1}\right) :\sigma \left( x\right) =\psi \left( x\right) \phi \left( x\right) ,\quad \psi ,\phi \in S_{\emptyset ,1}\left( \mathbb{R}^{1}\right) \right\} , $$ a vector space?
Finally, it is known that $S = SS$ e.g. Google "Weil-Schwartz envelopes for rapidly decreasing functions." on the Web.